4068 lines
115 KiB
C
4068 lines
115 KiB
C
/*
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* Copyright (c) 1995 Colin Plumb. All rights reserved.
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* For licensing and other legal details, see the file legal.c.
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*
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* lbn64.c - Low-level bignum routines, 64-bit version.
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*
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* NOTE: the magic constants "64" and "128" appear in many places in this
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* file, including inside identifiers. Because it is not possible to
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* ask "#ifdef" of a macro expansion, it is not possible to use the
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* preprocessor to conditionalize these properly. Thus, this file is
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* intended to be edited with textual search and replace to produce
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* alternate word size versions. Any reference to the number of bits
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* in a word must be the string "64", and that string must not appear
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* otherwise. Any reference to twice this number must appear as "128",
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* which likewise must not appear otherwise. Is that clear?
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*
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* Remember, when doubling the bit size replace the larger number (128)
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* first, then the smaller (64). When halving the bit size, do the
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* opposite. Otherwise, things will get wierd. Also, be sure to replace
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* every instance that appears. (:%s/foo/bar/g in vi)
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*
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* These routines work with a pointer to the least-significant end of
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* an array of WORD64s. The BIG(x), LITTLE(y) and BIGLTTLE(x,y) macros
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* defined in lbn.h (which expand to x on a big-edian machine and y on a
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* little-endian machine) are used to conditionalize the code to work
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* either way. If you have no assembly primitives, it doesn't matter.
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* Note that on a big-endian machine, the least-significant-end pointer
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* is ONE PAST THE END. The bytes are ptr[-1] through ptr[-len].
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* On little-endian, they are ptr[0] through ptr[len-1]. This makes
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* perfect sense if you consider pointers to point *between* bytes rather
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* than at them.
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*
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* Because the array index values are unsigned integers, ptr[-i]
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* may not work properly, since the index -i is evaluated as an unsigned,
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* and if pointers are wider, zero-extension will produce a positive
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* number rahter than the needed negative. The expression used in this
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* code, *(ptr-i) will, however, work. (The array syntax is equivalent
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* to *(ptr+-i), which is a pretty subtle difference.)
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*
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* Many of these routines will get very unhappy if fed zero-length inputs.
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* They use assert() to enforce this. An higher layer of code must make
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* sure that these aren't called with zero-length inputs.
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*
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* Any of these routines can be replaced with more efficient versions
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* elsewhere, by just #defining their names. If one of the names
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* is #defined, the C code is not compiled in and no declaration is
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* made. Use the BNINCLUDE file to do that. Typically, you compile
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* asm subroutines with the same name and just, e.g.
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* #define lbnMulAdd1_64 lbnMulAdd1_64
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*
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* If you want to write asm routines, start with lbnMulAdd1_64().
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* This is the workhorse of modular exponentiation. lbnMulN1_64() is
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* also used a fair bit, although not as much and it's defined in terms
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* of lbnMulAdd1_64 if that has a custom version. lbnMulSub1_64 and
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* lbnDiv21_64 are used in the usual division and remainder finding.
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* (Not the Montgomery reduction used in modular exponentiation, though.)
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* Once you have lbnMulAdd1_64 defined, writing the other two should
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* be pretty easy. (Just make sure you get the sign of the subtraction
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* in lbnMulSub1_64 right - it's dest = dest - source * k.)
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*
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* The only definitions that absolutely need a double-word (BNWORD128)
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* type are lbnMulAdd1_64 and lbnMulSub1_64; if those are provided,
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* the rest follows. lbnDiv21_64, however, is a lot slower unless you
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* have them, and lbnModQ_64 takes after it. That one is used quite a
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* bit for prime sieving.
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*/
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#ifndef HAVE_CONFIG_H
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#define HAVE_CONFIG_H 0
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#endif
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#if HAVE_CONFIG_H
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#include "bnconfig.h"
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#endif
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/*
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* Some compilers complain about #if FOO if FOO isn't defined,
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* so do the ANSI-mandated thing explicitly...
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*/
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#ifndef NO_ASSERT_H
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#define NO_ASSERT_H 0
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#endif
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#ifndef NO_STRING_H
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#define NO_STRING_H 0
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#endif
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#ifndef HAVE_STRINGS_H
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#define HAVE_STRINGS_H 0
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#endif
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#if !NO_ASSERT_H
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#include <assert.h>
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#else
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#define assert(x) (void)0
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#endif
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#if !NO_STRING_H
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#include <string.h> /* For memcpy */
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#elif HAVE_STRINGS_H
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#include <strings.h>
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#endif
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#include "lbn.h"
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#include "lbn64.h"
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#include "lbnmem.h"
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#include "kludge.h"
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#ifndef BNWORD64
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#error 64-bit bignum library requires a 64-bit data type
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#endif
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/* If this is defined, include bnYield() calls */
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#if BNYIELD
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extern int (*bnYield)(void); /* From bn.c */
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#endif
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/*
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* Most of the multiply (and Montgomery reduce) routines use an outer
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* loop that iterates over one of the operands - a so-called operand
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* scanning approach. One big advantage of this is that the assembly
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* support routines are simpler. The loops can be rearranged to have
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* an outer loop that iterates over the product, a so-called product
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* scanning approach. This has the advantage of writing less data
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* and doing fewer adds to memory, so is supposedly faster. Some
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* code has been written using a product-scanning approach, but
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* it appears to be slower, so it is turned off by default. Some
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* experimentation would be appreciated.
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*
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* (The code is also annoying to get right and not very well commented,
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* one of my pet peeves about math libraries. I'm sorry.)
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*/
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#ifndef PRODUCT_SCAN
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#define PRODUCT_SCAN 0
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#endif
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/*
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* Copy an array of words. <Marvin mode on> Thrilling, isn't it? </Marvin>
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* This is a good example of how the byte offsets and BIGLITTLE() macros work.
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* Another alternative would have been
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* memcpy(dest BIG(-len), src BIG(-len), len*sizeof(BNWORD64)), but I find that
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* putting operators into conditional macros is confusing.
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*/
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#ifndef lbnCopy_64
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void
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lbnCopy_64(BNWORD64 *dest, BNWORD64 const *src, unsigned len)
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{
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memcpy(BIGLITTLE(dest-len,dest), BIGLITTLE(src-len,src),
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len * sizeof(*src));
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}
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#endif /* !lbnCopy_64 */
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/*
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* Fill n words with zero. This does it manually rather than calling
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* memset because it can assume alignment to make things faster while
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* memset can't. Note how big-endian numbers are naturally addressed
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* using predecrement, while little-endian is postincrement.
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*/
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#ifndef lbnZero_64
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void
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lbnZero_64(BNWORD64 *num, unsigned len)
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{
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while (len--)
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BIGLITTLE(*--num,*num++) = 0;
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}
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#endif /* !lbnZero_64 */
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/*
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* Negate an array of words.
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* Negation is subtraction from zero. Negating low-order words
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* entails doing nothing until a non-zero word is hit. Once that
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* is negated, a borrow is generated and never dies until the end
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* of the number is hit. Negation with borrow, -x-1, is the same as ~x.
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* Repeat that until the end of the number.
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*
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* Doesn't return borrow out because that's pretty useless - it's
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* always set unless the input is 0, which is easy to notice in
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* normalized form.
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*/
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#ifndef lbnNeg_64
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void
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lbnNeg_64(BNWORD64 *num, unsigned len)
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{
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assert(len);
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/* Skip low-order zero words */
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while (BIGLITTLE(*--num,*num) == 0) {
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if (!--len)
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return;
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LITTLE(num++;)
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}
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/* Negate the lowest-order non-zero word */
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*num = -*num;
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/* Complement all the higher-order words */
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while (--len) {
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BIGLITTLE(--num,++num);
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*num = ~*num;
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}
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}
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#endif /* !lbnNeg_64 */
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/*
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* lbnAdd1_64: add the single-word "carry" to the given number.
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* Used for minor increments and propagating the carry after
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* adding in a shorter bignum.
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*
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* Technique: If we have a double-width word, presumably the compiler
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* can add using its carry in inline code, so we just use a larger
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* accumulator to compute the carry from the first addition.
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* If not, it's more complex. After adding the first carry, which may
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* be > 1, compare the sum and the carry. If the sum wraps (causing a
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* carry out from the addition), the result will be less than each of the
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* inputs, since the wrap subtracts a number (2^64) which is larger than
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* the other input can possibly be. If the sum is >= the carry input,
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* return success immediately.
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* In either case, if there is a carry, enter a loop incrementing words
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* until one does not wrap. Since we are adding 1 each time, the wrap
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* will be to 0 and we can test for equality.
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*/
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#ifndef lbnAdd1_64 /* If defined, it's provided as an asm subroutine */
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#ifdef BNWORD128
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BNWORD64
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lbnAdd1_64(BNWORD64 *num, unsigned len, BNWORD64 carry)
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{
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BNWORD128 t;
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assert(len > 0); /* Alternative: if (!len) return carry */
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t = (BNWORD128)BIGLITTLE(*--num,*num) + carry;
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BIGLITTLE(*num,*num++) = (BNWORD64)t;
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if ((t >> 64) == 0)
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return 0;
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while (--len) {
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if (++BIGLITTLE(*--num,*num++) != 0)
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return 0;
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}
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return 1;
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}
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#else /* no BNWORD128 */
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BNWORD64
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lbnAdd1_64(BNWORD64 *num, unsigned len, BNWORD64 carry)
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{
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assert(len > 0); /* Alternative: if (!len) return carry */
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if ((BIGLITTLE(*--num,*num++) += carry) >= carry)
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return 0;
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while (--len) {
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if (++BIGLITTLE(*--num,*num++) != 0)
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return 0;
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}
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return 1;
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}
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#endif
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#endif/* !lbnAdd1_64 */
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/*
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* lbnSub1_64: subtract the single-word "borrow" from the given number.
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* Used for minor decrements and propagating the borrow after
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* subtracting a shorter bignum.
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*
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* Technique: Similar to the add, above. If there is a double-length type,
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* use that to generate the first borrow.
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* If not, after subtracting the first borrow, which may be > 1, compare
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* the difference and the *negative* of the carry. If the subtract wraps
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* (causing a borrow out from the subtraction), the result will be at least
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* as large as -borrow. If the result < -borrow, then no borrow out has
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* appeared and we may return immediately, except when borrow == 0. To
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* deal with that case, use the identity that -x = ~x+1, and instead of
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* comparing < -borrow, compare for <= ~borrow.
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* Either way, if there is a borrow out, enter a loop decrementing words
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* until a non-zero word is reached.
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*
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* Note the cast of ~borrow to (BNWORD64). If the size of an int is larger
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* than BNWORD64, C rules say the number is expanded for the arithmetic, so
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* the inversion will be done on an int and the value won't be quite what
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* is expected.
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*/
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#ifndef lbnSub1_64 /* If defined, it's provided as an asm subroutine */
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#ifdef BNWORD128
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BNWORD64
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lbnSub1_64(BNWORD64 *num, unsigned len, BNWORD64 borrow)
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{
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BNWORD128 t;
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assert(len > 0); /* Alternative: if (!len) return borrow */
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t = (BNWORD128)BIGLITTLE(*--num,*num) - borrow;
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BIGLITTLE(*num,*num++) = (BNWORD64)t;
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if ((t >> 64) == 0)
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return 0;
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while (--len) {
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if ((BIGLITTLE(*--num,*num++))-- != 0)
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return 0;
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}
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return 1;
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}
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#else /* no BNWORD128 */
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BNWORD64
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lbnSub1_64(BNWORD64 *num, unsigned len, BNWORD64 borrow)
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{
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assert(len > 0); /* Alternative: if (!len) return borrow */
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if ((BIGLITTLE(*--num,*num++) -= borrow) <= (BNWORD64)~borrow)
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return 0;
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while (--len) {
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if ((BIGLITTLE(*--num,*num++))-- != 0)
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return 0;
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}
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return 1;
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}
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#endif
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#endif /* !lbnSub1_64 */
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/*
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* lbnAddN_64: add two bignums of the same length, returning the carry (0 or 1).
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* One of the building blocks, along with lbnAdd1, of adding two bignums of
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* differing lengths.
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*
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* Technique: Maintain a word of carry. If there is no double-width type,
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* use the same technique as in lbnAdd1, above, to maintain the carry by
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* comparing the inputs. Adding the carry sources is used as an OR operator;
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* at most one of the two comparisons can possibly be true. The first can
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* only be true if carry == 1 and x, the result, is 0. In that case the
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* second can't possibly be true.
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*/
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#ifndef lbnAddN_64
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#ifdef BNWORD128
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BNWORD64
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lbnAddN_64(BNWORD64 *num1, BNWORD64 const *num2, unsigned len)
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{
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BNWORD128 t;
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assert(len > 0);
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t = (BNWORD128)BIGLITTLE(*--num1,*num1) + BIGLITTLE(*--num2,*num2++);
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BIGLITTLE(*num1,*num1++) = (BNWORD64)t;
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while (--len) {
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t = (BNWORD128)BIGLITTLE(*--num1,*num1) +
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(BNWORD128)BIGLITTLE(*--num2,*num2++) + (t >> 64);
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BIGLITTLE(*num1,*num1++) = (BNWORD64)t;
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}
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return (BNWORD64)(t>>64);
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}
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#else /* no BNWORD128 */
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BNWORD64
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lbnAddN_64(BNWORD64 *num1, BNWORD64 const *num2, unsigned len)
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{
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BNWORD64 x, carry = 0;
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assert(len > 0); /* Alternative: change loop to test at start */
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do {
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x = BIGLITTLE(*--num2,*num2++);
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carry = (x += carry) < carry;
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carry += (BIGLITTLE(*--num1,*num1++) += x) < x;
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} while (--len);
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return carry;
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}
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#endif
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#endif /* !lbnAddN_64 */
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/*
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* lbnSubN_64: add two bignums of the same length, returning the carry (0 or 1).
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* One of the building blocks, along with subn1, of subtracting two bignums of
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* differing lengths.
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*
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* Technique: If no double-width type is availble, maintain a word of borrow.
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* First, add the borrow to the subtrahend (did you have to learn all those
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* awful words in elementary school, too?), and if it overflows, set the
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* borrow again. Then subtract the modified subtrahend from the next word
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* of input, using the same technique as in subn1, above.
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* Adding the borrows is used as an OR operator; at most one of the two
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* comparisons can possibly be true. The first can only be true if
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* borrow == 1 and x, the result, is 0. In that case the second can't
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* possibly be true.
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*
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* In the double-word case, (BNWORD64)-(t>>64) is subtracted, rather than
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* adding t>>64, because the shift would need to sign-extend and that's
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* not guaranteed to happen in ANSI C, even with signed types.
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*/
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#ifndef lbnSubN_64
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#ifdef BNWORD128
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BNWORD64
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lbnSubN_64(BNWORD64 *num1, BNWORD64 const *num2, unsigned len)
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{
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BNWORD128 t;
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assert(len > 0);
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t = (BNWORD128)BIGLITTLE(*--num1,*num1) - BIGLITTLE(*--num2,*num2++);
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BIGLITTLE(*num1,*num1++) = (BNWORD64)t;
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while (--len) {
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t = (BNWORD128)BIGLITTLE(*--num1,*num1) -
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(BNWORD128)BIGLITTLE(*--num2,*num2++) - (BNWORD64)-(t >> 64);
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BIGLITTLE(*num1,*num1++) = (BNWORD64)t;
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}
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return -(BNWORD64)(t>>64);
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}
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#else
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BNWORD64
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lbnSubN_64(BNWORD64 *num1, BNWORD64 const *num2, unsigned len)
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{
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BNWORD64 x, borrow = 0;
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assert(len > 0); /* Alternative: change loop to test at start */
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do {
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x = BIGLITTLE(*--num2,*num2++);
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borrow = (x += borrow) < borrow;
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borrow += (BIGLITTLE(*--num1,*num1++) -= x) > (BNWORD64)~x;
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} while (--len);
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return borrow;
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}
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#endif
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#endif /* !lbnSubN_64 */
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#ifndef lbnCmp_64
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/*
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* lbnCmp_64: compare two bignums of equal length, returning the sign of
|
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* num1 - num2. (-1, 0 or +1).
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*
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* Technique: Change the little-endian pointers to big-endian pointers
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* and compare from the most-significant end until a difference if found.
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* When it is, figure out the sign of the difference and return it.
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*/
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int
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lbnCmp_64(BNWORD64 const *num1, BNWORD64 const *num2, unsigned len)
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{
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BIGLITTLE(num1 -= len, num1 += len);
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BIGLITTLE(num2 -= len, num2 += len);
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while (len--) {
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if (BIGLITTLE(*num1++ != *num2++, *--num1 != *--num2)) {
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if (BIGLITTLE(num1[-1] < num2[-1], *num1 < *num2))
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return -1;
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else
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return 1;
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}
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}
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return 0;
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}
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#endif /* !lbnCmp_64 */
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/*
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* mul64_ppmmaa(ph,pl,x,y,a,b) is an optional routine that
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* computes (ph,pl) = x * y + a + b. mul64_ppmma and mul64_ppmm
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* are simpler versions. If you want to be lazy, all of these
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* can be defined in terms of the others, so here we create any
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* that have not been defined in terms of the ones that have been.
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*/
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/* Define ones with fewer a's in terms of ones with more a's */
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#if !defined(mul64_ppmma) && defined(mul64_ppmmaa)
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#define mul64_ppmma(ph,pl,x,y,a) mul64_ppmmaa(ph,pl,x,y,a,0)
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#endif
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#if !defined(mul64_ppmm) && defined(mul64_ppmma)
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#define mul64_ppmm(ph,pl,x,y) mul64_ppmma(ph,pl,x,y,0)
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#endif
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/*
|
|
* Use this definition to test the mul64_ppmm-based operations on machines
|
|
* that do not provide mul64_ppmm. Change the final "0" to a "1" to
|
|
* enable it.
|
|
*/
|
|
#if !defined(mul64_ppmm) && defined(BNWORD128) && 0 /* Debugging */
|
|
#define mul64_ppmm(ph,pl,x,y) \
|
|
({BNWORD128 _ = (BNWORD128)(x)*(y); (pl) = _; (ph) = _>>64;})
|
|
#endif
|
|
|
|
#if defined(mul64_ppmm) && !defined(mul64_ppmma)
|
|
#define mul64_ppmma(ph,pl,x,y,a) \
|
|
(mul64_ppmm(ph,pl,x,y), (ph) += ((pl) += (a)) < (a))
|
|
#endif
|
|
|
|
#if defined(mul64_ppmma) && !defined(mul64_ppmmaa)
|
|
#define mul64_ppmmaa(ph,pl,x,y,a,b) \
|
|
(mul64_ppmma(ph,pl,x,y,a), (ph) += ((pl) += (b)) < (b))
|
|
#endif
|
|
|
|
/*
|
|
* lbnMulN1_64: Multiply an n-word input by a 1-word input and store the
|
|
* n+1-word product. This uses either the mul64_ppmm and mul64_ppmma
|
|
* macros, or C multiplication with the BNWORD128 type. This uses mul64_ppmma
|
|
* if available, assuming you won't bother defining it unless you can do
|
|
* better than the normal multiplication.
|
|
*/
|
|
#ifndef lbnMulN1_64
|
|
#ifdef lbnMulAdd1_64 /* If we have this asm primitive, use it. */
|
|
void
|
|
lbnMulN1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
lbnZero_64(out, len);
|
|
BIGLITTLE(*(out-len-1),*(out+len)) = lbnMulAdd1_64(out, in, len, k);
|
|
}
|
|
#elif defined(mul64_ppmm)
|
|
void
|
|
lbnMulN1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
BNWORD64 carry, carryin;
|
|
|
|
assert(len > 0);
|
|
|
|
BIG(--out;--in;);
|
|
mul64_ppmm(carry, *out, *in, k);
|
|
LITTLE(out++;in++;)
|
|
|
|
while (--len) {
|
|
BIG(--out;--in;)
|
|
carryin = carry;
|
|
mul64_ppmma(carry, *out, *in, k, carryin);
|
|
LITTLE(out++;in++;)
|
|
}
|
|
BIGLITTLE(*--out,*out) = carry;
|
|
}
|
|
#elif defined(BNWORD128)
|
|
void
|
|
lbnMulN1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
BNWORD128 p;
|
|
|
|
assert(len > 0);
|
|
|
|
p = (BNWORD128)BIGLITTLE(*--in,*in++) * k;
|
|
BIGLITTLE(*--out,*out++) = (BNWORD64)p;
|
|
|
|
while (--len) {
|
|
p = (BNWORD128)BIGLITTLE(*--in,*in++) * k + (BNWORD64)(p >> 64);
|
|
BIGLITTLE(*--out,*out++) = (BNWORD64)p;
|
|
}
|
|
BIGLITTLE(*--out,*out) = (BNWORD64)(p >> 64);
|
|
}
|
|
#else
|
|
#error No 64x64 -> 128 multiply available for 64-bit bignum package
|
|
#endif
|
|
#endif /* lbnMulN1_64 */
|
|
|
|
/*
|
|
* lbnMulAdd1_64: Multiply an n-word input by a 1-word input and add the
|
|
* low n words of the product to the destination. *Returns the n+1st word
|
|
* of the product.* (That turns out to be more convenient than adding
|
|
* it into the destination and dealing with a possible unit carry out
|
|
* of *that*.) This uses either the mul64_ppmma and mul64_ppmmaa macros,
|
|
* or C multiplication with the BNWORD128 type.
