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Implement GCD using Stein's binary algorithm.

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Dag-Erling Smørgrav 2017-03-30 00:53:23 +02:00
parent a11c52e896
commit b4eb918dad
7 changed files with 403 additions and 1 deletions
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2
include/cryb/mpi.h
View file

@ -58,6 +58,7 @@ const char *cryb_mpi_version(void);
#define mpi_eq_i64 cryb_mpi_eq_i64
#define mpi_eq_u32 cryb_mpi_eq_u32
#define mpi_eq_u64 cryb_mpi_eq_u64
#define mpi_gcd_abs cryb_mpi_gcd_abs
#define mpi_grow cryb_mpi_grow
#define mpi_init cryb_mpi_init
#define mpi_load cryb_mpi_load
@ -118,6 +119,7 @@ int mpi_eq_i64(const cryb_mpi *, int64_t);
int mpi_eq_abs_u32(const cryb_mpi *, uint32_t);
int mpi_eq_u32(const cryb_mpi *, uint32_t);
int mpi_eq_i32(const cryb_mpi *, int32_t);
int mpi_gcd_abs(cryb_mpi *, const cryb_mpi *, const cryb_mpi *);
CRYB_END

1
lib/mpi/Makefile.am
View file

@ -15,6 +15,7 @@ libcryb_mpi_la_SOURCES = \
cryb_mpi_eq_abs.c \
cryb_mpi_eq_i32.c \
cryb_mpi_eq_i64.c \
cryb_mpi_gcd_abs.c \
cryb_mpi_grow.c \
cryb_mpi_init.c \
cryb_mpi_load.c \

116
lib/mpi/cryb_mpi_gcd_abs.c Normal file
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@ -0,0 +1,116 @@
/*
* Copyright (c) 2017 Dag-Erling Smørgrav
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. The name of the author may not be used to endorse or promote
* products derived from this software without specific prior written
* permission.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include "cryb/impl.h"
#include <stddef.h>
#include <stdint.h>
#include <stdio.h>
#include <cryb/assert.h>
#include <cryb/endian.h>
#include <cryb/mpi.h>
#include "cryb_mpi_impl.h"
/*
* Compute the greatest common denominator of the absolute values of A and
* B and store the result in X. This is an iterative implementation of
* Stein's binary GCD algorithm.
*
* TODO: replace with Lehmer's GCD algorithm, which should be
* significantly faster for large inputs.
*/
int
mpi_gcd_abs(cryb_mpi *X, const cryb_mpi *A, const cryb_mpi *B)
{
cryb_mpi TA = CRYB_MPI_ZERO, TB = CRYB_MPI_ZERO;
unsigned int ashift, bshift;
/* GCD(x, x) = x */
if (A == B || mpi_eq(A, B)) {
if (X != A && X != B && mpi_copy(X, A) != 0)
return (-1);
X->neg = 0;
return (0);
}
/* GCD(x, 0) = 0 */
if (A->msb == 0 || B->msb == 0) {
mpi_zero(X);
return (0);
}
/* Stein's algorithm is destructive, so we operate on copies */
if (mpi_copy(&TA, A) != 0 || mpi_copy(&TB, B) != 0)
goto fail;
fprintf(stderr, "%04x %08x %08x | %04x %08x %08x\n",
TA.words[2], TA.words[1], TA.words[0],
TB.words[2], TB.words[1], TB.words[0]);
fprintf(stderr, "-----------------------|-----------------------\n");
/* reduce each operand to its greatest odd denominator */
/* neither operand is zero, and mpi_rshift() cannot fail */
ashift = mpi_lsb(&TA) - 1;
if ((bshift = mpi_lsb(&TB) - 1) < ashift)
ashift = bshift;
(void)mpi_rshift(&TA, ashift);
(void)mpi_rshift(&TB, ashift);
while (TA.msb != 0) {
/* mpi_rshift() cannot fail */
if ((TA.words[0] & 1) == 0)
(void)mpi_rshift(&TA, mpi_lsb(&TA) - 1);
if ((TB.words[0] & 1) == 0)
(void)mpi_rshift(&TB, mpi_lsb(&TB) - 1);
if (mpi_cmp_abs(&TA, &TB) < 0)
mpi_swap(&TA, &TB);
/* mpi_sub_abs() cannot fail in this case */
(void)mpi_sub_abs(&TA, &TA, &TB);
/* mpi_rshift() cannot fail */
assert((TA.words[0] & 1) == 0);
(void)mpi_rshift(&TA, 1);
fprintf(stderr, "%04x %08x %08x | %04x %08x %08x\n",
TA.words[2], TA.words[1], TA.words[0],
TB.words[2], TB.words[1], TB.words[0]);
}
/* undo the initial reduction to greatest odd denominator */
if (mpi_copy(X, &TB) != 0 ||
mpi_lshift(X, ashift) != 0)
goto fail;
fprintf(stderr, "-----------------------|-----------------------\n");
fprintf(stderr, "%04x %08x %08x | %04x %08x %08x\n",
TA.words[2], TA.words[1], TA.words[0],
TB.words[2], TB.words[1], TB.words[0]);
X->neg = 0;
return (0);
fail:
mpi_destroy(&TA);
mpi_destroy(&TB);
return (-1);
}

1
t/.gitignore vendored
View file

@ -32,6 +32,7 @@
/t_mpi
/t_mpi_addsub
/t_mpi_compar
/t_mpi_gcd
/t_mpi_muldiv
/t_murmur3_32
/t_oath

3
t/Makefile.am
View file

@ -210,10 +210,11 @@ endif CRYB_MAC
# libcryb-mpi
if CRYB_MPI
TESTS += t_mpi t_mpi_addsub t_mpi_compar t_mpi_muldiv
TESTS += t_mpi t_mpi_addsub t_mpi_compar t_mpi_gcd t_mpi_muldiv
t_mpi_LDADD = $(libt) $(libmpi)
t_mpi_addsub_LDADD = $(libt) $(libmpi)
t_mpi_compar_LDADD = $(libt) $(libmpi)
t_mpi_gcd_LDADD = $(libt) $(libmpi)
t_mpi_muldiv_LDADD = $(libt) $(libmpi)
noinst_HEADER = t_mpi.h
endif CRYB_MPI

25
t/t_mpi.h
View file

@ -76,6 +76,7 @@ static int t_mpi_not_grown(const cryb_mpi *) CRYB_UNUSED;
static int t_mpi_grown(const cryb_mpi *) CRYB_UNUSED;
static int t_mpi_is_zero(const cryb_mpi *) CRYB_UNUSED;
static int t_compare_mpi(const cryb_mpi *, const cryb_mpi *) CRYB_UNUSED;
static int t_compare_mpi_u32(uint32_t, const cryb_mpi *) CRYB_UNUSED;
/*
* Verify that an MPI has never grown.
@ -154,6 +155,30 @@ t_compare_mpi(const cryb_mpi *e, const cryb_mpi *x)
return (ret);
}
/*
* Verify that an MPI has the expected value
*/
static int
t_compare_mpi_u32(uint32_t e, const cryb_mpi *x)
{
int ret = 1;
if (x == NULL)
return (0);
if (x->words == NULL) {
t_printv("uninitialized MPI\n");
return (0);
}
if (x->neg || x->msb > 32 || x->words[0] != e) {
t_printv("expected +%08x\n", e);
t_printv("received ");
t_printv_mpi(x);
t