|
|
*
|
|
* If you're going to write assembly primitives, this is the one to
|
|
* start with. It is by far the most commonly called function.
|
|
*/
|
|
#ifndef lbnMulAdd1_64
|
|
#if defined(mul64_ppmm)
|
|
BNWORD64
|
|
lbnMulAdd1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
BNWORD64 prod, carry, carryin;
|
|
|
|
assert(len > 0);
|
|
|
|
BIG(--out;--in;);
|
|
carryin = *out;
|
|
mul64_ppmma(carry, *out, *in, k, carryin);
|
|
LITTLE(out++;in++;)
|
|
|
|
while (--len) {
|
|
BIG(--out;--in;);
|
|
carryin = carry;
|
|
mul64_ppmmaa(carry, prod, *in, k, carryin, *out);
|
|
*out = prod;
|
|
LITTLE(out++;in++;)
|
|
}
|
|
|
|
return carry;
|
|
}
|
|
#elif defined(BNWORD128)
|
|
BNWORD64
|
|
lbnMulAdd1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
BNWORD128 p;
|
|
|
|
assert(len > 0);
|
|
|
|
p = (BNWORD128)BIGLITTLE(*--in,*in++) * k + BIGLITTLE(*--out,*out);
|
|
BIGLITTLE(*out,*out++) = (BNWORD64)p;
|
|
|
|
while (--len) {
|
|
p = (BNWORD128)BIGLITTLE(*--in,*in++) * k +
|
|
(BNWORD64)(p >> 64) + BIGLITTLE(*--out,*out);
|
|
BIGLITTLE(*out,*out++) = (BNWORD64)p;
|
|
}
|
|
|
|
return (BNWORD64)(p >> 64);
|
|
}
|
|
#else
|
|
#error No 64x64 -> 128 multiply available for 64-bit bignum package
|
|
#endif
|
|
#endif /* lbnMulAdd1_64 */
|
|
|
|
/*
|
|
* lbnMulSub1_64: Multiply an n-word input by a 1-word input and subtract the
|
|
* n-word product from the destination. Returns the n+1st word of the product.
|
|
* This uses either the mul64_ppmm and mul64_ppmma macros, or
|
|
* C multiplication with the BNWORD128 type.
|
|
*
|
|
* This is rather uglier than adding, but fortunately it's only used in
|
|
* division which is not used too heavily.
|
|
*/
|
|
#ifndef lbnMulSub1_64
|
|
#if defined(mul64_ppmm)
|
|
BNWORD64
|
|
lbnMulSub1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
BNWORD64 prod, carry, carryin;
|
|
|
|
assert(len > 0);
|
|
|
|
BIG(--in;)
|
|
mul64_ppmm(carry, prod, *in, k);
|
|
LITTLE(in++;)
|
|
carry += (BIGLITTLE(*--out,*out++) -= prod) > (BNWORD64)~prod;
|
|
|
|
while (--len) {
|
|
BIG(--in;);
|
|
carryin = carry;
|
|
mul64_ppmma(carry, prod, *in, k, carryin);
|
|
LITTLE(in++;)
|
|
carry += (BIGLITTLE(*--out,*out++) -= prod) > (BNWORD64)~prod;
|
|
}
|
|
|
|
return carry;
|
|
}
|
|
#elif defined(BNWORD128)
|
|
BNWORD64
|
|
lbnMulSub1_64(BNWORD64 *out, BNWORD64 const *in, unsigned len, BNWORD64 k)
|
|
{
|
|
BNWORD128 p;
|
|
BNWORD64 carry, t;
|
|
|
|
assert(len > 0);
|
|
|
|
p = (BNWORD128)BIGLITTLE(*--in,*in++) * k;
|
|
t = BIGLITTLE(*--out,*out);
|
|
carry = (BNWORD64)(p>>64) + ((BIGLITTLE(*out,*out++)=t-(BNWORD64)p) > t);
|
|
|
|
while (--len) {
|
|
p = (BNWORD128)BIGLITTLE(*--in,*in++) * k + carry;
|
|
t = BIGLITTLE(*--out,*out);
|
|
carry = (BNWORD64)(p>>64) +
|
|
( (BIGLITTLE(*out,*out++)=t-(BNWORD64)p) > t );
|
|
}
|
|
|
|
return carry;
|
|
}
|
|
#else
|
|
#error No 64x64 -> 128 multiply available for 64-bit bignum package
|
|
#endif
|
|
#endif /* !lbnMulSub1_64 */
|
|
|
|
/*
|
|
* Shift n words left "shift" bits. 0 < shift < 64. Returns the
|
|
* carry, any bits shifted off the left-hand side (0 <= carry < 2^shift).
|
|
*/
|
|
#ifndef lbnLshift_64
|
|
BNWORD64
|
|
lbnLshift_64(BNWORD64 *num, unsigned len, unsigned shift)
|
|
{
|
|
BNWORD64 x, carry;
|
|
|
|
assert(shift > 0);
|
|
assert(shift < 64);
|
|
|
|
carry = 0;
|
|
while (len--) {
|
|
BIG(--num;)
|
|
x = *num;
|
|
*num = (x<<shift) | carry;
|
|
LITTLE(num++;)
|
|
carry = x >> (64-shift);
|
|
}
|
|
return carry;
|
|
}
|
|
#endif /* !lbnLshift_64 */
|
|
|
|
/*
|
|
* An optimized version of the above, for shifts of 1.
|
|
* Some machines can use add-with-carry tricks for this.
|
|
*/
|
|
#ifndef lbnDouble_64
|
|
BNWORD64
|
|
lbnDouble_64(BNWORD64 *num, unsigned len)
|
|
{
|
|
BNWORD64 x, carry;
|
|
|
|
carry = 0;
|
|
while (len--) {
|
|
BIG(--num;)
|
|
x = *num;
|
|
*num = (x<<1) | carry;
|
|
LITTLE(num++;)
|
|
carry = x >> (64-1);
|
|
}
|
|
return carry;
|
|
}
|
|
#endif /* !lbnDouble_64 */
|
|
|
|
/*
|
|
* Shift n words right "shift" bits. 0 < shift < 64. Returns the
|
|
* carry, any bits shifted off the right-hand side (0 <= carry < 2^shift).
|
|
*/
|
|
#ifndef lbnRshift_64
|
|
BNWORD64
|
|
lbnRshift_64(BNWORD64 *num, unsigned len, unsigned shift)
|
|
{
|
|
BNWORD64 x, carry = 0;
|
|
|
|
assert(shift > 0);
|
|
assert(shift < 64);
|
|
|
|
BIGLITTLE(num -= len, num += len);
|
|
|
|
while (len--) {
|
|
LITTLE(--num;)
|
|
x = *num;
|
|
*num = (x>>shift) | carry;
|
|
BIG(num++;)
|
|
carry = x << (64-shift);
|
|
}
|
|
return carry >> (64-shift);
|
|
}
|
|
#endif /* !lbnRshift_64 */
|
|
|
|
/*
|
|
* Multiply two numbers of the given lengths. prod and num2 may overlap,
|
|
* provided that the low len1 bits of prod are free. (This corresponds
|
|
* nicely to the place the result is returned from lbnMontReduce_64.)
|
|
*
|
|
* TODO: Use Karatsuba multiply. The overlap constraints may have
|
|
* to get rewhacked.
|
|
*/
|
|
#ifndef lbnMul_64
|
|
void
|
|
lbnMul_64(BNWORD64 *prod, BNWORD64 const *num1, unsigned len1,
|
|
BNWORD64 const *num2, unsigned len2)
|
|
{
|
|
/* Special case of zero */
|
|
if (!len1 || !len2) {
|
|
lbnZero_64(prod, len1+len2);
|
|
return;
|
|
}
|
|
|
|
/* Multiply first word */
|
|
lbnMulN1_64(prod, num1, len1, BIGLITTLE(*--num2,*num2++));
|
|
|
|
/*
|
|
* Add in subsequent words, storing the most significant word,
|
|
* which is new each time.
|
|
*/
|
|
while (--len2) {
|
|
BIGLITTLE(--prod,prod++);
|
|
BIGLITTLE(*(prod-len1-1),*(prod+len1)) =
|
|
lbnMulAdd1_64(prod, num1, len1, BIGLITTLE(*--num2,*num2++));
|
|
}
|
|
}
|
|
#endif /* !lbnMul_64 */
|
|
|
|
/*
|
|
* lbnMulX_64 is a square multiply - both inputs are the same length.
|
|
* It's normally just a macro wrapper around the general multiply,
|
|
* but might be implementable in assembly more efficiently (such as
|
|
* when product scanning).
|
|
*/
|
|
#ifndef lbnMulX_64
|
|
#if defined(BNWORD128) && PRODUCT_SCAN
|
|
/*
|
|
* Test code to see whether product scanning is any faster. It seems
|
|
* to make the C code slower, so PRODUCT_SCAN is not defined.
|
|
*/
|
|
static void
|
|
lbnMulX_64(BNWORD64 *prod, BNWORD64 const *num1, BNWORD64 const *num2,
|
|
unsigned len)
|
|
{
|
|
BNWORD128 x, y;
|
|
BNWORD64 const *p1, *p2;
|
|
unsigned carry;
|
|
unsigned i, j;
|
|
|
|
/* Special case of zero */
|
|
if (!len)
|
|
return;
|
|
|
|
x = (BNWORD128)BIGLITTLE(num1[-1] * num2[-1], num1[0] * num2[0]);
|
|
BIGLITTLE(*--prod, *prod++) = (BNWORD64)x;
|
|
x >>= 64;
|
|
|
|
for (i = 1; i < len; i++) {
|
|
carry = 0;
|
|
p1 = num1;
|
|
p2 = BIGLITTLE(num2-i-1,num2+i+1);
|
|
for (j = 0; j <= i; j++) {
|
|
BIG(y = (BNWORD128)*--p1 * *p2++;)
|
|
LITTLE(y = (BNWORD128)*p1++ * *--p2;)
|
|
x += y;
|
|
carry += (x < y);
|
|
}
|
|
BIGLITTLE(*--prod,*prod++) = (BNWORD64)x;
|
|
x = (x >> 64) | (BNWORD128)carry << 64;
|
|
}
|
|
for (i = 1; i < len; i++) {
|
|
carry = 0;
|
|
p1 = BIGLITTLE(num1-i,num1+i);
|
|
p2 = BIGLITTLE(num2-len,num2+len);
|
|
for (j = i; j < len; j++) {
|
|
BIG(y = (BNWORD128)*--p1 * *p2++;)
|
|
LITTLE(y = (BNWORD128)*p1++ * *--p2;)
|
|
x += y;
|
|
carry += (x < y);
|
|
}
|
|
BIGLITTLE(*--prod,*prod++) = (BNWORD64)x;
|
|
x = (x >> 64) | (BNWORD128)carry << 64;
|
|
}
|
|
|
|
BIGLITTLE(*--prod,*prod) = (BNWORD64)x;
|
|
}
|
|
#else /* !defined(BNWORD128) || !PRODUCT_SCAN */
|
|
/* Default trivial macro definition */
|
|
#define lbnMulX_64(prod, num1, num2, len) lbnMul_64(prod, num1, len, num2, len)
|
|
#endif /* !defined(BNWORD128) || !PRODUCT_SCAN */
|
|
#endif /* !lbmMulX_64 */
|
|
|
|
#if !defined(lbnMontMul_64) && defined(BNWORD128) && PRODUCT_SCAN
|
|
/*
|
|
* Test code for product-scanning multiply. This seems to slow the C
|
|
* code down rather than speed it up.
|
|
* This does a multiply and Montgomery reduction together, using the
|
|
* same loops. The outer loop scans across the product, twice.
|
|
* The first pass computes the low half of the product and the
|
|
* Montgomery multipliers. These are stored in the product array,
|
|
* which contains no data as of yet. x and carry add up the columns
|
|
* and propagate carries forward.
|
|
*
|
|
* The second half multiplies the upper half, adding in the modulus
|
|
* times the Montgomery multipliers. The results of this multiply
|
|
* are stored.
|
|
*/
|
|
static void
|
|
lbnMontMul_64(BNWORD64 *prod, BNWORD64 const *num1, BNWORD64 const *num2,
|
|
BNWORD64 const *mod, unsigned len, BNWORD64 inv)
|
|
{
|
|
BNWORD128 x, y;
|
|
BNWORD64 const *p1, *p2, *pm;
|
|
BNWORD64 *pp;
|
|
BNWORD64 t;
|
|
unsigned carry;
|
|
unsigned i, j;
|
|
|
|
/* Special case of zero */
|
|
if (!len)
|
|
return;
|
|
|
|
/*
|
|
* This computes directly into the high half of prod, so just
|
|
* shift the pointer and consider prod only "len" elements long
|
|
* for the rest of the code.
|
|
*/
|
|
BIGLITTLE(prod -= len, prod += len);
|
|
|
|
/* Pass 1 - compute Montgomery multipliers */
|
|
/* First iteration can have certain simplifications. */
|
|
x = (BNWORD128)BIGLITTLE(num1[-1] * num2[-1], num1[0] * num2[0]);
|
|
BIGLITTLE(prod[-1], prod[0]) = t = inv * (BNWORD64)x;
|
|
y = (BNWORD128)t * BIGLITTLE(mod[-1],mod[0]);
|
|
x += y;
|
|
/* Note: GCC 2.6.3 has a bug if you try to eliminate "carry" */
|
|
carry = (x < y);
|
|
assert((BNWORD64)x == 0);
|
|
x = x >> 64 | (BNWORD128)carry << 64;
|
|
|
|
for (i = 1; i < len; i++) {
|
|
carry = 0;
|
|
p1 = num1;
|
|
p2 = BIGLITTLE(num2-i-1,num2+i+1);
|
|
pp = prod;
|
|
pm = BIGLITTLE(mod-i-1,mod+i+1);
|
|
for (j = 0; j < i; j++) {
|
|
y = (BNWORD128)BIGLITTLE(*--p1 * *p2++, *p1++ * *--p2);
|
|
x += y;
|
|
carry += (x < y);
|
|
y = (BNWORD128)BIGLITTLE(*--pp * *pm++, *pp++ * *--pm);
|
|
x += y;
|
|
carry += (x < y);
|
|
}
|
|
y = (BNWORD128)BIGLITTLE(p1[-1] * p2[0], p1[0] * p2[-1]);
|
|
x += y;
|
|
carry += (x < y);
|
|
assert(BIGLITTLE(pp == prod-i, pp == prod+i));
|
|
BIGLITTLE(pp[-1], pp[0]) = t = inv * (BNWORD64)x;
|
|
assert(BIGLITTLE(pm == mod-1, pm == mod+1));
|
|
y = (BNWORD128)t * BIGLITTLE(pm[0],pm[-1]);
|
|
x += y;
|
|
carry += (x < y);
|
|
assert((BNWORD64)x == 0);
|
|
x = x >> 64 | (BNWORD128)carry << 64;
|
|
}
|
|
|
|
/* Pass 2 - compute reduced product and store */
|
|
for (i = 1; i < len; i++) {
|
|
carry = 0;
|
|
p1 = BIGLITTLE(num1-i,num1+i);
|
|
p2 = BIGLITTLE(num2-len,num2+len);
|
|
pm = BIGLITTLE(mod-i,mod+i);
|
|
pp = BIGLITTLE(prod-len,prod+len);
|
|
for (j = i; j < len; j++) {
|
|
y = (BNWORD128)BIGLITTLE(*--p1 * *p2++, *p1++ * *--p2);
|
|
x += y;
|
|
carry += (x < y);
|
|
y = (BNWORD128)BIGLITTLE(*--pm * *pp++, *pm++ * *--pp);
|
|
x += y;
|
|
carry += (x < y);
|
|
}
|
|
assert(BIGLITTLE(pm == mod-len, pm == mod+len));
|
|
assert(BIGLITTLE(pp == prod-i, pp == prod+i));
|
|
BIGLITTLE(pp[0],pp[-1]) = (BNWORD64)x;
|
|
x = (x >> 64) | (BNWORD128)carry << 64;
|
|
}
|
|
|
|
/* Last round of second half, simplified. */
|
|
BIGLITTLE(*(prod-len),*(prod+len-1)) = (BNWORD64)x;
|
|
carry = (x >> 64);
|
|
|
|
while (carry)
|
|
carry -= lbnSubN_64(prod, mod, len);
|
|
while (lbnCmp_64(prod, mod, len) >= 0)
|
|
(void)lbnSubN_64(prod, mod, len);
|
|
}
|
|
/* Suppress later definition */
|
|
#define lbnMontMul_64 lbnMontMul_64
|
|
#endif
|
|
|
|
#if !defined(lbnSquare_64) && defined(BNWORD128) && PRODUCT_SCAN
|
|
/*
|
|
* Trial code for product-scanning squaring. This seems to slow the C
|
|
* code down rather than speed it up.
|
|
*/
|
|
void
|
|
lbnSquare_64(BNWORD64 *prod, BNWORD64 const *num, unsigned len)
|
|
{
|
|
BNWORD128 x, y, z;
|
|
BNWORD64 const *p1, *p2;
|
|
unsigned carry;
|
|
unsigned i, j;
|
|
|
|
/* Special case of zero */
|
|
if (!len)
|
|
return;
|
|
|
|
/* Word 0 of product */
|
|
x = (BNWORD128)BIGLITTLE(num[-1] * num[-1], num[0] * num[0]);
|
|
BIGLITTLE(*--prod, *prod++) = (BNWORD64)x;
|
|
x >>= 64;
|
|
|
|
/* Words 1 through len-1 */
|
|
for (i = 1; i < len; i++) {
|
|
carry = 0;
|
|
y = 0;
|
|
p1 = num;
|
|
p2 = BIGLITTLE(num-i-1,num+i+1);
|
|
for (j = 0; j < (i+1)/2; j++) {
|
|
BIG(z = (BNWORD128)*--p1 * *p2++;)
|
|
LITTLE(z = (BNWORD128)*p1++ * *--p2;)
|
|
y += z;
|
|
carry += (y < z);
|
|
}
|
|
y += z = y;
|
|
carry += carry + (y < z);
|
|
if ((i & 1) == 0) {
|
|
assert(BIGLITTLE(--p1 == p2, p1 == --p2));
|
|
BIG(z = (BNWORD128)*p2 * *p2;)
|
|
LITTLE(z = (BNWORD128)*p1 * *p1;)
|
|
y += z;
|
|
carry += (y < z);
|
|
}
|
|
x += y;
|
|
carry += (x < y);
|
|
BIGLITTLE(*--prod,*prod++) = (BNWORD64)x;
|
|
x = (x >> 64) | (BNWORD128)carry << 64;
|
|
}
|
|
/* Words len through 2*len-2 */
|
|
for (i = 1; i < len; i++) {
|
|
carry = 0;
|
|
y = 0;
|
|
p1 = BIGLITTLE(num-i,num+i);
|
|
p2 = BIGLITTLE(num-len,num+len);
|
|
for (j = 0; j < (len-i)/2; j++) {
|
|
BIG(z = (BNWORD128)*--p1 * *p2++;)
|
|
LITTLE(z = (BNWORD128)*p1++ * *--p2;)
|
|
y += z;
|
|
carry += (y < z);
|
|
}
|
|
y += z = y;
|
|
carry += carry + (y < z);
|
|
if ((len-i) & 1) {
|
|
assert(BIGLITTLE(--p1 == p2, p1 == --p2));
|
|
BIG(z = (BNWORD128)*p2 * *p2;)
|
|
LITTLE(z = (BNWORD128)*p1 * *p1;)
|
|
y += z;
|
|
carry += (y < z);
|
|
}
|
|
x += y;
|
|
carry += (x < y);
|
|
BIGLITTLE(*--prod,*prod++) = (BNWORD64)x;
|
|
x = (x >> 64) | (BNWORD128)carry << 64;
|
|
}
|
|
|
|
/* Word 2*len-1 */
|
|
BIGLITTLE(*--prod,*prod) = (BNWORD64)x;
|
|
}
|
|
/* Suppress later definition */
|
|
#define lbnSquare_64 lbnSquare_64
|
|
#endif
|
|
|
|
/*
|
|
* Square a number, using optimized squaring to reduce the number of
|
|
* primitive multiples that are executed. There may not be any
|
|
* overlap of the input and output.
|
|
*
|
|
* Technique: Consider the partial products in the multiplication
|
|
* of "abcde" by itself:
|
|
*
|
|
* a b c d e
|
|
* * a b c d e
|
|
* ==================
|
|
* ae be ce de ee
|
|
* ad bd cd dd de
|
|
* ac bc cc cd ce
|
|
* ab bb bc bd be
|
|
* aa ab ac ad ae
|
|
*
|
|
* Note that everything above the main diagonal:
|
|
* ae be ce de = (abcd) * e
|
|
* ad bd cd = (abc) * d
|
|
* ac bc = (ab) * c
|
|
* ab = (a) * b
|
|
*
|
|
* is a copy of everything below the main diagonal:
|
|
* de
|
|
* cd ce
|
|
* bc bd be
|
|
* ab ac ad ae
|
|
*
|
|
* Thus, the sum is 2 * (off the diagonal) + diagonal.
|
|
*
|
|
* This is accumulated beginning with the diagonal (which
|
|
* consist of the squares of the digits of the input), which is then
|
|
* divided by two, the off-diagonal added, and multiplied by two
|
|
* again. The low bit is simply a copy of the low bit of the
|
|
* input, so it doesn't need special care.
|
|
*
|
|
* TODO: Merge the shift by 1 with the squaring loop.
|
|
* TODO: Use Karatsuba. (a*W+b)^2 = a^2 * (W^2+W) + b^2 * (W+1) - (a-b)^2 * W.
|
|
*/
|
|
#ifndef lbnSquare_64
|
|
void
|
|
lbnSquare_64(BNWORD64 *prod, BNWORD64 const *num, unsigned len)
|
|
{
|
|
BNWORD64 t;
|
|
BNWORD64 *prodx = prod; /* Working copy of the argument */
|
|
BNWORD64 const *numx = num; /* Working copy of the argument */
|
|
unsigned lenx = len; /* Working copy of the argument */
|
|
|
|
if (!len)
|
|
return;
|
|
|
|
/* First, store all the squares */
|
|
while (lenx--) {
|
|
#ifdef mul64_ppmm
|
|
BNWORD64 ph, pl;
|
|
t = BIGLITTLE(*--numx,*numx++);
|
|
mul64_ppmm(ph,pl,t,t);
|
|
BIGLITTLE(*--prodx,*prodx++) = pl;
|
|
BIGLITTLE(*--prodx,*prodx++) = ph;
|
|
#elif defined(BNWORD128) /* use BNWORD128 */
|
|
BNWORD128 p;
|
|
t = BIGLITTLE(*--numx,*numx++);
|
|
p = (BNWORD128)t * t;
|
|
BIGLITTLE(*--prodx,*prodx++) = (BNWORD64)p;
|
|
BIGLITTLE(*--prodx,*prodx++) = (BNWORD64)(p>>64);
|
|
#else /* Use lbnMulN1_64 */
|
|
t = BIGLITTLE(numx[-1],*numx);
|
|
lbnMulN1_64(prodx, numx, 1, t);
|
|
BIGLITTLE(--numx,numx++);
|
|
BIGLITTLE(prodx -= 2, prodx += 2);
|
|
#endif
|
|
}
|
|
/* Then, shift right 1 bit */
|
|
(void)lbnRshift_64(prod, 2*len, 1);
|
|
|
|
/* Then, add in the off-diagonal sums */
|
|
lenx = len;
|
|
numx = num;
|
|
prodx = prod;
|
|
while (--lenx) {
|
|
t = BIGLITTLE(*--numx,*numx++);
|
|
BIGLITTLE(--prodx,prodx++);
|
|
t = lbnMulAdd1_64(prodx, numx, lenx, t);
|
|
lbnAdd1_64(BIGLITTLE(prodx-lenx,prodx+lenx), lenx+1, t);
|
|
BIGLITTLE(--prodx,prodx++);
|
|
}
|
|
|
|
/* Shift it back up */
|
|
lbnDouble_64(prod, 2*len);
|
|
|
|
/* And set the low bit appropriately */
|
|
BIGLITTLE(prod[-1],prod[0]) |= BIGLITTLE(num[-1],num[0]) & 1;
|
|
}
|
|
#endif /* !lbnSquare_64 */
|
|
|
|
/*
|
|
* lbnNorm_64 - given a number, return a modified length such that the
|
|
* most significant digit is non-zero. Zero-length input is okay.
|
|
*/
|
|
#ifndef lbnNorm_64
|
|
unsigned
|
|
lbnNorm_64(BNWORD64 const *num, unsigned len)
|
|
{
|
|
BIGLITTLE(num -= len,num += len);
|
|
while (len && BIGLITTLE(*num++,*--num) == 0)
|
|
--len;
|
|
return len;
|
|
}
|
|
#endif /* lbnNorm_64 */
|
|
|
|
/*
|
|
* lbnBits_64 - return the number of significant bits in the array.
|
|
* It starts by normalizing the array. Zero-length input is okay.
|
|
* Then assuming there's anything to it, it fetches the high word,
|
|
* generates a bit length by multiplying the word length by 64, and
|
|
* subtracts off 64/2, 64/4, 64/8, ... bits if the high bits are clear.
|
|
*/
|
|
#ifndef lbnBits_64
|
|
unsigned
|
|
lbnBits_64(BNWORD64 const *num, unsigned len)
|
|
{
|
|
BNWORD64 t;
|
|
unsigned i;
|
|
|
|
len = lbnNorm_64(num, len);
|
|
if (len) {
|
|
t = BIGLITTLE(*(num-len),*(num+(len-1)));
|
|
assert(t);
|
|
len *= 64;
|
|
i = 64/2;
|
|
do {
|
|
if (t >> i)
|
|
t >>= i;
|
|
else
|
|
len -= i;
|
|
} while ((i /= 2) != 0);
|
|
}
|
|
return len;
|
|
}
|
|
#endif /* lbnBits_64 */
|
|
|
|
/*
|
|
* If defined, use hand-rolled divide rather than compiler's native.
|
|
* If the machine doesn't do it in line, the manual code is probably
|
|
* faster, since it can assume normalization and the fact that the
|
|
* quotient will fit into 64 bits, which a general 128-bit divide
|
|
* in a compiler's run-time library can't do.
|
|
*/
|
|
#ifndef BN_SLOW_DIVIDE_128
|
|
/* Assume that divisors of more than thirty-two bits are slow */
|
|
#define BN_SLOW_DIVIDE_128 (128 > 0x20)
|
|
#endif
|
|
|
|
/*
|
|
* Return (nh<<64|nl) % d, and place the quotient digit into *q.
|
|
* It is guaranteed that nh < d, and that d is normalized (with its high
|
|
* bit set). If we have a double-width type, it's easy. If not, ooh,
|
|
* yuk!
|
|
*/
|
|
#ifndef lbnDiv21_64
|
|
#if defined(BNWORD128) && !BN_SLOW_DIVIDE_128
|
|
BNWORD64
|
|
lbnDiv21_64(BNWORD64 *q, BNWORD64 nh, BNWORD64 nl, BNWORD64 d)
|
|
{
|
|
BNWORD128 n = (BNWORD128)nh << 64 | nl;
|
|
|
|
/* Divisor must be normalized */
|
|
assert(d >> (64-1) == 1);
|
|
|
|
*q = n / d;
|
|
return n % d;
|
|
}
|
|
#else
|
|
/*
|
|
* This is where it gets ugly.
|
|
*
|
|
* Do the division in two halves, using Algorithm D from section 4.3.1
|
|
* of Knuth. Note Theorem B from that section, that the quotient estimate
|
|
* is never more than the true quotient, and is never more than two
|
|
* too low.
|
|
*
|
|
* The mapping onto conventional long division is (everything a half word):
|
|
* _____________qh___ql_
|
|
* dh dl ) nh.h nh.l nl.h nl.l
|
|
* - (qh * d)
|
|
* -----------
|
|
* rrrr rrrr nl.l
|
|
* - (ql * d)
|
|
* -----------
|
|
* rrrr rrrr
|
|
*
|
|
* The implicit 3/2-digit d*qh and d*ql subtractors are computed this way:
|
|
* First, estimate a q digit so that nh/dh works. Subtracting qh*dh from
|
|
* the (nh.h nh.l) list leaves a 1/2-word remainder r. Then compute the
|
|
* low part of the subtractor, qh * dl. This also needs to be subtracted
|
|
* from (nh.h nh.l nl.h) to get the final remainder. So we take the
|
|
* remainder, which is (nh.h nh.l) - qh*dl, shift it and add in nl.h, and
|
|
* try to subtract qh * dl from that. Since the remainder is 1/2-word
|
|
* long, shifting and adding nl.h results in a single word r.
|
|
* It is possible that the remainder we're working with, r, is less than
|
|
* the product qh * dl, if we estimated qh too high. The estimation
|
|
* technique can produce a qh that is too large (never too small), leading
|
|
* to r which is too small. In that case, decrement the digit qh, add
|
|
* shifted dh to r (to correct for that error), and subtract dl from the
|
|
* product we're comparing r with. That's the "correct" way to do it, but
|
|
* just adding dl to r instead of subtracting it from the product is
|
|
* equivalent and a lot simpler. You just have to watch out for overflow.
|
|
*
|
|
* The process is repeated with (rrrr rrrr nl.l) for the low digit of the
|
|
* quotient ql.
|
|
*
|
|
* The various uses of 64/2 for shifts are because of the note about
|
|
* automatic editing of this file at the very top of the file.
|
|
*/
|
|
#define highhalf(x) ( (x) >> 64/2 )
|
|
#define lowhalf(x) ( (x) & (((BNWORD64)1 << 64/2)-1) )
|
|
BNWORD64
|
|
lbnDiv21_64(BNWORD64 *q, BNWORD64 nh, BNWORD64 nl, BNWORD64 d)
|
|
{
|
|
BNWORD64 dh = highhalf(d), dl = lowhalf(d);
|
|
BNWORD64 qh, ql, prod, r;
|
|
|
|
/* Divisor must be normalized */
|
|
assert((d >> (64-1)) == 1);
|
|
|
|
/* Do first half-word of division */
|
|
qh = nh / dh;
|
|
r = nh % dh;
|
|
prod = qh * dl;
|
|
|
|
/*
|
|
* Add next half-word of numerator to remainder and correct.
|
|
* qh may be up to two too large.
|
|
*/
|
|
r = (r << (64/2)) | highhalf(nl);
|
|
if (r < prod) {
|
|
--qh; r += d;
|
|
if (r >= d && r < prod) {
|
|
--qh; r += d;
|
|
}
|
|
}
|
|
r -= prod;
|
|
|
|
/* Do second half-word of division */
|
|
ql = r / dh;
|
|
r = r % dh;
|
|
prod = ql * dl;
|
|
|
|
r = (r << (64/2)) | lowhalf(nl);
|
|
if (r < prod) {
|
|
--ql; r += d;
|
|
if (r >= d && r < prod) {
|
|
--ql; r += d;
|
|
}
|
|
}
|
|
r -= prod;
|
|
|
|
*q = (qh << (64/2)) | ql;
|
|
|
|
return r;
|
|
}
|
|
#endif
|
|
#endif /* lbnDiv21_64 */
|
|
|
|
|
|
/*
|
|
* In the division functions, the dividend and divisor are referred to
|
|
* as "n" and "d", which stand for "numerator" and "denominator".
|
|
*
|
|
* The quotient is (nlen-dlen+1) digits long. It may be overlapped with
|
|
* the high (nlen-dlen) words of the dividend, but one extra word is needed
|
|
* on top to hold the top word.
|
|
*/
|
|
|
|
/*
|
|
* Divide an n-word number by a 1-word number, storing the remainder
|
|
* and n-1 words of the n-word quotient. The high word is returned.
|
|
* It IS legal for rem to point to the same address as n, and for
|
|
* q to point one word higher.
|
|
*
|
|
* TODO: If BN_SLOW_DIVIDE_128, add a divnhalf_64 which uses 64-bit
|
|
* dividends if the divisor is half that long.
|
|
* TODO: Shift the dividend on the fly to avoid the last division and
|
|
* instead have a remainder that needs shifting.
|
|
* TODO: Use reciprocals rather than dividing.
|
|
*/
|
|
#ifndef lbnDiv1_64
|
|
BNWORD64
|
|
lbnDiv1_64(BNWORD64 *q, BNWORD64 *rem, BNWORD64 const *n, unsigned len,
|
|
BNWORD64 d)
|
|
{
|
|
unsigned shift;
|
|
unsigned xlen;
|
|
BNWORD64 r;
|
|
BNWORD64 qhigh;
|
|
|
|
assert(len > 0);
|
|
assert(d);
|
|
|
|
if (len == 1) {
|
|
r = *n;
|
|
*rem = r%d;
|
|
return r/d;
|
|
}
|
|
|
|
shift = 0;
|
|
r = d;
|
|
xlen = 64/2;
|
|
do {
|
|
if (r >> xlen)
|
|
r >>= xlen;
|
|
else
|
|
shift += xlen;
|
|
} while ((xlen /= 2) != 0);
|
|
assert((d >> (64-1-shift)) == 1);
|
|
d <<= shift;
|
|
|
|
BIGLITTLE(q -= len-1,q += len-1);
|
|
BIGLITTLE(n -= len,n += len);
|
|
|
|
r = BIGLITTLE(*n++,*--n);
|
|
if (r < d) {
|
|
qhigh = 0;
|
|
} else {
|
|
qhigh = r/d;
|
|
r %= d;
|
|
}
|
|
|
|
xlen = len;
|
|
while (--xlen)
|
|
r = lbnDiv21_64(BIGLITTLE(q++,--q), r, BIGLITTLE(*n++,*--n), d);
|
|
|
|
/*
|
|
* Final correction for shift - shift the quotient up "shift"
|
|
* bits, and merge in the extra bits of quotient. Then reduce
|
|
* the final remainder mod the real d.
|
|
*/
|
|
if (shift) {
|
|
d >>= shift;
|
|
qhigh = (qhigh << shift) | lbnLshift_64(q, len-1, shift);
|
|
BIGLITTLE(q[-1],*q) |= r/d;
|
|
r %= d;
|
|
}
|
|
*rem = r;
|
|
|
|
return qhigh;
|
|
}
|
|
#endif
|
|
|
|
/*
|
|
* This function performs a "quick" modulus of a number with a divisor
|
|
* d which is guaranteed to be at most sixteen bits, i.e. less than 65536.
|
|
* This applies regardless of the word size the library is compiled with.
|
|
*
|
|
* This function is important to prime generation, for sieving.
|
|
*/
|
|
#ifndef lbnModQ_64
|
|
/* If there's a custom lbnMod21_64, no normalization needed */
|
|
#ifdef lbnMod21_64
|
|
unsigned
|
|
lbnModQ_64(BNWORD64 const *n, unsigned len, unsigned d)
|
|
{
|
|
unsigned i, shift;
|
|
BNWORD64 r;
|
|
|
|
assert(len > 0);
|
|
|
|
BIGLITTLE(n -= len,n += len);
|
|
|
|
/* Try using a compare to avoid the first divide */
|
|
r = BIGLITTLE(*n++,*--n);
|
|
if (r >= d)
|
|
r %= d;
|
|
while (--len)
|
|
r = lbnMod21_64(r, BIGLITTLE(*n++,*--n), d);
|
|
|
|
return r;
|
|
}
|
|
#elif defined(BNWORD128) && !BN_SLOW_DIVIDE_128
|
|
unsigned
|
|
lbnModQ_64(BNWORD64 const *n, unsigned len, unsigned d)
|
|
{
|
|
BNWORD64 r;
|
|
|
|
if (!--len)
|
|
return BIGLITTLE(n[-1],n[0]) % d;
|
|
|
|
BIGLITTLE(n -= len,n += len);
|
|
r = BIGLITTLE(n[-1],n[0]);
|
|
|
|
do {
|
|
r = (BNWORD64)((((BNWORD128)r<<64) | BIGLITTLE(*n++,*--n)) % d);
|
|
} while (--len);
|
|
|
|
return r;
|
|
}
|
|
#elif 64 >= 0x20
|
|
/*
|
|
* If the single word size can hold 65535*65536, then this function
|
|
* is avilable.
|
|
*/
|
|
#ifndef highhalf
|
|
#define highhalf(x) ( (x) >> 64/2 )
|
|
#define lowhalf(x) ( (x) & ((1 << 64/2)-1) )
|
|
#endif
|
|
unsigned
|
|
lbnModQ_64(BNWORD64 const *n, unsigned len, unsigned d)
|
|
{
|
|
BNWORD64 r, x;
|
|
|
|
BIGLITTLE(n -= len,n += len);
|
|
|
|
r = BIGLITTLE(*n++,*--n);
|
|
while (--len) {
|
|
x = BIGLITTLE(*n++,*--n);
|
|
r = (r%d << 64/2) | highhalf(x);
|
|
r = (r%d << 64/2) | lowhalf(x);
|
|
}
|
|
|
|
return r%d;
|
|
}
|
|
#else
|
|
/* Default case - use lbnDiv21_64 */
|
|
unsigned
|
|
lbnModQ_64(BNWORD64 const *n, unsigned len, unsigned d)
|
|
{
|
|
unsigned i, shift;
|
|
BNWORD64 r;
|
|
BNWORD64 q;
|
|
|
|
assert(len > 0);
|
|
|
|
shift = 0;
|
|
r = d;
|
|
i = 64;
|
|
while (i /= 2) {
|
|
if (r >> i)
|
|
r >>= i;
|
|
else
|
|
shift += i;
|
|
}
|
|
assert(d >> (64-1-shift) == 1);
|
|
d <<= shift;
|
|
|
|
BIGLITTLE(n -= len,n += len);
|
|
|
|
r = BIGLITTLE(*n++,*--n);
|
|
if (r >= d)
|
|
r %= d;
|
|
|
|
while (--len)
|
|
r = lbnDiv21_64(&q, r, BIGLITTLE(*n++,*--n), d);
|
|
|
|
/*
|
|
* Final correction for shift - shift the quotient up "shift"
|
|
* bits, and merge in the extra bits of quotient. Then reduce
|
|
* the final remainder mod the real d.
|
|
*/
|
|
if (shift)
|
|
r %= d >> shift;
|
|
|
|
return r;
|
|
}
|
|
#endif
|
|
#endif /* lbnModQ_64 */
|
|
|
|
/*
|
|
* Reduce n mod d and return the quotient. That is, find:
|
|
* q = n / d;
|
|
* n = n % d;
|
|
* d is altered during the execution of this subroutine by normalizing it.
|
|
* It must already have its most significant word non-zero; it is shifted
|
|
* so its most significant bit is non-zero.
|
|
*
|
|
* The quotient q is nlen-dlen+1 words long. To make it possible to
|
|
* overlap the quptient with the input (you can store it in the high dlen
|
|
* words), the high word of the quotient is *not* stored, but is returned.
|
|
* (If all you want is the remainder, you don't care about it, anyway.)
|
|
*
|
|
* This uses algorithm D from Knuth (4.3.1), except that we do binary
|
|
* (shift) normalization of the divisor. WARNING: This is hairy!
|
|
*
|
|
* This function is used for some modular reduction, but it is not used in
|
|
* the modular exponentiation loops; they use Montgomery form and the
|
|
* corresponding, more efficient, Montgomery reduction. This code
|
|
* is needed for the conversion to Montgomery form, however, so it
|
|
* has to be here and it might as well be reasonably efficient.
|
|
*
|
|
* The overall operation is as follows ("top" and "up" refer to the
|
|
* most significant end of the number; "bottom" and "down", the least):
|
|
*
|
|
* - Shift the divisor up until the most significant bit is set.
|
|
* - Shift the dividend up the same amount. This will produce the
|
|
* correct quotient, and the remainder can be recovered by shifting
|
|
* it back down the same number of bits. This may produce an overflow
|
|
* word, but the word is always strictly less than the most significant
|
|
* divisor word.
|
|
* - Estimate the first quotient digit qhat:
|
|
* - First take the top two words (one of which is the overflow) of the
|
|
* dividend and divide by the top word of the divisor:
|
|
* qhat = (nh,nm)/dh. This qhat is >= the correct quotient digit
|
|
* and, since dh is normalized, it is at most two over.
|
|
* - Second, correct by comparing the top three words. If
|
|
* (dh,dl) * qhat > (nh,nm,ml), decrease qhat and try again.
|
|
* The second iteration can be simpler because there can't be a third.
|
|
* The computation can be simplified by subtracting dh*qhat from
|
|
* both sides, suitably shifted. This reduces the left side to
|
|
* dl*qhat. On the right, (nh,nm)-dh*qhat is simply the
|
|
* remainder r from (nh,nm)%dh, so the right is (r,nl).
|
|
* This produces qhat that is almost always correct and at
|
|
* most (prob ~ 2/2^64) one too high.
|
|
* - Subtract qhat times the divisor (suitably shifted) from the dividend.
|
|
* If there is a borrow, qhat was wrong, so decrement it
|
|
* and add the divisor back in (once).
|
|
* - Store the final quotient digit qhat in the quotient array q.
|
|
*
|
|
* Repeat the quotient digit computation for successive digits of the
|
|
* quotient until the whole quotient has been computed. Then shift the
|
|
* divisor and the remainder down to correct for the normalization.
|
|
*
|
|
* TODO: Special case 2-word divisors.
|
|
* TODO: Use reciprocals rather than dividing.
|
|
*/
|
|
#ifndef divn_64
|
|
BNWORD64
|
|
lbnDiv_64(BNWORD64 *q, BNWORD64 *n, unsigned nlen, BNWORD64 *d, unsigned dlen)
|
|
{
|
|
BNWORD64 nh,nm,nl; /* Top three words of the dividend */
|
|
BNWORD64 dh,dl; /* Top two words of the divisor */
|
|
BNWORD64 qhat; /* Extimate of quotient word */
|
|
BNWORD64 r; /* Remainder from quotient estimate division */
|
|
BNWORD64 qhigh; /* High word of quotient */
|
|
unsigned i; /* Temp */
|
|
unsigned shift; /* Bits shifted by normalization */
|
|
unsigned qlen = nlen-dlen; /* Size of quotient (less 1) */
|
|
#ifdef mul64_ppmm
|
|
BNWORD64 t64;
|
|
#elif defined(BNWORD128)
|
|
BNWORD128 t128;
|
|
#else /* use lbnMulN1_64 */
|
|
BNWORD64 t2[2];
|
|
#define t2high BIGLITTLE(t2[0],t2[1])
|
|
#define t2low BIGLITTLE(t2[1],t2[0])
|
|
#endif
|
|
|
|
assert(dlen);
|
|
assert(nlen >= dlen);
|
|
|
|
/*
|
|
* Special cases for short divisors. The general case uses the
|
|
* top top 2 digits of the divisor (d) to estimate a quotient digit,
|
|
* so it breaks if there are fewer digits available. Thus, we need
|
|
* special cases for a divisor of length 1. A divisor of length
|
|
* 2 can have a *lot* of administrivia overhead removed removed,
|
|
* so it's probably worth special-casing that case, too.
|
|
*/
|
|
if (dlen == 1)
|
|
return lbnDiv1_64(q, BIGLITTLE(n-1,n), n, nlen,
|
|
BIGLITTLE(d[-1],d[0]));
|
|
|
|
#if 0
|
|
/*
|
|
* @@@ This is not yet written... The general loop will do,
|
|
* albeit less efficiently
|
|
*/
|
|
if (dlen == 2) {
|
|
/*
|
|
* divisor two digits long:
|
|
* use the 3/2 technique from Knuth, but we know
|
|
* it's exact.
|
|
*/
|
|
dh = BIGLITTLE(d[-1],d[0]);
|
|
dl = BIGLITTLE(d[-2],d[1]);
|
|
shift = 0;
|
|
if ((sh & ((BNWORD64)1 << 64-1-shift)) == 0) {
|
|
do {
|
|
shift++;
|
|
} while (dh & (BNWORD64)1<<64-1-shift) == 0);
|
|
dh = dh << shift | dl >> (64-shift);
|
|
dl <<= shift;
|
|
|
|
|
|
}
|
|
|
|
|
|
for (shift = 0; (dh & (BNWORD64)1 << 64-1-shift)) == 0; shift++)
|
|
;
|
|
if (shift) {
|
|
}
|
|
dh = dh << shift | dl >> (64-shift);
|
|
shift = 0;
|
|
while (dh
|
|
}
|
|
#endif
|
|
|
|
dh = BIGLITTLE(*(d-dlen),*(d+(dlen-1)));
|
|
assert(dh);
|
|
|
|
/* Normalize the divisor */
|
|
shift = 0;
|
|
r = dh;
|
|
i = 64/2;
|
|
do {
|
|
if (r >> i)
|
|
r >>= i;
|
|
else
|
|
shift += i;
|
|
} while ((i /= 2) != 0);
|
|
|
|
nh = 0;
|
|
if (shift) {
|
|
lbnLshift_64(d, dlen, shift);
|
|
dh = BIGLITTLE(*(d-dlen),*(d+(dlen-1)));
|
|
nh = lbnLshift_64(n, nlen, shift);
|
|
}
|
|
|
|
/* Assert that dh is now normalized */
|
|
assert(dh >> (64-1));
|
|
|
|
/* Also get the second-most significant word of the divisor */
|
|
dl = BIGLITTLE(*(d-(dlen-1)),*(d+(dlen-2)));
|
|
|
|
/*
|
|
* Adjust pointers: n to point to least significant end of first
|
|
* first subtract, and q to one the most-significant end of the
|
|
* quotient array.
|
|
*/
|
|
BIGLITTLE(n -= qlen,n += qlen);
|
|
BIGLITTLE(q -= qlen,q += qlen);
|
|
|
|
/* Fetch the most significant stored word of the dividend */
|
|
nm = BIGLITTLE(*(n-dlen),*(n+(dlen-1)));
|
|
|
|
/*
|
|
* Compute the first digit of the quotient, based on the
|
|
* first two words of the dividend (the most significant of which
|
|
* is the overflow word h).
|
|
*/
|
|
if (nh) {
|
|
assert(nh < dh);
|
|
r = lbnDiv21_64(&qhat, nh, nm, dh);
|
|
} else if (nm >= dh) {
|
|
qhat = nm/dh;
|
|
r = nm % dh;
|
|
} else { /* Quotient is zero */
|
|
qhigh = 0;
|
|
goto divloop;
|
|
}
|
|
|
|
/* Now get the third most significant word of the dividend */
|
|
nl = BIGLITTLE(*(n-(dlen-1)),*(n+(dlen-2)));
|
|
|
|
/*
|
|
* Correct qhat, the estimate of quotient digit.
|
|
* qhat can only be high, and at most two words high,
|
|
* so the loop can be unrolled and abbreviated.
|
|
*/
|
|
#ifdef mul64_ppmm
|
|
mul64_ppmm(nm, t64, qhat, dl);
|
|
if (nm > r || (nm == r && t64 > nl)) {
|
|
/* Decrement qhat and adjust comparison parameters */
|
|
qhat--;
|
|
if ((r += dh) >= dh) {
|
|
nm -= (t64 < dl);
|
|
t64 -= dl;
|
|
if (nm > r || (nm == r && t64 > nl))
|
|
qhat--;
|
|
}
|
|
}
|
|
#elif defined(BNWORD128)
|
|
t128 = (BNWORD128)qhat * dl;
|
|
if (t128 > ((BNWORD128)r << 64) + nl) {
|
|
/* Decrement qhat and adjust comparison parameters */
|
|
qhat--;
|
|
if ((r += dh) > dh) {
|
|
t128 -= dl;
|
|
if (t128 > ((BNWORD128)r << 64) + nl)
|
|
qhat--;
|
|
}
|
|
}
|
|
#else /* Use lbnMulN1_64 */
|
|
lbnMulN1_64(BIGLITTLE(t2+2,t2), &dl, 1, qhat);
|
|
if (t2high > r || (t2high == r && t2low > nl)) {
|
|
/* Decrement qhat and adjust comparison parameters */
|
|
qhat--;
|
|
if ((r += dh) >= dh) {
|
|
t2high -= (t2low < dl);
|
|
t2low -= dl;
|
|
if (t2high > r || (t2high == r && t2low > nl))
|
|
qhat--;
|
|
}
|
|
}
|
|
#endif
|
|
|
|
/* Do the multiply and subtract */
|
|
r = lbnMulSub1_64(n, d, dlen, qhat);
|
|
/* If there was a borrow, add back once. */
|
|
if (r > nh) { /* Borrow? */
|
|
(void)lbnAddN_64(n, d, dlen);
|
|
qhat--;
|
|
}
|
|
|
|
/* Remember the first quotient digit. */
|
|
qhigh = qhat;
|
|
|
|
/* Now, the main division loop: */
|
|
divloop:
|
|
while (qlen--) {
|
|
|
|
/* Advance n */
|
|
nh = BIGLITTLE(*(n-dlen),*(n+(dlen-1)));
|
|
BIGLITTLE(++n,--n);
|
|
nm = BIGLITTLE(*(n-dlen),*(n+(dlen-1)));
|
|
|
|
if (nh == dh) {
|
|
qhat = ~(BNWORD64)0;
|
|
/* Optimized computation of r = (nh,nm) - qhat * dh */
|
|
r = nh + nm;
|
|
if (r < nh)
|
|
goto subtract;
|
|
} else {
|
|
assert(nh < dh);
|
|
r = lbnDiv21_64(&qhat, nh, nm, dh);
|
|
}
|
|
|
|
nl = BIGLITTLE(*(n-(dlen-1)),*(n+(dlen-2)));
|
|
#ifdef mul64_ppmm
|
|
mul64_ppmm(nm, t64, qhat, dl);
|
|
if (nm > r || (nm == r && t64 > nl)) {
|
|
/* Decrement qhat and adjust comparison parameters */
|
|
qhat--;
|
|
if ((r += dh) >= dh) {
|
|
nm -= (t64 < dl);
|
|
t64 -= dl;
|
|
if (nm > r || (nm == r && t64 > nl))
|
|
qhat--;
|
|
}
|
|
}
|
|
#elif defined(BNWORD128)
|
|
t128 = (BNWORD128)qhat * dl;
|
|
if (t128 > ((BNWORD128)r<<64) + nl) {
|
|
/* Decrement qhat and adjust comparison parameters */
|
|
qhat--;
|
|
if ((r += dh) >= dh) {
|
|
t128 -= dl;
|
|
if (t128 > ((BNWORD128)r << 64) + nl)
|
|
qhat--;
|
|
}
|
|
}
|
|
#else /* Use lbnMulN1_64 */
|
|
lbnMulN1_64(BIGLITTLE(t2+2,t2), &dl, 1, qhat);
|
|
if (t2high > r || (t2high == r && t2low > nl)) {
|
|
/* Decrement qhat and adjust comparison parameters */
|
|
qhat--;
|
|
if ((r += dh) >= dh) {
|
|
t2high -= (t2low < dl);
|
|
t2low -= dl;
|
|
if (t2high > r || (t2high == r && t2low > nl))
|
|
qhat--;
|
|
}
|
|
}
|
|
#endif
|
|
|
|
/*
|
|
* As a point of interest, note that it is not worth checking
|
|
* for qhat of 0 or 1 and installing special-case code. These
|
|
* occur with probability 2^-64, so spending 1 cycle to check
|
|
* for them is only worth it if we save more than 2^15 cycles,
|
|
* and a multiply-and-subtract for numbers in the 1024-bit
|
|
* range just doesn't take that long.
|
|
*/
|
|
subtract:
|
|
/*
|
|
* n points to the least significant end of the substring
|
|
* of n to be subtracted from. qhat is either exact or
|
|
* one too large. If the subtract gets a borrow, it was
|
|
* one too large and the divisor is added back in. It's
|
|
* a dlen+1 word add which is guaranteed to produce a
|
|
* carry out, so it can be done very simply.
|
|
*/
|
|
r = lbnMulSub1_64(n, d, dlen, qhat);
|
|
if (r > nh) { /* Borrow? */
|
|
(void)lbnAddN_64(n, d, dlen);
|
|
qhat--;
|
|
}
|
|
/* Store the quotient digit */
|
|
BIGLITTLE(*q++,*--q) = qhat;
|
|
}
|
|
/* Tah dah! */
|
|
|
|
if (shift) {
|
|
lbnRshift_64(d, dlen, shift);
|
|
lbnRshift_64(n, dlen, shift);
|
|
}
|
|
|
|
return qhigh;
|
|
}
|
|
#endif
|
|
|
|
/*
|
|
* Find the negative multiplicative inverse of x (x must be odd!) modulo 2^64.
|
|
*
|
|
* This just performs Newton's iteration until it gets the
|
|
* inverse. The initial estimate is always correct to 3 bits, and
|
|
* sometimes 4. The number of valid bits doubles each iteration.
|
|
* (To prove it, assume x * y == 1 (mod 2^n), and introduce a variable
|
|
* for the error mod 2^2n. x * y == 1 + k*2^n (mod 2^2n) and follow
|
|
* the iteration through.)
|
|
*/
|
|
#ifndef lbnMontInv1_64
|
|
BNWORD64
|
|
lbnMontInv1_64(BNWORD64 const x)
|
|
{
|
|
BNWORD64 y = x, z;
|
|
|
|
assert(x & 1);
|
|
|
|
while ((z = x*y) != 1)
|
|
y *= 2 - z;
|
|
return -y;
|
|
}
|
|
#endif /* !lbnMontInv1_64 */
|
|
|
|
#if defined(BNWORD128) && PRODUCT_SCAN
|
|
/*
|
|
* Test code for product-scanning Montgomery reduction.
|
|
* This seems to slow the C code down rather than speed it up.
|
|
*
|
|
* The first loop computes the Montgomery multipliers, storing them over
|
|
* the low half of the number n.
|
|
*
|
|
* The second half multiplies the upper half, adding in the modulus
|
|
* times the Montgomery multipliers. The results of this multiply
|
|
* are stored.
|
|
*/
|
|
void
|
|
lbnMontReduce_64(BNWORD64 *n, BNWORD64 const *mod, unsigned mlen, BNWORD64 inv)
|
|
{
|
|
BNWORD128 x, y;
|
|
BNWORD64 const *pm;
|
|
BNWORD64 *pn;
|
|
BNWORD64 t;
|
|
unsigned carry;
|
|
unsigned i, j;
|
|
|
|
/* Special case of zero */
|
|
if (!mlen)
|
|
return;
|
|
|
|
/* Pass 1 - compute Montgomery multipliers */
|
|
/* First iteration can have certain simplifications. */
|
|
t = BIGLITTLE(n[-1],n[0]);
|
|
x = t;
|
|
t *= inv;
|
|
BIGLITTLE(n[-1], n[0]) = t;
|
|
x += (BNWORD128)t * BIGLITTLE(mod[-1],mod[0]); /* Can't overflow */
|
|
assert((BNWORD64)x == 0);
|
|
x = x >> 64;
|
|
|
|
for (i = 1; i < mlen; i++) {
|
|
carry = 0;
|
|
pn = n;
|
|
pm = BIGLITTLE(mod-i-1,mod+i+1);
|
|
for (j = 0; j < i; j++) {
|
|
y = (BNWORD128)BIGLITTLE(*--pn * *pm++, *pn++ * *--pm);
|
|
x += y;
|
|
carry += (x < y);
|
|
}
|
|
assert(BIGLITTLE(pn == n-i, pn == n+i));
|
|
y = t = BIGLITTLE(pn[-1], pn[0]);
|
|
x += y;
|
|
carry += (x < y);
|
|
BIGLITTLE(pn[-1], pn[0]) = t = inv * (BNWORD64)x;
|
|
assert(BIGLITTLE(pm == mod-1, pm == mod+1));
|
|
y = (BNWORD128)t * BIGLITTLE(pm[0],pm[-1]);
|
|
x += y;
|
|
carry += (x < y);
|
|
assert((BNWORD64)x == 0);
|
|
x = x >> 64 | (BNWORD128)carry << 64;
|
|
}
|
|
|
|
BIGLITTLE(n -= mlen, n += mlen);
|
|
|
|
/* Pass 2 - compute upper words and add to n */
|
|
for (i = 1; i < mlen; i++) {
|
|
carry = 0;
|
|
pm = BIGLITTLE(mod-i,mod+i);
|
|
pn = n;
|
|
for (j = i; j < mlen; j++) {
|
|
y = (BNWORD128)BIGLITTLE(*--pm * *pn++, *pm++ * *--pn);
|
|
x += y;
|
|
carry += (x < y);
|
|
}
|
|
assert(BIGLITTLE(pm == mod-mlen, pm == mod+mlen));
|
|
assert(BIGLITTLE(pn == n+mlen-i, pn == n-mlen+i));
|
|
y = t = BIGLITTLE(*(n-i),*(n+i-1));
|
|
x += y;
|
|
carry += (x < y);
|
|
BIGLITTLE(*(n-i),*(n+i-1)) = (BNWORD64)x;
|
|
x = (x >> 64) | (BNWORD128)carry << 64;
|
|
}
|
|
|
|
/* Last round of second half, simplified. */
|
|
t = BIGLITTLE(*(n-mlen),*(n+mlen-1));
|
|
x += t;
|
|
BIGLITTLE(*(n-mlen),*(n+mlen-1)) = (BNWORD64)x;
|
|
carry = (unsigned)(x >> 64);
|
|
|
|
while (carry)
|
|
carry -= lbnSubN_64(n, mod, mlen);
|
|
while (lbnCmp_64(n, mod, mlen) >= 0)
|
|
(void)lbnSubN_64(n, mod, mlen);
|
|
}
|
|
#define lbnMontReduce_64 lbnMontReduce_64
|
|
#endif
|
|
|
|
/*
|
|
* Montgomery reduce n, modulo mod. This reduces modulo mod and divides by
|
|
* 2^(64*mlen). Returns the result in the *top* mlen words of the argument n.
|
|
* This is ready for another multiplication using lbnMul_64.
|
|
*
|
|
* Montgomery representation is a very useful way to encode numbers when
|
|
* you're doing lots of modular reduction. What you do is pick a multiplier
|
|
* R which is relatively prime to the modulus and very easy to divide by.
|
|
* Since the modulus is odd, R is closen as a power of 2, so the division
|
|
* is a shift. In fact, it's a shift of an integral number of words,
|
|
* so the shift can be implicit - just drop the low-order words.
|
|
*
|
|
* Now, choose R *larger* than the modulus m, 2^(64*mlen). Then convert
|
|
* all numbers a, b, etc. to Montgomery form M(a), M(b), etc using the
|
|
* relationship M(a) = a*R mod m, M(b) = b*R mod m, etc. Note that:
|
|
* - The Montgomery form of a number depends on the modulus m.
|
|
* A fixed modulus m is assumed throughout this discussion.
|
|
* - Since R is relaitvely prime to m, multiplication by R is invertible;
|
|
* no information about the numbers is lost, they're just scrambled.
|
|
* - Adding (and subtracting) numbers in this form works just as usual.
|
|
* M(a+b) = (a+b)*R mod m = (a*R + b*R) mod m = (M(a) + M(b)) mod m
|
|
* - Multiplying numbers in this form produces a*b*R*R. The problem
|
|
* is to divide out the excess factor of R, modulo m as well as to
|
|
* reduce to the given length mlen. It turns out that this can be
|
|
* done *faster* than a normal divide, which is where the speedup
|
|
* in Montgomery division comes from.
|
|
*
|
|
* Normal reduction chooses a most-significant quotient digit q and then
|
|
* subtracts q*m from the number to be reduced. Choosing q is tricky
|
|
* and involved (just look at lbnDiv_64 to see!) and is usually
|
|
* imperfect, requiring a check for correction after the subtraction.
|
|
*
|
|
* Montgomery reduction *adds* a multiple of m to the *low-order* part
|
|
* of the number to be reduced. This multiple is chosen to make the
|
|
* low-order part of the number come out to zero. This can be done
|
|
* with no trickery or error using a precomputed inverse of the modulus.
|
|
* In this code, the "part" is one word, but any width can be used.
|
|
*
|
|
* Repeating this step sufficiently often results in a value which
|
|
* is a multiple of R (a power of two, remember) but is still (since
|
|
* the additions were to the low-order part and thus did not increase
|
|
* the value of the number being reduced very much) still not much
|
|
* larger than m*R. Then implicitly divide by R and subtract off
|
|
* m until the result is in the correct range.
|
|
*
|
|
* Since the low-order part being cancelled is less than R, the
|
|
* multiple of m added must have a multiplier which is at most R-1.
|
|
* Assuming that the input is at most m*R-1, the final number is
|
|
* at most m*(2*R-1)-1 = 2*m*R - m - 1, so subtracting m once from
|
|
* the high-order part, equivalent to subtracting m*R from the
|
|
* while number, produces a result which is at most m*R - m - 1,
|
|
* which divided by R is at most m-1.
|
|
*
|
|
* To convert *to* Montgomery form, you need a regular remainder
|
|
* routine, although you can just compute R*R (mod m) and do the
|
|
* conversion using Montgomery multiplication. To convert *from*
|
|
* Montgomery form, just Montgomery reduce the number to
|
|
* remove the extra factor of R.
|
|
*
|
|
* TODO: Change to a full inverse and use Karatsuba's multiplication
|
|
* rather than this word-at-a-time.
|
|
*/
|
|
#ifndef lbnMontReduce_64
|
|
void
|
|
lbnMontReduce_64(BNWORD64 *n, BNWORD64 const *mod, unsigned const mlen,
|
|
BNWORD64 inv)
|
|
{
|
|
BNWORD64 t;
|
|
BNWORD64 c = 0;
|
|
unsigned len = mlen;
|
|
|
|
/* inv must be the negative inverse of mod's least significant word */
|
|
assert((BNWORD64)(inv * BIGLITTLE(mod[-1],mod[0])) == (BNWORD64)-1);
|
|
|
|
assert(len);
|
|
|
|
do {
|
|
t = lbnMulAdd1_64(n, mod, mlen, inv * BIGLITTLE(n[-1],n[0]));
|
|
c += lbnAdd1_64(BIGLITTLE(n-mlen,n+mlen), len, t);
|
|
BIGLITTLE(--n,++n);
|
|
} while (--len);
|
|
|
|
/*
|
|
* All that adding can cause an overflow past the modulus size,
|
|
* but it's unusual, and never by much, so a subtraction loop
|
|
* is the right way to deal with it.
|
|
* This subtraction happens infrequently - I've only ever seen it
|
|
* invoked once per reduction, and then just under 22.5% of the time.
|
|
*/
|
|
while (c)
|
|
c -= lbnSubN_64(n, mod, mlen);
|
|
while (lbnCmp_64(n, mod, mlen) >= 0)
|
|
(void)lbnSubN_64(n, mod, mlen);
|
|
}
|
|
#endif /* !lbnMontReduce_64 */
|
|
|
|
/*
|
|
* A couple of helpers that you might want to implement atomically
|
|
* in asm sometime.
|
|
*/
|
|
#ifndef lbnMontMul_64
|
|
/*
|
|
* Multiply "num1" by "num2", modulo "mod", all of length "len", and
|
|
* place the result in the high half of "prod". "inv" is the inverse
|
|
* of the least-significant word of the modulus, modulo 2^64.
|
|
* This uses numbers in Montgomery form. Reduce using "len" and "inv".
|
|
*
|
|
* This is implemented as a macro to win on compilers that don't do
|
|
* inlining, since it's so trivial.
|
|
*/
|
|
#define lbnMontMul_64(prod, n1, n2, mod, len, inv) \
|
|
(lbnMulX_64(prod, n1, n2, len), lbnMontReduce_64(prod, mod, len, inv))
|
|
#endif /* !lbnMontMul_64 */
|
|
|
|
#ifndef lbnMontSquare_64
|
|
/*
|
|
* Square "n", modulo "mod", both of length "len", and place the result
|
|
* in the high half of "prod". "inv" is the inverse of the least-significant
|
|
* word of the modulus, modulo 2^64.
|
|
* This uses numbers in Montgomery form. Reduce using "len" and "inv".
|
|
*
|
|
* This is implemented as a macro to win on compilers that don't do
|
|
* inlining, since it's so trivial.
|
|
*/
|
|
#define lbnMontSquare_64(prod, n, mod, len, inv) \
|
|
(lbnSquare_64(prod, n, len), lbnMontReduce_64(prod, mod, len, inv))
|
|
|
|
#endif /* !lbnMontSquare_64 */
|
|
|
|
/*
|
|
* Convert a number to Montgomery form - requires mlen + nlen words
|
|
* of memory in "n".
|
|
*/
|
|
void
|
|
lbnToMont_64(BNWORD64 *n, unsigned nlen, BNWORD64 *mod, unsigned mlen)
|
|
{
|
|
/* Move n up "mlen" words */
|
|
lbnCopy_64(BIGLITTLE(n-mlen,n+mlen), n, nlen);
|
|
lbnZero_64(n, mlen);
|
|
/* Do the division - dump the quotient in the high-order words */
|
|
(void)lbnDiv_64(BIGLITTLE(n-mlen,n+mlen), n, mlen+nlen, mod, mlen);
|
|
}
|
|
|
|
/*
|
|
* Convert from Montgomery form. Montgomery reduction is all that is
|
|
* needed.
|
|
*/
|
|
void
|
|
lbnFromMont_64(BNWORD64 *n, BNWORD64 *mod, unsigned len)
|
|
{
|
|
/* Zero the high words of n */
|
|
lbnZero_64(BIGLITTLE(n-len,n+len), len);
|
|
lbnMontReduce_64(n, mod, len, lbnMontInv1_64(mod[BIGLITTLE(-1,0)]));
|
|
/* Move n down len words */
|
|
lbnCopy_64(n, BIGLITTLE(n-len,n+len), len);
|
|
}
|
|
|
|
/*
|
|
* The windowed exponentiation algorithm, precomputes a table of odd
|
|
* powers of n up to 2^k. See the comment in bnExpMod_64 below for
|
|
* an explanation of how it actually works works.
|
|
*
|
|
* It takes 2^(k-1)-1 multiplies to compute the table, and (e-1)/(k+1)
|
|
* multiplies (on average) to perform the exponentiation. To minimize
|
|
* the sum, k must vary with e. The optimal window sizes vary with the
|
|
* exponent length. Here are some selected values and the boundary cases.
|
|
* (An underscore _ has been inserted into some of the numbers to ensure
|
|
* that magic strings like 64 do not appear in this table. It should be
|
|
* ignored.)
|
|
*
|
|
* At e = 1 bits, k=1 (0.000000) is best
|
|
* At e = 2 bits, k=1 (0.500000) is best
|
|
* At e = 4 bits, k=1 (1.500000) is best
|
|
* At e = 8 bits, k=2 (3.333333) < k=1 (3.500000)
|
|
* At e = 1_6 bits, k=2 (6.000000) is best
|
|
* At e = 26 bits, k=3 (9.250000) < k=2 (9.333333)
|
|
* At e = 3_2 bits, k=3 (10.750000) is best
|
|
* At e = 6_4 bits, k=3 (18.750000) is best
|
|
* At e = 82 bits, k=4 (23.200000) < k=3 (23.250000)
|
|
* At e = 128 bits, k=4 (3_2.400000) is best
|
|
* At e = 242 bits, k=5 (55.1_66667) < k=4 (55.200000)
|
|
* At e = 256 bits, k=5 (57.500000) is best
|
|
* At e = 512 bits, k=5 (100.1_66667) is best
|
|
* At e = 674 bits, k=6 (127.142857) < k=5 (127.1_66667)
|
|
* At e = 1024 bits, k=6 (177.142857) is best
|
|
* At e = 1794 bits, k=7 (287.125000) < k=6 (287.142857)
|
|
* At e = 2048 bits, k=7 (318.875000) is best
|
|
* At e = 4096 bits, k=7 (574.875000) is best
|
|
*
|
|
* The numbers in parentheses are the expected number of multiplications
|
|
* needed to do the computation. The normal russian-peasant modular
|
|
* exponentiation technique always uses (e-1)/2. For exponents as
|
|
* small as 192 bits (below the range of current factoring algorithms),
|
|
* half of the multiplies are eliminated, 45.2 as opposed to the naive
|
|
* 95.5. Counting the 191 squarings as 3/4 a multiply each (squaring
|
|
* proper is just over half of multiplying, but the Montgomery
|
|
* reduction in each case is also a multiply), that's 143.25
|
|
* multiplies, for totals of 188.45 vs. 238.75 - a 21% savings.
|
|
* For larger exponents (like 512 bits), it's 483.92 vs. 639.25, a
|
|
* 24.3% savings. It asymptotically approaches 25%.
|
|
*
|
|
* Um, actually there's a slightly more accurate way to count, which
|
|
* really is the average number of multiplies required, averaged
|
|
* uniformly over all 2^(e-1) e-bit numbers, from 2^(e-1) to (2^e)-1.
|
|
* It's based on the recurrence that for the last b bits, b <= k, at
|
|
* most one multiply is needed (and none at all 1/2^b of the time),
|
|
* while when b > k, the odds are 1/2 each way that the bit will be
|
|
* 0 (meaning no multiplies to reduce it to the b-1-bit case) and
|
|
* 1/2 that the bit will be 1, starting a k-bit window and requiring
|
|
* 1 multiply beyond the b-k-bit case. Since the most significant
|
|
* bit is always 1, a k-bit window always starts there, and that
|
|
* multiply is by 1, so it isn't a multiply at all. Thus, the
|
|
* number of multiplies is simply that needed for the last e-k bits.
|
|
* This recurrence produces:
|
|
*
|
|
* At e = 1 bits, k=1 (0.000000) is best
|
|
* At e = 2 bits, k=1 (0.500000) is best
|
|
* At e = 4 bits, k=1 (1.500000) is best
|
|
* At e = 6 bits, k=2 (2.437500) < k=1 (2.500000)
|
|
* At e = 8 bits, k=2 (3.109375) is best
|
|
* At e = 1_6 bits, k=2 (5.777771) is best
|
|
* At e = 24 bits, k=3 (8.437629) < k=2 (8.444444)
|
|
* At e = 3_2 bits, k=3 (10.437492) is best
|
|
* At e = 6_4 bits, k=3 (18.437500) is best
|
|
* At e = 81 bits, k=4 (22.6_40000) < k=3 (22.687500)
|
|
* At e = 128 bits, k=4 (3_2.040000) is best
|
|
* At e = 241 bits, k=5 (54.611111) < k=4 (54.6_40000)
|
|
* At e = 256 bits, k=5 (57.111111) is best
|
|
* At e = 512 bits, k=5 (99.777778) is best
|
|
* At e = 673 bits, k=6 (126.591837) < k=5 (126.611111)
|
|
* At e = 1024 bits, k=6 (176.734694) is best
|
|
* At e = 1793 bits, k=7 (286.578125) < k=6 (286.591837)
|
|
* At e = 2048 bits, k=7 (318.453125) is best
|
|
* At e = 4096 bits, k=7 (574.453125) is best
|
|
*
|
|
* This has the rollover points at 6, 24, 81, 241, 673 and 1793 instead
|
|
* of 8, 26, 82, 242, 674, and 1794. Not a very big difference.
|
|
* (The numbers past that are k=8 at 4609 and k=9 at 11521,
|
|
* vs. one more in each case for the approximation.)
|
|
*
|
|
* Given that exponents for which k>7 are useful are uncommon,
|
|
* a fixed size table for k <= 7 is used for simplicity.
|
|
*
|
|
* The basic number of squarings needed is e-1, although a k-bit
|
|
* window (for k > 1) can save, on average, k-2 of those, too.
|
|
* That savings currently isn't counted here. It would drive the
|
|
* crossover points slightly lower.
|
|
* (Actually, this win is also reduced in the DoubleExpMod case,
|
|
* meaning we'd have to split the tables. Except for that, the
|
|
* multiplies by powers of the two bases are independent, so
|
|
* the same logic applies to each as the single case.)
|
|
*
|
|
* Table entry i is the largest number of bits in an exponent to
|
|
* process with a window size of i+1. Entry 6 is the largest
|
|
* possible unsigned number, so the window will never be more
|
|
* than 7 bits, requiring 2^6 = 0x40 slots.
|
|
*/
|
|
#define BNEXPMOD_MAX_WINDOW 7
|
|
static unsigned const bnExpModThreshTable[BNEXPMOD_MAX_WINDOW] = {
|
|
5, 23, 80, 240, 672, 1792, (unsigned)-1
|
|
/* 7, 25, 81, 241, 673, 1793, (unsigned)-1 ### The old approximations */
|
|
};
|
|
|
|
/*
|
|
* Perform modular exponentiation, as fast as possible! This uses
|
|
* Montgomery reduction, optimized squaring, and windowed exponentiation.
|
|
* The modulus "mod" MUST be odd!
|
|
*
|
|
* This returns 0 on success, -1 on out of memory.
|
|
*
|
|
* The window algorithm:
|
|
* The idea is to keep a running product of b1 = n^(high-order bits of exp),
|
|
* and then keep appending exponent bits to it. The following patterns
|
|
* apply to a 3-bit window (k = 3):
|
|
* To append 0: square
|
|
* To append 1: square, multiply by n^1
|
|
* To append 10: square, multiply by n^1, square
|
|
* To append 11: square, square, multiply by n^3
|
|
* To append 100: square, multiply by n^1, square, square
|
|
* To append 101: square, square, square, multiply by n^5
|
|
* To append 110: square, square, multiply by n^3, square
|
|
* To append 111: square, square, square, multiply by n^7
|
|
*
|
|
* Since each pattern involves only one multiply, the longer the pattern
|
|
* the better, except that a 0 (no multiplies) can be appended directly.
|
|
* We precompute a table of odd powers of n, up to 2^k, and can then
|
|
* multiply k bits of exponent at a time. Actually, assuming random
|
|
* exponents, there is on average one zero bit between needs to
|
|
* multiply (1/2 of the time there's none, 1/4 of the time there's 1,
|
|
* 1/8 of the time, there's 2, 1/64 of the time, there's 3, etc.), so
|
|
* you have to do one multiply per k+1 bits of exponent.
|
|
*
|
|
* The loop walks down the exponent, squaring the result buffer as
|
|
* it goes. There is a wbits+1 bit lookahead buffer, buf, that is
|
|
* filled with the upcoming exponent bits. (What is read after the
|
|
* end of the exponent is unimportant, but it is filled with zero here.)
|
|
* When the most-significant bit of this buffer becomes set, i.e.
|
|
* (buf & tblmask) != 0, we have to decide what pattern to multiply
|
|
* by, and when to do it. We decide, remember to do it in future
|
|
* after a suitable number of squarings have passed (e.g. a pattern
|
|
* of "100" in the buffer requires that we multiply by n^1 immediately;
|
|
* a pattern of "110" calls for multiplying by n^3 after one more
|
|
* squaring), clear the buffer, and continue.
|
|
*
|
|
* When we start, there is one more optimization: the result buffer
|
|
* is implcitly one, so squaring it or multiplying by it can be
|
|
* optimized away. Further, if we start with a pattern like "100"
|
|
* in the lookahead window, rather than placing n into the buffer
|
|
* and then starting to square it, we have already computed n^2
|
|
* to compute the odd-powers table, so we can place that into
|
|
* the buffer and save a squaring.
|
|
*
|
|
* This means that if you have a k-bit window, to compute n^z,
|
|
* where z is the high k bits of the exponent, 1/2 of the time
|
|
* it requires no squarings. 1/4 of the time, it requires 1
|
|
* squaring, ... 1/2^(k-1) of the time, it reqires k-2 squarings.
|
|
* And the remaining 1/2^(k-1) of the time, the top k bits are a
|
|
* 1 followed by k-1 0 bits, so it again only requires k-2
|
|
* squarings, not k-1. The average of these is 1. Add that
|
|
* to the one squaring we have to do to compute the table,
|
|
* and you'll see that a k-bit window saves k-2 squarings
|
|
* as well as reducing the multiplies. (It actually doesn't
|
|
* hurt in the case k = 1, either.)
|
|
*
|
|
* n must have mlen words allocated. Although fewer may be in use
|
|
* when n is passed in, all are in use on exit.
|
|
*/
|
|
int
|
|
lbnExpMod_64(BNWORD64 *result, BNWORD64 const *n, unsigned nlen,
|
|
BNWORD64 const *e, unsigned elen, BNWORD64 *mod, unsigned mlen)
|
|
{
|
|
BNWORD64 *table[1 << (BNEXPMOD_MAX_WINDOW-1)];
|
|
/* Table of odd powers of n */
|
|
unsigned ebits; /* Exponent bits */
|
|
unsigned wbits; /* Window size */
|
|
unsigned tblmask; /* Mask of exponentiation window */
|
|
BNWORD64 bitpos; /* Mask of current look-ahead bit */
|
|
unsigned buf; /* Buffer of exponent bits */
|
|
unsigned multpos; /* Where to do pending multiply */
|
|
BNWORD64 const *mult; /* What to multiply by */
|
|
unsigned i; /* Loop counter */
|
|
int isone; /* Flag: accum. is implicitly one */
|
|
BNWORD64 *a, *b; /* Working buffers/accumulators */
|
|
BNWORD64 *t; /* Pointer into the working buffers */
|
|
BNWORD64 inv; /* mod^-1 modulo 2^64 */
|
|
int y; /* bnYield() result */
|
|
|
|
assert(mlen);
|
|
assert(nlen <= mlen);
|
|
|
|
/* First, a couple of trivial cases. */
|
|
elen = lbnNorm_64(e, elen);
|
|
if (!elen) {
|
|
/* x ^ 0 == 1 */
|
|
lbnZero_64(result, mlen);
|
|
BIGLITTLE(result[-1],result[0]) = 1;
|
|
return 0;
|
|
}
|
|
ebits = lbnBits_64(e, elen);
|
|
if (ebits == 1) {
|
|
/* x ^ 1 == x */
|
|
if (n != result)
|
|
lbnCopy_64(result, n, nlen);
|
|
if (mlen > nlen)
|
|
lbnZero_64(BIGLITTLE(result-nlen,result+nlen),
|
|
mlen-nlen);
|
|
return 0;
|
|
}
|
|
|
|
/* Okay, now move the exponent pointer to the most-significant word */
|
|
e = BIGLITTLE(e-elen, e+elen-1);
|
|
|
|
/* Look up appropriate k-1 for the exponent - tblmask = 1<<(k-1) */
|
|
wbits = 0;
|
|
while (ebits > bnExpModThreshTable[wbits])
|
|
wbits++;
|
|
|
|
/* Allocate working storage: two product buffers and the tables. */
|
|
LBNALLOC(a, BNWORD64, 2*mlen);
|
|
if (!a)
|
|
return -1;
|
|
LBNALLOC(b, BNWORD64, 2*mlen);
|
|
if (!b) {
|
|
LBNFREE(a, 2*mlen);
|
|
return -1;
|
|
}
|
|
|
|
/* Convert to the appropriate table size: tblmask = 1<<(k-1) */
|
|
tblmask = 1u << wbits;
|
|
|
|
/* We have the result buffer available, so use it. */
|
|
table[0] = result;
|
|
|
|
/*
|
|
* Okay, we now have a minimal-sized table - expand it.
|
|
* This is allowed to fail! If so, scale back the table size
|
|
* and proceed.
|
|
*/
|
|
for (i = 1; i < tblmask; i++) {
|
|
LBNALLOC(t, BNWORD64, mlen);
|
|
if (!t) /* Out of memory! Quit the loop. */
|
|
break;
|
|
table[i] = t;
|
|
}
|
|
|
|
/* If we stopped, with i < tblmask, shrink the tables appropriately */
|
|
while (tblmask > i) {
|
|
wbits--;
|
|
tblmask >>= 1;
|
|
}
|
|
/* Free up our overallocations */
|
|
while (--i > tblmask)
|
|
LBNFREE(table[i], mlen);
|
|
|
|
/* Okay, fill in the table */
|
|
|
|
/* Compute the necessary modular inverse */
|
|
inv = lbnMontInv1_64(mod[BIGLITTLE(-1,0)]); /* LSW of modulus */
|
|
|
|
/* Convert n to Montgomery form */
|
|
|
|
/* Move n up "mlen" words into a */
|
|
t = BIGLITTLE(a-mlen, a+mlen);
|
|
lbnCopy_64(t, n, nlen);
|
|
lbnZero_64(a, mlen);
|
|
/* Do the division - lose the quotient into the high-order words */
|
|
(void)lbnDiv_64(t, a, mlen+nlen, mod, mlen);
|
|
/* Copy into first table entry */
|
|
lbnCopy_64(table[0], a, mlen);
|
|
|
|
/* Square a into b */
|
|
lbnMontSquare_64(b, a, mod, mlen, inv);
|
|
|
|
/* Use high half of b to initialize the table */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
for (i = 1; i < tblmask; i++) {
|
|
lbnMontMul_64(a, t, table[i-1], mod, mlen, inv);
|
|
lbnCopy_64(table[i], BIGLITTLE(a-mlen, a+mlen), mlen);
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
goto yield;
|
|
#endif
|
|
}
|
|
|
|
/* We might use b = n^2 later... */
|
|
|
|
/* Initialze the fetch pointer */
|
|
bitpos = (BNWORD64)1 << ((ebits-1) & (64-1)); /* Initialize mask */
|
|
|
|
/* This should point to the msbit of e */
|
|
assert((*e & bitpos) != 0);
|
|
|
|
/*
|
|
* Pre-load the window. Becuase the window size is
|
|
* never larger than the exponent size, there is no need to
|
|
* detect running off the end of e in here.
|
|
*
|
|
* The read-ahead is controlled by elen and the bitpos mask.
|
|
* Note that this is *ahead* of ebits, which tracks the
|
|
* most significant end of the window. The purpose of this
|
|
* initialization is to get the two wbits+1 bits apart,
|
|
* like they should be.
|
|
*
|
|
* Note that bitpos and e1len together keep track of the
|
|
* lookahead read pointer in the exponent that is used here.
|
|
*/
|
|
buf = 0;
|
|
for (i = 0; i <= wbits; i++) {
|
|
buf = (buf << 1) | ((*e & bitpos) != 0);
|
|
bitpos >>= 1;
|
|
if (!bitpos) {
|
|
BIGLITTLE(e++,e--);
|
|
bitpos = (BNWORD64)1 << (64-1);
|
|
elen--;
|
|
}
|
|
}
|
|
assert(buf & tblmask);
|
|
|
|
/*
|
|
* Set the pending multiply positions to a location that will
|
|
* never be encountered, thus ensuring that nothing will happen
|
|
* until the need for a multiply appears and one is scheduled.
|
|
*/
|
|
multpos = ebits; /* A NULL value */
|
|
mult = 0; /* Force a crash if we use these */
|
|
|
|
/*
|
|
* Okay, now begins the real work. The first step is
|
|
* slightly magic, so it's done outside the main loop,
|
|
* but it's very similar to what's inside.
|
|
*/
|
|
ebits--; /* Start processing the first bit... */
|
|
isone = 1;
|
|
|
|
/*
|
|
* This is just like the multiply in the loop, except that
|
|
* - We know the msbit of buf is set, and
|
|
* - We have the extra value n^2 floating around.
|
|
* So, do the usual computation, and if the result is that
|
|
* the buffer should be multiplied by n^1 immediately
|
|
* (which we'd normally then square), we multiply it
|
|
* (which reduces to a copy, which reduces to setting a flag)
|
|
* by n^2 and skip the squaring. Thus, we do the
|
|
* multiply and the squaring in one step.
|
|
*/
|
|
assert(buf & tblmask);
|
|
multpos = ebits - wbits;
|
|
while ((buf & 1) == 0) {
|
|
buf >>= 1;
|
|
multpos++;
|
|
}
|
|
/* Intermediates can wrap, but final must NOT */
|
|
assert(multpos <= ebits);
|
|
mult = table[buf>>1];
|
|
buf = 0;
|
|
|
|
/* Special case: use already-computed value sitting in buffer */
|
|
if (multpos == ebits)
|
|
isone = 0;
|
|
|
|
/*
|
|
* At this point, the buffer (which is the high half of b) holds
|
|
* either 1 (implicitly, as the "isone" flag is set), or n^2.
|
|
*/
|
|
|
|
/*
|
|
* The main loop. The procedure is:
|
|
* - Advance the window
|
|
* - If the most-significant bit of the window is set,
|
|
* schedule a multiply for the appropriate time in the
|
|
* future (may be immediately)
|
|
* - Perform any pending multiples
|
|
* - Check for termination
|
|
* - Square the buffer
|
|
*
|
|
* At any given time, the acumulated product is held in
|
|
* the high half of b.
|
|
*/
|
|
for (;;) {
|
|
ebits--;
|
|
|
|
/* Advance the window */
|
|
assert(buf < tblmask);
|
|
buf <<= 1;
|
|
/*
|
|
* This reads ahead of the current exponent position
|
|
* (controlled by ebits), so we have to be able to read
|
|
* past the lsb of the exponents without error.
|
|
*/
|
|
if (elen) {
|
|
buf |= ((*e & bitpos) != 0);
|
|
bitpos >>= 1;
|
|
if (!bitpos) {
|
|
BIGLITTLE(e++,e--);
|
|
bitpos = (BNWORD64)1 << (64-1);
|
|
elen--;
|
|
}
|
|
}
|
|
|
|
/* Examine the window for pending multiplies */
|
|
if (buf & tblmask) {
|
|
multpos = ebits - wbits;
|
|
while ((buf & 1) == 0) {
|
|
buf >>= 1;
|
|
multpos++;
|
|
}
|
|
/* Intermediates can wrap, but final must NOT */
|
|
assert(multpos <= ebits);
|
|
mult = table[buf>>1];
|
|
buf = 0;
|
|
}
|
|
|
|
/* If we have a pending multiply, do it */
|
|
if (ebits == multpos) {
|
|
/* Multiply by the table entry remembered previously */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
if (isone) {
|
|
/* Multiply by 1 is a trivial case */
|
|
lbnCopy_64(t, mult, mlen);
|
|
isone = 0;
|
|
} else {
|
|
lbnMontMul_64(a, t, mult, mod, mlen, inv);
|
|
/* Swap a and b */
|
|
t = a; a = b; b = t;
|
|
}
|
|
}
|
|
|
|
/* Are we done? */
|
|
if (!ebits)
|
|
break;
|
|
|
|
/* Square the input */
|
|
if (!isone) {
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
lbnMontSquare_64(a, t, mod, mlen, inv);
|
|
/* Swap a and b */
|
|
t = a; a = b; b = t;
|
|
}
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
goto yield;
|
|
#endif
|
|
} /* for (;;) */
|
|
|
|
assert(!isone);
|
|
assert(!buf);
|
|
|
|
/* DONE! */
|
|
|
|
/* Convert result out of Montgomery form */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
lbnCopy_64(b, t, mlen);
|
|
lbnZero_64(t, mlen);
|
|
lbnMontReduce_64(b, mod, mlen, inv);
|
|
lbnCopy_64(result, t, mlen);
|
|
/*
|
|
* Clean up - free intermediate storage.
|
|
* Do NOT free table[0], which is the result
|
|
* buffer.
|
|
*/
|
|
y = 0;
|
|
#if BNYIELD
|
|
yield:
|
|
#endif
|
|
while (--tblmask)
|
|
LBNFREE(table[tblmask], mlen);
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
|
|
return y; /* Success */
|
|
}
|
|
|
|
/*
|
|
* Compute and return n1^e1 * n2^e2 mod "mod".
|
|
* result may be either input buffer, or something separate.
|
|
* It must be "mlen" words long.
|
|
*
|
|
* There is a current position in the exponents, which is kept in e1bits.
|
|
* (The exponents are swapped if necessary so e1 is the longer of the two.)
|
|
* At any given time, the value in the accumulator is
|
|
* n1^(e1>>e1bits) * n2^(e2>>e1bits) mod "mod".
|
|
* As e1bits is counted down, this is updated, by squaring it and doing
|
|
* any necessary multiplies.
|
|
* To decide on the necessary multiplies, two windows, each w1bits+1 bits
|
|
* wide, are maintained in buf1 and buf2, which read *ahead* of the
|
|
* e1bits position (with appropriate handling of the case when e1bits
|
|
* drops below w1bits+1). When the most-significant bit of either window
|
|
* becomes set, indicating that something needs to be multiplied by
|
|
* the accumulator or it will get out of sync, the window is examined
|
|
* to see which power of n1 or n2 to multiply by, and when (possibly
|
|
* later, if the power is greater than 1) the multiply should take
|
|
* place. Then the multiply and its location are remembered and the
|
|
* window is cleared.
|
|
*
|
|
* If we had every power of n1 in the table, the multiply would always
|
|
* be w1bits steps in the future. But we only keep the odd powers,
|
|
* so instead of waiting w1bits squarings and then multiplying
|
|
* by n1^k, we wait w1bits-k squarings and multiply by n1.
|
|
*
|
|
* Actually, w2bits can be less than w1bits, but the window is the same
|
|
* size, to make it easier to keep track of where we're reading. The
|
|
* appropriate number of low-order bits of the window are just ignored.
|
|
*/
|
|
int
|
|
lbnDoubleExpMod_64(BNWORD64 *result,
|
|
BNWORD64 const *n1, unsigned n1len,
|
|
BNWORD64 const *e1, unsigned e1len,
|
|
BNWORD64 const *n2, unsigned n2len,
|
|
BNWORD64 const *e2, unsigned e2len,
|
|
BNWORD64 *mod, unsigned mlen)
|
|
{
|
|
BNWORD64 *table1[1 << (BNEXPMOD_MAX_WINDOW-1)];
|
|
/* Table of odd powers of n1 */
|
|
BNWORD64 *table2[1 << (BNEXPMOD_MAX_WINDOW-1)];
|
|
/* Table of odd powers of n2 */
|
|
unsigned e1bits, e2bits; /* Exponent bits */
|
|
unsigned w1bits, w2bits; /* Window sizes */
|
|
unsigned tblmask; /* Mask of exponentiation window */
|
|
BNWORD64 bitpos; /* Mask of current look-ahead bit */
|
|
unsigned buf1, buf2; /* Buffer of exponent bits */
|
|
unsigned mult1pos, mult2pos; /* Where to do pending multiply */
|
|
BNWORD64 const *mult1, *mult2; /* What to multiply by */
|
|
unsigned i; /* Loop counter */
|
|
int isone; /* Flag: accum. is implicitly one */
|
|
BNWORD64 *a, *b; /* Working buffers/accumulators */
|
|
BNWORD64 *t; /* Pointer into the working buffers */
|
|
BNWORD64 inv; /* mod^-1 modulo 2^64 */
|
|
int y; /* bnYield() result */
|
|
|
|
assert(mlen);
|
|
assert(n1len <= mlen);
|
|
assert(n2len <= mlen);
|
|
|
|
/* First, a couple of trivial cases. */
|
|
e1len = lbnNorm_64(e1, e1len);
|
|
e2len = lbnNorm_64(e2, e2len);
|
|
|
|
/* Ensure that the first exponent is the longer */
|
|
e1bits = lbnBits_64(e1, e1len);
|
|
e2bits = lbnBits_64(e2, e2len);
|
|
if (e1bits < e2bits) {
|
|
i = e1len; e1len = e2len; e2len = i;
|
|
i = e1bits; e1bits = e2bits; e2bits = i;
|
|
t = (BNWORD64 *)n1; n1 = n2; n2 = t;
|
|
t = (BNWORD64 *)e1; e1 = e2; e2 = t;
|
|
}
|
|
assert(e1bits >= e2bits);
|
|
|
|
/* Handle a trivial case */
|
|
if (!e2len)
|
|
return lbnExpMod_64(result, n1, n1len, e1, e1len, mod, mlen);
|
|
assert(e2bits);
|
|
|
|
/* The code below fucks up if the exponents aren't at least 2 bits */
|
|
if (e1bits == 1) {
|
|
assert(e2bits == 1);
|
|
|
|
LBNALLOC(a, BNWORD64, n1len+n2len);
|
|
if (!a)
|
|
return -1;
|
|
|
|
lbnMul_64(a, n1, n1len, n2, n2len);
|
|
/* Do a direct modular reduction */
|
|
if (n1len + n2len >= mlen)
|
|
(void)lbnDiv_64(a+mlen, a, n1len+n2len, mod, mlen);
|
|
lbnCopy_64(result, a, mlen);
|
|
LBNFREE(a, n1len+n2len);
|
|
return 0;
|
|
}
|
|
|
|
/* Okay, now move the exponent pointers to the most-significant word */
|
|
e1 = BIGLITTLE(e1-e1len, e1+e1len-1);
|
|
e2 = BIGLITTLE(e2-e2len, e2+e2len-1);
|
|
|
|
/* Look up appropriate k-1 for the exponent - tblmask = 1<<(k-1) */
|
|
w1bits = 0;
|
|
while (e1bits > bnExpModThreshTable[w1bits])
|
|
w1bits++;
|
|
w2bits = 0;
|
|
while (e2bits > bnExpModThreshTable[w2bits])
|
|
w2bits++;
|
|
|
|
assert(w1bits >= w2bits);
|
|
|
|
/* Allocate working storage: two product buffers and the tables. */
|
|
LBNALLOC(a, BNWORD64, 2*mlen);
|
|
if (!a)
|
|
return -1;
|
|
LBNALLOC(b, BNWORD64, 2*mlen);
|
|
if (!b) {
|
|
LBNFREE(a, 2*mlen);
|
|
return -1;
|
|
}
|
|
|
|
/* Convert to the appropriate table size: tblmask = 1<<(k-1) */
|
|
tblmask = 1u << w1bits;
|
|
/* Use buf2 for its size, temporarily */
|
|
buf2 = 1u << w2bits;
|
|
|
|
LBNALLOC(t, BNWORD64, mlen);
|
|
if (!t) {
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
return -1;
|
|
}
|
|
table1[0] = t;
|
|
table2[0] = result;
|
|
|
|
/*
|
|
* Okay, we now have some minimal-sized tables - expand them.
|
|
* This is allowed to fail! If so, scale back the table sizes
|
|
* and proceed. We allocate both tables at the same time
|
|
* so if it fails partway through, they'll both be a reasonable
|
|
* size rather than one huge and one tiny.
|
|
* When i passes buf2 (the number of entries in the e2 window,
|
|
* which may be less than the number of entries in the e1 window),
|
|
* stop allocating e2 space.
|
|
*/
|
|
for (i = 1; i < tblmask; i++) {
|
|
LBNALLOC(t, BNWORD64, mlen);
|
|
if (!t) /* Out of memory! Quit the loop. */
|
|
break;
|
|
table1[i] = t;
|
|
if (i < buf2) {
|
|
LBNALLOC(t, BNWORD64, mlen);
|
|
if (!t) {
|
|
LBNFREE(table1[i], mlen);
|
|
break;
|
|
}
|
|
table2[i] = t;
|
|
}
|
|
}
|
|
|
|
/* If we stopped, with i < tblmask, shrink the tables appropriately */
|
|
while (tblmask > i) {
|
|
w1bits--;
|
|
tblmask >>= 1;
|
|
}
|
|
/* Free up our overallocations */
|
|
while (--i > tblmask) {
|
|
if (i < buf2)
|
|
LBNFREE(table2[i], mlen);
|
|
LBNFREE(table1[i], mlen);
|
|
}
|
|
/* And shrink the second window too, if needed */
|
|
if (w2bits > w1bits) {
|
|
w2bits = w1bits;
|
|
buf2 = tblmask;
|
|
}
|
|
|
|
/*
|
|
* From now on, use the w2bits variable for the difference
|
|
* between w1bits and w2bits.
|
|
*/
|
|
w2bits = w1bits-w2bits;
|
|
|
|
/* Okay, fill in the tables */
|
|
|
|
/* Compute the necessary modular inverse */
|
|
inv = lbnMontInv1_64(mod[BIGLITTLE(-1,0)]); /* LSW of modulus */
|
|
|
|
/* Convert n1 to Montgomery form */
|
|
|
|
/* Move n1 up "mlen" words into a */
|
|
t = BIGLITTLE(a-mlen, a+mlen);
|
|
lbnCopy_64(t, n1, n1len);
|
|
lbnZero_64(a, mlen);
|
|
/* Do the division - lose the quotient into the high-order words */
|
|
(void)lbnDiv_64(t, a, mlen+n1len, mod, mlen);
|
|
/* Copy into first table entry */
|
|
lbnCopy_64(table1[0], a, mlen);
|
|
|
|
/* Square a into b */
|
|
lbnMontSquare_64(b, a, mod, mlen, inv);
|
|
|
|
/* Use high half of b to initialize the first table */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
for (i = 1; i < tblmask; i++) {
|
|
lbnMontMul_64(a, t, table1[i-1], mod, mlen, inv);
|
|
lbnCopy_64(table1[i], BIGLITTLE(a-mlen, a+mlen), mlen);
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
goto yield;
|
|
#endif
|
|
}
|
|
|
|
/* Convert n2 to Montgomery form */
|
|
|
|
t = BIGLITTLE(a-mlen, a+mlen);
|
|
/* Move n2 up "mlen" words into a */
|
|
lbnCopy_64(t, n2, n2len);
|
|
lbnZero_64(a, mlen);
|
|
/* Do the division - lose the quotient into the high-order words */
|
|
(void)lbnDiv_64(t, a, mlen+n2len, mod, mlen);
|
|
/* Copy into first table entry */
|
|
lbnCopy_64(table2[0], a, mlen);
|
|
|
|
/* Square it into a */
|
|
lbnMontSquare_64(a, table2[0], mod, mlen, inv);
|
|
/* Copy to b, low half */
|
|
lbnCopy_64(b, t, mlen);
|
|
|
|
/* Use b to initialize the second table */
|
|
for (i = 1; i < buf2; i++) {
|
|
lbnMontMul_64(a, b, table2[i-1], mod, mlen, inv);
|
|
lbnCopy_64(table2[i], t, mlen);
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
goto yield;
|
|
#endif
|
|
}
|
|
|
|
/*
|
|
* Okay, a recap: at this point, the low part of b holds
|
|
* n2^2, the high part holds n1^2, and the tables are
|
|
* initialized with the odd powers of n1 and n2 from 1
|
|
* through 2*tblmask-1 and 2*buf2-1.
|
|
*
|
|
* We might use those squares in b later, or we might not.
|
|
*/
|
|
|
|
/* Initialze the fetch pointer */
|
|
bitpos = (BNWORD64)1 << ((e1bits-1) & (64-1)); /* Initialize mask */
|
|
|
|
/* This should point to the msbit of e1 */
|
|
assert((*e1 & bitpos) != 0);
|
|
|
|
/*
|
|
* Pre-load the windows. Becuase the window size is
|
|
* never larger than the exponent size, there is no need to
|
|
* detect running off the end of e1 in here.
|
|
*
|
|
* The read-ahead is controlled by e1len and the bitpos mask.
|
|
* Note that this is *ahead* of e1bits, which tracks the
|
|
* most significant end of the window. The purpose of this
|
|
* initialization is to get the two w1bits+1 bits apart,
|
|
* like they should be.
|
|
*
|
|
* Note that bitpos and e1len together keep track of the
|
|
* lookahead read pointer in the exponent that is used here.
|
|
* e2len is not decremented, it is only ever compared with
|
|
* e1len as *that* is decremented.
|
|
*/
|
|
buf1 = buf2 = 0;
|
|
for (i = 0; i <= w1bits; i++) {
|
|
buf1 = (buf1 << 1) | ((*e1 & bitpos) != 0);
|
|
if (e1len <= e2len)
|
|
buf2 = (buf2 << 1) | ((*e2 & bitpos) != 0);
|
|
bitpos >>= 1;
|
|
if (!bitpos) {
|
|
BIGLITTLE(e1++,e1--);
|
|
if (e1len <= e2len)
|
|
BIGLITTLE(e2++,e2--);
|
|
bitpos = (BNWORD64)1 << (64-1);
|
|
e1len--;
|
|
}
|
|
}
|
|
assert(buf1 & tblmask);
|
|
|
|
/*
|
|
* Set the pending multiply positions to a location that will
|
|
* never be encountered, thus ensuring that nothing will happen
|
|
* until the need for a multiply appears and one is scheduled.
|
|
*/
|
|
mult1pos = mult2pos = e1bits; /* A NULL value */
|
|
mult1 = mult2 = 0; /* Force a crash if we use these */
|
|
|
|
/*
|
|
* Okay, now begins the real work. The first step is
|
|
* slightly magic, so it's done outside the main loop,
|
|
* but it's very similar to what's inside.
|
|
*/
|
|
isone = 1; /* Buffer is implicitly 1, so replace * by copy */
|
|
e1bits--; /* Start processing the first bit... */
|
|
|
|
/*
|
|
* This is just like the multiply in the loop, except that
|
|
* - We know the msbit of buf1 is set, and
|
|
* - We have the extra value n1^2 floating around.
|
|
* So, do the usual computation, and if the result is that
|
|
* the buffer should be multiplied by n1^1 immediately
|
|
* (which we'd normally then square), we multiply it
|
|
* (which reduces to a copy, which reduces to setting a flag)
|
|
* by n1^2 and skip the squaring. Thus, we do the
|
|
* multiply and the squaring in one step.
|
|
*/
|
|
assert(buf1 & tblmask);
|
|
mult1pos = e1bits - w1bits;
|
|
while ((buf1 & 1) == 0) {
|
|
buf1 >>= 1;
|
|
mult1pos++;
|
|
}
|
|
/* Intermediates can wrap, but final must NOT */
|
|
assert(mult1pos <= e1bits);
|
|
mult1 = table1[buf1>>1];
|
|
buf1 = 0;
|
|
|
|
/* Special case: use already-computed value sitting in buffer */
|
|
if (mult1pos == e1bits)
|
|
isone = 0;
|
|
|
|
/*
|
|
* The first multiply by a power of n2. Similar, but
|
|
* we might not even want to schedule a multiply if e2 is
|
|
* shorter than e1, and the window might be shorter so
|
|
* we have to leave the low w2bits bits alone.
|
|
*/
|
|
if (buf2 & tblmask) {
|
|
/* Remember low-order bits for later */
|
|
i = buf2 & ((1u << w2bits) - 1);
|
|
buf2 >>= w2bits;
|
|
mult2pos = e1bits - w1bits + w2bits;
|
|
while ((buf2 & 1) == 0) {
|
|
buf2 >>= 1;
|
|
mult2pos++;
|
|
}
|
|
assert(mult2pos <= e1bits);
|
|
mult2 = table2[buf2>>1];
|
|
buf2 = i;
|
|
|
|
if (mult2pos == e1bits) {
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
if (isone) {
|
|
lbnCopy_64(t, b, mlen); /* Copy low to high */
|
|
isone = 0;
|
|
} else {
|
|
lbnMontMul_64(a, t, b, mod, mlen, inv);
|
|
t = a; a = b; b = t;
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
* At this point, the buffer (which is the high half of b)
|
|
* holds either 1 (implicitly, as the "isone" flag is set),
|
|
* n1^2, n2^2 or n1^2 * n2^2.
|
|
*/
|
|
|
|
/*
|
|
* The main loop. The procedure is:
|
|
* - Advance the windows
|
|
* - If the most-significant bit of a window is set,
|
|
* schedule a multiply for the appropriate time in the
|
|
* future (may be immediately)
|
|
* - Perform any pending multiples
|
|
* - Check for termination
|
|
* - Square the buffers
|
|
*
|
|
* At any given time, the acumulated product is held in
|
|
* the high half of b.
|
|
*/
|
|
for (;;) {
|
|
e1bits--;
|
|
|
|
/* Advance the windows */
|
|
assert(buf1 < tblmask);
|
|
buf1 <<= 1;
|
|
assert(buf2 < tblmask);
|
|
buf2 <<= 1;
|
|
/*
|
|
* This reads ahead of the current exponent position
|
|
* (controlled by e1bits), so we have to be able to read
|
|
* past the lsb of the exponents without error.
|
|
*/
|
|
if (e1len) {
|
|
buf1 |= ((*e1 & bitpos) != 0);
|
|
if (e1len <= e2len)
|
|
buf2 |= ((*e2 & bitpos) != 0);
|
|
bitpos >>= 1;
|
|
if (!bitpos) {
|
|
BIGLITTLE(e1++,e1--);
|
|
if (e1len <= e2len)
|
|
BIGLITTLE(e2++,e2--);
|
|
bitpos = (BNWORD64)1 << (64-1);
|
|
e1len--;
|
|
}
|
|
}
|
|
|
|
/* Examine the first window for pending multiplies */
|
|
if (buf1 & tblmask) {
|
|
mult1pos = e1bits - w1bits;
|
|
while ((buf1 & 1) == 0) {
|
|
buf1 >>= 1;
|
|
mult1pos++;
|
|
}
|
|
/* Intermediates can wrap, but final must NOT */
|
|
assert(mult1pos <= e1bits);
|
|
mult1 = table1[buf1>>1];
|
|
buf1 = 0;
|
|
}
|
|
|
|
/*
|
|
* Examine the second window for pending multiplies.
|
|
* Window 2 can be smaller than window 1, but we
|
|
* keep the same number of bits in buf2, so we need
|
|
* to ignore any low-order bits in the buffer when
|
|
* computing what to multiply by, and recompute them
|
|
* later.
|
|
*/
|
|
if (buf2 & tblmask) {
|
|
/* Remember low-order bits for later */
|
|
i = buf2 & ((1u << w2bits) - 1);
|
|
buf2 >>= w2bits;
|
|
mult2pos = e1bits - w1bits + w2bits;
|
|
while ((buf2 & 1) == 0) {
|
|
buf2 >>= 1;
|
|
mult2pos++;
|
|
}
|
|
assert(mult2pos <= e1bits);
|
|
mult2 = table2[buf2>>1];
|
|
buf2 = i;
|
|
}
|
|
|
|
|
|
/* If we have a pending multiply for e1, do it */
|
|
if (e1bits == mult1pos) {
|
|
/* Multiply by the table entry remembered previously */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
if (isone) {
|
|
/* Multiply by 1 is a trivial case */
|
|
lbnCopy_64(t, mult1, mlen);
|
|
isone = 0;
|
|
} else {
|
|
lbnMontMul_64(a, t, mult1, mod, mlen, inv);
|
|
/* Swap a and b */
|
|
t = a; a = b; b = t;
|
|
}
|
|
}
|
|
|
|
/* If we have a pending multiply for e2, do it */
|
|
if (e1bits == mult2pos) {
|
|
/* Multiply by the table entry remembered previously */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
if (isone) {
|
|
/* Multiply by 1 is a trivial case */
|
|
lbnCopy_64(t, mult2, mlen);
|
|
isone = 0;
|
|
} else {
|
|
lbnMontMul_64(a, t, mult2, mod, mlen, inv);
|
|
/* Swap a and b */
|
|
t = a; a = b; b = t;
|
|
}
|
|
}
|
|
|
|
/* Are we done? */
|
|
if (!e1bits)
|
|
break;
|
|
|
|
/* Square the buffer */
|
|
if (!isone) {
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
lbnMontSquare_64(a, t, mod, mlen, inv);
|
|
/* Swap a and b */
|
|
t = a; a = b; b = t;
|
|
}
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
goto yield;
|
|
#endif
|
|
} /* for (;;) */
|
|
|
|
assert(!isone);
|
|
assert(!buf1);
|
|
assert(!buf2);
|
|
|
|
/* DONE! */
|
|
|
|
/* Convert result out of Montgomery form */
|
|
t = BIGLITTLE(b-mlen, b+mlen);
|
|
lbnCopy_64(b, t, mlen);
|
|
lbnZero_64(t, mlen);
|
|
lbnMontReduce_64(b, mod, mlen, inv);
|
|
lbnCopy_64(result, t, mlen);
|
|
|
|
/* Clean up - free intermediate storage */
|
|
y = 0;
|
|
#if BNYIELD
|
|
yield:
|
|
#endif
|
|
buf2 = tblmask >> w2bits;
|
|
while (--tblmask) {
|
|
if (tblmask < buf2)
|
|
LBNFREE(table2[tblmask], mlen);
|
|
LBNFREE(table1[tblmask], mlen);
|
|
}
|
|
t = table1[0];
|
|
LBNFREE(t, mlen);
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
|
|
return y; /* Success */
|
|
}
|
|
|
|
/*
|
|
* 2^exp (mod mod). This is an optimized version for use in Fermat
|
|
* tests. The input value of n is ignored; it is returned with
|
|
* "mlen" words valid.
|
|
*/
|
|
int
|
|
lbnTwoExpMod_64(BNWORD64 *n, BNWORD64 const *exp, unsigned elen,
|
|
BNWORD64 *mod, unsigned mlen)
|
|
{
|
|
unsigned e; /* Copy of high words of the exponent */
|
|
unsigned bits; /* Assorted counter of bits */
|
|
BNWORD64 const *bitptr;
|
|
BNWORD64 bitword, bitpos;
|
|
BNWORD64 *a, *b, *a1;
|
|
BNWORD64 inv;
|
|
int y; /* Result of bnYield() */
|
|
|
|
assert(mlen);
|
|
|
|
bitptr = BIGLITTLE(exp-elen, exp+elen-1);
|
|
bitword = *bitptr;
|
|
assert(bitword);
|
|
|
|
/* Clear n for future use. */
|
|
lbnZero_64(n, mlen);
|
|
|
|
bits = lbnBits_64(exp, elen);
|
|
|
|
/* First, a couple of trivial cases. */
|
|
if (bits <= 1) {
|
|
/* 2 ^ 0 == 1, 2 ^ 1 == 2 */
|
|
BIGLITTLE(n[-1],n[0]) = (BNWORD64)1<<elen;
|
|
return 0;
|
|
}
|
|
|
|
/* Set bitpos to the most significant bit */
|
|
bitpos = (BNWORD64)1 << ((bits-1) & (64-1));
|
|
|
|
/* Now, count the bits in the modulus. */
|
|
bits = lbnBits_64(mod, mlen);
|
|
assert(bits > 1); /* a 1-bit modulus is just stupid... */
|
|
|
|
/*
|
|
* We start with 1<<e, where "e" is as many high bits of the
|
|
* exponent as we can manage without going over the modulus.
|
|
* This first loop finds "e".
|
|
*/
|
|
e = 1;
|
|
while (elen) {
|
|
/* Consume the first bit */
|
|
bitpos >>= 1;
|
|
if (!bitpos) {
|
|
if (!--elen)
|
|
break;
|
|
bitword = BIGLITTLE(*++bitptr,*--bitptr);
|
|
bitpos = (BNWORD64)1<<(64-1);
|
|
}
|
|
e = (e << 1) | ((bitpos & bitword) != 0);
|
|
if (e >= bits) { /* Overflow! Back out. */
|
|
e >>= 1;
|
|
break;
|
|
}
|
|
}
|
|
/*
|
|
* The bit in "bitpos" being examined by the bit buffer has NOT
|
|
* been consumed yet. This may be past the end of the exponent,
|
|
* in which case elen == 1.
|
|
*/
|
|
|
|
/* Okay, now, set bit "e" in n. n is already zero. */
|
|
inv = (BNWORD64)1 << (e & (64-1));
|
|
e /= 64;
|
|
BIGLITTLE(n[-e-1],n[e]) = inv;
|
|
/*
|
|
* The effective length of n in words is now "e+1".
|
|
* This is used a little bit later.
|
|
*/
|
|
|
|
if (!elen)
|
|
return 0; /* That was easy! */
|
|
|
|
/*
|
|
* We have now processed the first few bits. The next step
|
|
* is to convert this to Montgomery form for further squaring.
|
|
*/
|
|
|
|
/* Allocate working storage: two product buffers */
|
|
LBNALLOC(a, BNWORD64, 2*mlen);
|
|
if (!a)
|
|
return -1;
|
|
LBNALLOC(b, BNWORD64, 2*mlen);
|
|
if (!b) {
|
|
LBNFREE(a, 2*mlen);
|
|
return -1;
|
|
}
|
|
|
|
/* Convert n to Montgomery form */
|
|
inv = BIGLITTLE(mod[-1],mod[0]); /* LSW of modulus */
|
|
assert(inv & 1); /* Modulus must be odd */
|
|
inv = lbnMontInv1_64(inv);
|
|
/* Move n (length e+1, remember?) up "mlen" words into b */
|
|
/* Note that we lie about a1 for a bit - it's pointing to b */
|
|
a1 = BIGLITTLE(b-mlen,b+mlen);
|
|
lbnCopy_64(a1, n, e+1);
|
|
lbnZero_64(b, mlen);
|
|
/* Do the division - dump the quotient into the high-order words */
|
|
(void)lbnDiv_64(a1, b, mlen+e+1, mod, mlen);
|
|
/*
|
|
* Now do the first squaring and modular reduction to put
|
|
* the number up in a1 where it belongs.
|
|
*/
|
|
lbnMontSquare_64(a, b, mod, mlen, inv);
|
|
/* Fix up a1 to point to where it should go. */
|
|
a1 = BIGLITTLE(a-mlen,a+mlen);
|
|
|
|
/*
|
|
* Okay, now, a1 holds the number being accumulated, and
|
|
* b is a scratch register. Start working:
|
|
*/
|
|
for (;;) {
|
|
/*
|
|
* Is the bit set? If so, double a1 as well.
|
|
* A modular doubling like this is very cheap.
|
|
*/
|
|
if (bitpos & bitword) {
|
|
/*
|
|
* Double the number. If there was a carry out OR
|
|
* the result is greater than the modulus, subract
|
|
* the modulus.
|
|
*/
|
|
if (lbnDouble_64(a1, mlen) ||
|
|
lbnCmp_64(a1, mod, mlen) > 0)
|
|
(void)lbnSubN_64(a1, mod, mlen);
|
|
}
|
|
|
|
/* Advance to the next exponent bit */
|
|
bitpos >>= 1;
|
|
if (!bitpos) {
|
|
if (!--elen)
|
|
break; /* Done! */
|
|
bitword = BIGLITTLE(*++bitptr,*--bitptr);
|
|
bitpos = (BNWORD64)1<<(64-1);
|
|
}
|
|
|
|
/*
|
|
* The elen/bitword/bitpos bit buffer is known to be
|
|
* non-empty, i.e. there is at least one more unconsumed bit.
|
|
* Thus, it's safe to square the number.
|
|
*/
|
|
lbnMontSquare_64(b, a1, mod, mlen, inv);
|
|
/* Rename result (in b) back to a (a1, really). */
|
|
a1 = b; b = a; a = a1;
|
|
a1 = BIGLITTLE(a-mlen,a+mlen);
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
goto yield;
|
|
#endif
|
|
}
|
|
|
|
/* DONE! Just a little bit of cleanup... */
|
|
|
|
/*
|
|
* Convert result out of Montgomery form... this is
|
|
* just a Montgomery reduction.
|
|
*/
|
|
lbnCopy_64(a, a1, mlen);
|
|
lbnZero_64(a1, mlen);
|
|
lbnMontReduce_64(a, mod, mlen, inv);
|
|
lbnCopy_64(n, a1, mlen);
|
|
|
|
/* Clean up - free intermediate storage */
|
|
y = 0;
|
|
#if BNYIELD
|
|
yield:
|
|
#endif
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
|
|
return y; /* Success */
|
|
}
|
|
|
|
|
|
/*
|
|
* Returns a substring of the big-endian array of bytes representation
|
|
* of the bignum array based on two parameters, the least significant
|
|
* byte number (0 to start with the least significant byte) and the
|
|
* length. I.e. the number returned is a representation of
|
|
* (bn / 2^(8*lsbyte)) % 2 ^ (8*buflen).
|
|
*
|
|
* It is an error if the bignum is not at least buflen + lsbyte bytes
|
|
* long.
|
|
*
|
|
* This code assumes that the compiler has the minimal intelligence
|
|
* neded to optimize divides and modulo operations on an unsigned data
|
|
* type with a power of two.
|
|
*/
|
|
void
|
|
lbnExtractBigBytes_64(BNWORD64 const *n, unsigned char *buf,
|
|
unsigned lsbyte, unsigned buflen)
|
|
{
|
|
BNWORD64 t = 0; /* Needed to shut up uninitialized var warnings */
|
|
unsigned shift;
|
|
|
|
lsbyte += buflen;
|
|
|
|
shift = (8 * lsbyte) % 64;
|
|
lsbyte /= (64/8); /* Convert to word offset */
|
|
BIGLITTLE(n -= lsbyte, n += lsbyte);
|
|
|
|
if (shift)
|
|
t = BIGLITTLE(n[-1],n[0]);
|
|
|
|
while (buflen--) {
|
|
if (!shift) {
|
|
t = BIGLITTLE(*n++,*--n);
|
|
shift = 64;
|
|
}
|
|
shift -= 8;
|
|
*buf++ = (unsigned char)(t>>shift);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Merge a big-endian array of bytes into a bignum array.
|
|
* The array had better be big enough. This is
|
|
* equivalent to extracting the entire bignum into a
|
|
* large byte array, copying the input buffer into the
|
|
* middle of it, and converting back to a bignum.
|
|
*
|
|
* The buf is "len" bytes long, and its *last* byte is at
|
|
* position "lsbyte" from the end of the bignum.
|
|
*
|
|
* Note that this is a pain to get right. Fortunately, it's hardly
|
|
* critical for efficiency.
|
|
*/
|
|
void
|
|
lbnInsertBigBytes_64(BNWORD64 *n, unsigned char const *buf,
|
|
unsigned lsbyte, unsigned buflen)
|
|
{
|
|
BNWORD64 t = 0; /* Shut up uninitialized varibale warnings */
|
|
|
|
lsbyte += buflen;
|
|
|
|
BIGLITTLE(n -= lsbyte/(64/8), n += lsbyte/(64/8));
|
|
|
|
/* Load up leading odd bytes */
|
|
if (lsbyte % (64/8)) {
|
|
t = BIGLITTLE(*--n,*n++);
|
|
t >>= (lsbyte * 8) % 64;
|
|
}
|
|
|
|
/* The main loop - merge into t, storing at each word boundary. */
|
|
while (buflen--) {
|
|
t = (t << 8) | *buf++;
|
|
if ((--lsbyte % (64/8)) == 0)
|
|
BIGLITTLE(*n++,*--n) = t;
|
|
}
|
|
|
|
/* Merge odd bytes in t into last word */
|
|
lsbyte = (lsbyte * 8) % 64;
|
|
if (lsbyte) {
|
|
t <<= lsbyte;
|
|
t |= (((BNWORD64)1 << lsbyte) - 1) & BIGLITTLE(n[0],n[-1]);
|
|
BIGLITTLE(n[0],n[-1]) = t;
|
|
}
|
|
|
|
return;
|
|
}
|
|
|
|
/*
|
|
* Returns a substring of the little-endian array of bytes representation
|
|
* of the bignum array based on two parameters, the least significant
|
|
* byte number (0 to start with the least significant byte) and the
|
|
* length. I.e. the number returned is a representation of
|
|
* (bn / 2^(8*lsbyte)) % 2 ^ (8*buflen).
|
|
*
|
|
* It is an error if the bignum is not at least buflen + lsbyte bytes
|
|
* long.
|
|
*
|
|
* This code assumes that the compiler has the minimal intelligence
|
|
* neded to optimize divides and modulo operations on an unsigned data
|
|
* type with a power of two.
|
|
*/
|
|
void
|
|
lbnExtractLittleBytes_64(BNWORD64 const *n, unsigned char *buf,
|
|
unsigned lsbyte, unsigned buflen)
|
|
{
|
|
BNWORD64 t = 0; /* Needed to shut up uninitialized var warnings */
|
|
|
|
BIGLITTLE(n -= lsbyte/(64/8), n += lsbyte/(64/8));
|
|
|
|
if (lsbyte % (64/8)) {
|
|
t = BIGLITTLE(*--n,*n++);
|
|
t >>= (lsbyte % (64/8)) * 8 ;
|
|
}
|
|
|
|
while (buflen--) {
|
|
if ((lsbyte++ % (64/8)) == 0)
|
|
t = BIGLITTLE(*--n,*n++);
|
|
*buf++ = (unsigned char)t;
|
|
t >>= 8;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Merge a little-endian array of bytes into a bignum array.
|
|
* The array had better be big enough. This is
|
|
* equivalent to extracting the entire bignum into a
|
|
* large byte array, copying the input buffer into the
|
|
* middle of it, and converting back to a bignum.
|
|
*
|
|
* The buf is "len" bytes long, and its first byte is at
|
|
* position "lsbyte" from the end of the bignum.
|
|
*
|
|
* Note that this is a pain to get right. Fortunately, it's hardly
|
|
* critical for efficiency.
|
|
*/
|
|
void
|
|
lbnInsertLittleBytes_64(BNWORD64 *n, unsigned char const *buf,
|
|
unsigned lsbyte, unsigned buflen)
|
|
{
|
|
BNWORD64 t = 0; /* Shut up uninitialized varibale warnings */
|
|
|
|
/* Move to most-significant end */
|
|
lsbyte += buflen;
|
|
buf += buflen;
|
|
|
|
BIGLITTLE(n -= lsbyte/(64/8), n += lsbyte/(64/8));
|
|
|
|
/* Load up leading odd bytes */
|
|
if (lsbyte % (64/8)) {
|
|
t = BIGLITTLE(*--n,*n++);
|
|
t >>= (lsbyte * 8) % 64;
|
|
}
|
|
|
|
/* The main loop - merge into t, storing at each word boundary. */
|
|
while (buflen--) {
|
|
t = (t << 8) | *--buf;
|
|
if ((--lsbyte % (64/8)) == 0)
|
|
BIGLITTLE(*n++,*--n) = t;
|
|
}
|
|
|
|
/* Merge odd bytes in t into last word */
|
|
lsbyte = (lsbyte * 8) % 64;
|
|
if (lsbyte) {
|
|
t <<= lsbyte;
|
|
t |= (((BNWORD64)1 << lsbyte) - 1) & BIGLITTLE(n[0],n[-1]);
|
|
BIGLITTLE(n[0],n[-1]) = t;
|
|
}
|
|
|
|
return;
|
|
}
|
|
|
|
#ifdef DEADCODE /* This was a precursor to the more flexible lbnExtractBytes */
|
|
/*
|
|
* Convert a big-endian array of bytes to a bignum.
|
|
* Returns the number of words in the bignum.
|
|
* Note the expression "64/8" for the number of bytes per word.
|
|
* This is so the word-size adjustment will work.
|
|
*/
|
|
unsigned
|
|
lbnFromBytes_64(BNWORD64 *a, unsigned char const *b, unsigned blen)
|
|
{
|
|
BNWORD64 t;
|
|
unsigned alen = (blen + (64/8-1))/(64/8);
|
|
BIGLITTLE(a -= alen, a += alen);
|
|
|
|
while (blen) {
|
|
t = 0;
|
|
do {
|
|
t = t << 8 | *b++;
|
|
} while (--blen & (64/8-1));
|
|
BIGLITTLE(*a++,*--a) = t;
|
|
}
|
|
return alen;
|
|
}
|
|
#endif
|
|
|
|
/*
|
|
* Computes the GCD of a and b. Modifies both arguments; when it returns,
|
|
* one of them is the GCD and the other is trash. The return value
|
|
* indicates which: 0 for a, and 1 for b. The length of the retult is
|
|
* returned in rlen. Both inputs must have one extra word of precision.
|
|
* alen must be >= blen.
|
|
*
|
|
* TODO: use the binary algorithm (Knuth section 4.5.2, algorithm B).
|
|
* This is based on taking out common powers of 2, then repeatedly:
|
|
* gcd(2*u,v) = gcd(u,2*v) = gcd(u,v) - isolated powers of 2 can be deleted.
|
|
* gcd(u,v) = gcd(u-v,v) - the numbers can be easily reduced.
|
|
* It gets less reduction per step, but the steps are much faster than
|
|
* the division case.
|
|
*/
|
|
int
|
|
lbnGcd_64(BNWORD64 *a, unsigned alen, BNWORD64 *b, unsigned blen,
|
|
unsigned *rlen)
|
|
{
|
|
#if BNYIELD
|
|
int y;
|
|
#endif
|
|
assert(alen >= blen);
|
|
|
|
while (blen != 0) {
|
|
(void)lbnDiv_64(BIGLITTLE(a-blen,a+blen), a, alen, b, blen);
|
|
alen = lbnNorm_64(a, blen);
|
|
if (alen == 0) {
|
|
*rlen = blen;
|
|
return 1;
|
|
}
|
|
(void)lbnDiv_64(BIGLITTLE(b-alen,b+alen), b, blen, a, alen);
|
|
blen = lbnNorm_64(b, alen);
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield()) < 0)
|
|
return y;
|
|
#endif
|
|
}
|
|
*rlen = alen;
|
|
return 0;
|
|
}
|
|
|
|
/*
|
|
* Invert "a" modulo "mod" using the extended Euclidean algorithm.
|
|
* Note that this only computes one of the cosequences, and uses the
|
|
* theorem that the signs flip every step and the absolute value of
|
|
* the cosequence values are always bounded by the modulus to avoid
|
|
* having to work with negative numbers.
|
|
* gcd(a,mod) had better equal 1. Returns 1 if the GCD is NOT 1.
|
|
* a must be one word longer than "mod". It is overwritten with the
|
|
* result.
|
|
* TODO: Use Richard Schroeppel's *much* faster algorithm.
|
|
*/
|
|
int
|
|
lbnInv_64(BNWORD64 *a, unsigned alen, BNWORD64 const *mod, unsigned mlen)
|
|
{
|
|
BNWORD64 *b; /* Hold a copy of mod during GCD reduction */
|
|
BNWORD64 *p; /* Temporary for products added to t0 and t1 */
|
|
BNWORD64 *t0, *t1; /* Inverse accumulators */
|
|
BNWORD64 cy;
|
|
unsigned blen, t0len, t1len, plen;
|
|
int y;
|
|
|
|
alen = lbnNorm_64(a, alen);
|
|
if (!alen)
|
|
return 1; /* No inverse */
|
|
|
|
mlen = lbnNorm_64(mod, mlen);
|
|
|
|
assert (alen <= mlen);
|
|
|
|
/* Inverse of 1 is 1 */
|
|
if (alen == 1 && BIGLITTLE(a[-1],a[0]) == 1) {
|
|
lbnZero_64(BIGLITTLE(a-alen,a+alen), mlen-alen);
|
|
return 0;
|
|
}
|
|
|
|
/* Allocate a pile of space */
|
|
LBNALLOC(b, BNWORD64, mlen+1);
|
|
if (b) {
|
|
/*
|
|
* Although products are guaranteed to always be less than the
|
|
* modulus, it can involve multiplying two 3-word numbers to
|
|
* get a 5-word result, requiring a 6th word to store a 0
|
|
* temporarily. Thus, mlen + 1.
|
|
*/
|
|
LBNALLOC(p, BNWORD64, mlen+1);
|
|
if (p) {
|
|
LBNALLOC(t0, BNWORD64, mlen);
|
|
if (t0) {
|
|
LBNALLOC(t1, BNWORD64, mlen);
|
|
if (t1)
|
|
goto allocated;
|
|
LBNFREE(t0, mlen);
|
|
}
|
|
LBNFREE(p, mlen+1);
|
|
}
|
|
LBNFREE(b, mlen+1);
|
|
}
|
|
return -1;
|
|
|
|
allocated:
|
|
|
|
/* Set t0 to 1 */
|
|
t0len = 1;
|
|
BIGLITTLE(t0[-1],t0[0]) = 1;
|
|
|
|
/* b = mod */
|
|
lbnCopy_64(b, mod, mlen);
|
|
/* blen = mlen (implicitly) */
|
|
|
|
/* t1 = b / a; b = b % a */
|
|
cy = lbnDiv_64(t1, b, mlen, a, alen);
|
|
*(BIGLITTLE(t1-(mlen-alen)-1,t1+(mlen-alen))) = cy;
|
|
t1len = lbnNorm_64(t1, mlen-alen+1);
|
|
blen = lbnNorm_64(b, alen);
|
|
|
|
/* while (b > 1) */
|
|
while (blen > 1 || BIGLITTLE(b[-1],b[0]) != (BNWORD64)1) {
|
|
/* q = a / b; a = a % b; */
|
|
if (alen < blen || (alen == blen && lbnCmp_64(a, a, alen) < 0))
|
|
assert(0);
|
|
cy = lbnDiv_64(BIGLITTLE(a-blen,a+blen), a, alen, b, blen);
|
|
*(BIGLITTLE(a-alen-1,a+alen)) = cy;
|
|
plen = lbnNorm_64(BIGLITTLE(a-blen,a+blen), alen-blen+1);
|
|
assert(plen);
|
|
alen = lbnNorm_64(a, blen);
|
|
if (!alen)
|
|
goto failure; /* GCD not 1 */
|
|
|
|
/* t0 += q * t1; */
|
|
assert(plen+t1len <= mlen+1);
|
|
lbnMul_64(p, BIGLITTLE(a-blen,a+blen), plen, t1, t1len);
|
|
plen = lbnNorm_64(p, plen + t1len);
|
|
assert(plen <= mlen);
|
|
if (plen > t0len) {
|
|
lbnZero_64(BIGLITTLE(t0-t0len,t0+t0len), plen-t0len);
|
|
t0len = plen;
|
|
}
|
|
cy = lbnAddN_64(t0, p, plen);
|
|
if (cy) {
|
|
if (t0len > plen) {
|
|
cy = lbnAdd1_64(BIGLITTLE(t0-plen,t0+plen),
|
|
t0len-plen, cy);
|
|
}
|
|
if (cy) {
|
|
BIGLITTLE(t0[-t0len-1],t0[t0len]) = cy;
|
|
t0len++;
|
|
}
|
|
}
|
|
|
|
/* if (a <= 1) return a ? t0 : FAIL; */
|
|
if (alen <= 1 && BIGLITTLE(a[-1],a[0]) == (BNWORD64)1) {
|
|
if (alen == 0)
|
|
goto failure; /* FAIL */
|
|
assert(t0len <= mlen);
|
|
lbnCopy_64(a, t0, t0len);
|
|
lbnZero_64(BIGLITTLE(a-t0len, a+t0len), mlen-t0len);
|
|
goto success;
|
|
}
|
|
|
|
/* q = b / a; b = b % a; */
|
|
if (blen < alen || (blen == alen && lbnCmp_64(b, a, alen) < 0))
|
|
assert(0);
|
|
cy = lbnDiv_64(BIGLITTLE(b-alen,b+alen), b, blen, a, alen);
|
|
*(BIGLITTLE(b-blen-1,b+blen)) = cy;
|
|
plen = lbnNorm_64(BIGLITTLE(b-alen,b+alen), blen-alen+1);
|
|
assert(plen);
|
|
blen = lbnNorm_64(b, alen);
|
|
if (!blen)
|
|
goto failure; /* GCD not 1 */
|
|
|
|
/* t1 += q * t0; */
|
|
assert(plen+t0len <= mlen+1);
|
|
lbnMul_64(p, BIGLITTLE(b-alen,b+alen), plen, t0, t0len);
|
|
plen = lbnNorm_64(p, plen + t0len);
|
|
assert(plen <= mlen);
|
|
if (plen > t1len) {
|
|
lbnZero_64(BIGLITTLE(t1-t1len,t1+t1len), plen-t1len);
|
|
t1len = plen;
|
|
}
|
|
cy = lbnAddN_64(t1, p, plen);
|
|
if (cy) {
|
|
if (t1len > plen) {
|
|
cy = lbnAdd1_64(BIGLITTLE(t1-plen,t0+plen),
|
|
t1len-plen, cy);
|
|
}
|
|
if (cy) {
|
|
BIGLITTLE(t1[-t1len-1],t1[t1len]) = cy;
|
|
t1len++;
|
|
}
|
|
}
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield() < 0))
|
|
goto yield;
|
|
#endif
|
|
}
|
|
|
|
if (!blen)
|
|
goto failure; /* gcd(a, mod) != 1 -- FAIL */
|
|
|
|
/* return mod-t1 */
|
|
lbnCopy_64(a, mod, mlen);
|
|
assert(t1len <= mlen);
|
|
cy = lbnSubN_64(a, t1, t1len);
|
|
if (cy) {
|
|
assert(mlen > t1len);
|
|
cy = lbnSub1_64(BIGLITTLE(a-t1len, a+t1len), mlen-t1len, cy);
|
|
assert(!cy);
|
|
}
|
|
|
|
success:
|
|
LBNFREE(t1, mlen);
|
|
LBNFREE(t0, mlen);
|
|
LBNFREE(p, mlen+1);
|
|
LBNFREE(b, mlen+1);
|
|
|
|
return 0;
|
|
|
|
failure: /* GCD is not 1 - no inverse exists! */
|
|
y = 1;
|
|
#if BNYIELD
|
|
yield:
|
|
#endif
|
|
LBNFREE(t1, mlen);
|
|
LBNFREE(t0, mlen);
|
|
LBNFREE(p, mlen+1);
|
|
LBNFREE(b, mlen+1);
|
|
|
|
return y;
|
|
}
|
|
|
|
/*
|
|
* Precompute powers of "a" mod "mod". Compute them every "bits"
|
|
* for "n" steps. This is sufficient to compute powers of g with
|
|
* exponents up to n*bits bits long, i.e. less than 2^(n*bits).
|
|
*
|
|
* This assumes that the caller has already initialized "array" to point
|
|
* to "n" buffers of size "mlen".
|
|
*/
|
|
int
|
|
lbnBasePrecompBegin_64(BNWORD64 **array, unsigned n, unsigned bits,
|
|
BNWORD64 const *g, unsigned glen, BNWORD64 *mod, unsigned mlen)
|
|
{
|
|
BNWORD64 *a, *b; /* Temporary double-width accumulators */
|
|
BNWORD64 *a1; /* Pointer to high half of a*/
|
|
BNWORD64 inv; /* Montgomery inverse of LSW of mod */
|
|
BNWORD64 *t;
|
|
unsigned i;
|
|
|
|
glen = lbnNorm_64(g, glen);
|
|
assert(glen);
|
|
|
|
assert (mlen == lbnNorm_64(mod, mlen));
|
|
assert (glen <= mlen);
|
|
|
|
/* Allocate two temporary buffers, and the array slots */
|
|
LBNALLOC(a, BNWORD64, mlen*2);
|
|
if (!a)
|
|
return -1;
|
|
LBNALLOC(b, BNWORD64, mlen*2);
|
|
if (!b) {
|
|
LBNFREE(a, 2*mlen);
|
|
return -1;
|
|
}
|
|
|
|
/* Okay, all ready */
|
|
|
|
/* Convert n to Montgomery form */
|
|
inv = BIGLITTLE(mod[-1],mod[0]); /* LSW of modulus */
|
|
assert(inv & 1); /* Modulus must be odd */
|
|
inv = lbnMontInv1_64(inv);
|
|
/* Move g up "mlen" words into a (clearing the low mlen words) */
|
|
a1 = BIGLITTLE(a-mlen,a+mlen);
|
|
lbnCopy_64(a1, g, glen);
|
|
lbnZero_64(a, mlen);
|
|
|
|
/* Do the division - dump the quotient into the high-order words */
|
|
(void)lbnDiv_64(a1, a, mlen+glen, mod, mlen);
|
|
|
|
/* Copy the first value into the array */
|
|
t = *array;
|
|
lbnCopy_64(t, a, mlen);
|
|
a1 = a; /* This first value is *not* shifted up */
|
|
|
|
/* Now compute the remaining n-1 array entries */
|
|
assert(bits);
|
|
assert(n);
|
|
while (--n) {
|
|
i = bits;
|
|
do {
|
|
/* Square a1 into b1 */
|
|
lbnMontSquare_64(b, a1, mod, mlen, inv);
|
|
t = b; b = a; a = t;
|
|
a1 = BIGLITTLE(a-mlen, a+mlen);
|
|
} while (--i);
|
|
t = *++array;
|
|
lbnCopy_64(t, a1, mlen);
|
|
}
|
|
|
|
/* Hooray, we're done. */
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
return 0;
|
|
}
|
|
|
|
/*
|
|
* result = base^exp (mod mod). "array" is a an array of pointers
|
|
* to procomputed powers of base, each 2^bits apart. (I.e. array[i]
|
|
* is base^(2^(i*bits))).
|
|
*
|
|
* The algorithm consists of:
|
|
* a = b = (powers of g to be raised to the power 2^bits-1)
|
|
* a *= b *= (powers of g to be raised to the power 2^bits-2)
|
|
* ...
|
|
* a *= b *= (powers of g to be raised to the power 1)
|
|
*
|
|
* All we do is walk the exponent 2^bits-1 times in groups of "bits" bits,
|
|
*/
|
|
int
|
|
lbnBasePrecompExp_64(BNWORD64 *result, BNWORD64 const * const *array,
|
|
unsigned bits, BNWORD64 const *exp, unsigned elen,
|
|
BNWORD64 const *mod, unsigned mlen)
|
|
{
|
|
BNWORD64 *a, *b, *c, *t;
|
|
BNWORD64 *a1, *b1;
|
|
int anull, bnull; /* Null flags: values are implicitly 1 */
|
|
unsigned i, j; /* Loop counters */
|
|
unsigned mask; /* Exponent bits to examime */
|
|
BNWORD64 const *eptr; /* Pointer into exp */
|
|
BNWORD64 buf, curbits, nextword; /* Bit-buffer varaibles */
|
|
BNWORD64 inv; /* Inverse of LSW of modulus */
|
|
unsigned ewords; /* Words of exponent left */
|
|
int bufbits; /* Number of valid bits */
|
|
int y = 0;
|
|
|
|
mlen = lbnNorm_64(mod, mlen);
|
|
assert (mlen);
|
|
|
|
elen = lbnNorm_64(exp, elen);
|
|
if (!elen) {
|
|
lbnZero_64(result, mlen);
|
|
BIGLITTLE(result[-1],result[0]) = 1;
|
|
return 0;
|
|
}
|
|
/*
|
|
* This could be precomputed, but it's so cheap, and it would require
|
|
* making the precomputation structure word-size dependent.
|
|
*/
|
|
inv = lbnMontInv1_64(mod[BIGLITTLE(-1,0)]); /* LSW of modulus */
|
|
|
|
assert(elen);
|
|
|
|
/*
|
|
* Allocate three temporary buffers. The current numbers generally
|
|
* live in the upper halves of these buffers.
|
|
*/
|
|
LBNALLOC(a, BNWORD64, mlen*2);
|
|
if (a) {
|
|
LBNALLOC(b, BNWORD64, mlen*2);
|
|
if (b) {
|
|
LBNALLOC(c, BNWORD64, mlen*2);
|
|
if (c)
|
|
goto allocated;
|
|
LBNFREE(b, 2*mlen);
|
|
}
|
|
LBNFREE(a, 2*mlen);
|
|
}
|
|
return -1;
|
|
|
|
allocated:
|
|
|
|
anull = bnull = 1;
|
|
|
|
mask = (1u<<bits) - 1;
|
|
for (i = mask; i; --i) {
|
|
/* Set up bit buffer for walking the exponent */
|
|
eptr = exp;
|
|
buf = BIGLITTLE(*--eptr, *eptr++);
|
|
ewords = elen-1;
|
|
bufbits = 64;
|
|
for (j = 0; ewords || buf; j++) {
|
|
/* Shift down current buffer */
|
|
curbits = buf;
|
|
buf >>= bits;
|
|
/* If necessary, add next word */
|
|
bufbits -= bits;
|
|
if (bufbits < 0 && ewords > 0) {
|
|
nextword = BIGLITTLE(*--eptr, *eptr++);
|
|
ewords--;
|
|
curbits |= nextword << (bufbits+bits);
|
|
buf = nextword >> -bufbits;
|
|
bufbits += 64;
|
|
}
|
|
/* If appropriate, multiply b *= array[j] */
|
|
if ((curbits & mask) == i) {
|
|
BNWORD64 const *d = array[j];
|
|
|
|
b1 = BIGLITTLE(b-mlen-1,b+mlen);
|
|
if (bnull) {
|
|
lbnCopy_64(b1, d, mlen);
|
|
bnull = 0;
|
|
} else {
|
|
lbnMontMul_64(c, b1, d, mod, mlen, inv);
|
|
t = c; c = b; b = t;
|
|
}
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield() < 0))
|
|
goto yield;
|
|
#endif
|
|
}
|
|
}
|
|
|
|
/* Multiply a *= b */
|
|
if (!bnull) {
|
|
a1 = BIGLITTLE(a-mlen-1,a+mlen);
|
|
b1 = BIGLITTLE(b-mlen-1,b+mlen);
|
|
if (anull) {
|
|
lbnCopy_64(a1, b1, mlen);
|
|
anull = 0;
|
|
} else {
|
|
lbnMontMul_64(c, a1, b1, mod, mlen, inv);
|
|
t = c; c = a; a = t;
|
|
}
|
|
}
|
|
}
|
|
|
|
assert(!anull); /* If it were, elen would have been 0 */
|
|
|
|
/* Convert out of Montgomery form and return */
|
|
a1 = BIGLITTLE(a-mlen-1,a+mlen);
|
|
lbnCopy_64(a, a1, mlen);
|
|
lbnZero_64(a1, mlen);
|
|
lbnMontReduce_64(a, mod, mlen, inv);
|
|
lbnCopy_64(result, a1, mlen);
|
|
|
|
#if BNYIELD
|
|
yield:
|
|
#endif
|
|
LBNFREE(c, 2*mlen);
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
|
|
return y;
|
|
}
|
|
|
|
/*
|
|
* result = base1^exp1 *base2^exp2 (mod mod). "array1" and "array2" are
|
|
* arrays of pointers to procomputed powers of the corresponding bases,
|
|
* each 2^bits apart. (I.e. array1[i] is base1^(2^(i*bits))).
|
|
*
|
|
* Bits must be the same in both. (It could be made adjustable, but it's
|
|
* a bit of a pain. Just make them both equal to the larger one.)
|
|
*
|
|
* The algorithm consists of:
|
|
* a = b = (powers of base1 and base2 to be raised to the power 2^bits-1)
|
|
* a *= b *= (powers of base1 and base2 to be raised to the power 2^bits-2)
|
|
* ...
|
|
* a *= b *= (powers of base1 and base2 to be raised to the power 1)
|
|
*
|
|
* All we do is walk the exponent 2^bits-1 times in groups of "bits" bits,
|
|
*/
|
|
int
|
|
lbnDoubleBasePrecompExp_64(BNWORD64 *result, unsigned bits,
|
|
BNWORD64 const * const *array1, BNWORD64 const *exp1, unsigned elen1,
|
|
BNWORD64 const * const *array2, BNWORD64 const *exp2,
|
|
unsigned elen2, BNWORD64 const *mod, unsigned mlen)
|
|
{
|
|
BNWORD64 *a, *b, *c, *t;
|
|
BNWORD64 *a1, *b1;
|
|
int anull, bnull; /* Null flags: values are implicitly 1 */
|
|
unsigned i, j, k; /* Loop counters */
|
|
unsigned mask; /* Exponent bits to examime */
|
|
BNWORD64 const *eptr; /* Pointer into exp */
|
|
BNWORD64 buf, curbits, nextword; /* Bit-buffer varaibles */
|
|
BNWORD64 inv; /* Inverse of LSW of modulus */
|
|
unsigned ewords; /* Words of exponent left */
|
|
int bufbits; /* Number of valid bits */
|
|
int y = 0;
|
|
BNWORD64 const * const *array;
|
|
|
|
mlen = lbnNorm_64(mod, mlen);
|
|
assert (mlen);
|
|
|
|
elen1 = lbnNorm_64(exp1, elen1);
|
|
if (!elen1) {
|
|
return lbnBasePrecompExp_64(result, array2, bits, exp2, elen2,
|
|
mod, mlen);
|
|
}
|
|
elen2 = lbnNorm_64(exp2, elen2);
|
|
if (!elen2) {
|
|
return lbnBasePrecompExp_64(result, array1, bits, exp1, elen1,
|
|
mod, mlen);
|
|
}
|
|
/*
|
|
* This could be precomputed, but it's so cheap, and it would require
|
|
* making the precomputation structure word-size dependent.
|
|
*/
|
|
inv = lbnMontInv1_64(mod[BIGLITTLE(-1,0)]); /* LSW of modulus */
|
|
|
|
assert(elen1);
|
|
assert(elen2);
|
|
|
|
/*
|
|
* Allocate three temporary buffers. The current numbers generally
|
|
* live in the upper halves of these buffers.
|
|
*/
|
|
LBNALLOC(a, BNWORD64, mlen*2);
|
|
if (a) {
|
|
LBNALLOC(b, BNWORD64, mlen*2);
|
|
if (b) {
|
|
LBNALLOC(c, BNWORD64, mlen*2);
|
|
if (c)
|
|
goto allocated;
|
|
LBNFREE(b, 2*mlen);
|
|
}
|
|
LBNFREE(a, 2*mlen);
|
|
}
|
|
return -1;
|
|
|
|
allocated:
|
|
|
|
anull = bnull = 1;
|
|
|
|
mask = (1u<<bits) - 1;
|
|
for (i = mask; i; --i) {
|
|
/* Walk each exponent in turn */
|
|
for (k = 0; k < 2; k++) {
|
|
/* Set up the exponent for walking */
|
|
array = k ? array2 : array1;
|
|
eptr = k ? exp2 : exp1;
|
|
ewords = (k ? elen2 : elen1) - 1;
|
|
/* Set up bit buffer for walking the exponent */
|
|
buf = BIGLITTLE(*--eptr, *eptr++);
|
|
bufbits = 64;
|
|
for (j = 0; ewords || buf; j++) {
|
|
/* Shift down current buffer */
|
|
curbits = buf;
|
|
buf >>= bits;
|
|
/* If necessary, add next word */
|
|
bufbits -= bits;
|
|
if (bufbits < 0 && ewords > 0) {
|
|
nextword = BIGLITTLE(*--eptr, *eptr++);
|
|
ewords--;
|
|
curbits |= nextword << (bufbits+bits);
|
|
buf = nextword >> -bufbits;
|
|
bufbits += 64;
|
|
}
|
|
/* If appropriate, multiply b *= array[j] */
|
|
if ((curbits & mask) == i) {
|
|
BNWORD64 const *d = array[j];
|
|
|
|
b1 = BIGLITTLE(b-mlen-1,b+mlen);
|
|
if (bnull) {
|
|
lbnCopy_64(b1, d, mlen);
|
|
bnull = 0;
|
|
} else {
|
|
lbnMontMul_64(c, b1, d, mod, mlen, inv);
|
|
t = c; c = b; b = t;
|
|
}
|
|
#if BNYIELD
|
|
if (bnYield && (y = bnYield() < 0))
|
|
goto yield;
|
|
#endif
|
|
}
|
|
}
|
|
}
|
|
|
|
/* Multiply a *= b */
|
|
if (!bnull) {
|
|
a1 = BIGLITTLE(a-mlen-1,a+mlen);
|
|
b1 = BIGLITTLE(b-mlen-1,b+mlen);
|
|
if (anull) {
|
|
lbnCopy_64(a1, b1, mlen);
|
|
anull = 0;
|
|
} else {
|
|
lbnMontMul_64(c, a1, b1, mod, mlen, inv);
|
|
t = c; c = a; a = t;
|
|
}
|
|
}
|
|
}
|
|
|
|
assert(!anull); /* If it were, elen would have been 0 */
|
|
|
|
/* Convert out of Montgomery form and return */
|
|
a1 = BIGLITTLE(a-mlen-1,a+mlen);
|
|
lbnCopy_64(a, a1, mlen);
|
|
lbnZero_64(a1, mlen);
|
|
lbnMontReduce_64(a, mod, mlen, inv);
|
|
lbnCopy_64(result, a1, mlen);
|
|
|
|
#if BNYIELD
|
|
yield:
|
|
#endif
|
|
LBNFREE(c, 2*mlen);
|
|
LBNFREE(b, 2*mlen);
|
|
LBNFREE(a, 2*mlen);
|
|
|
|
return y;
|
|
}
|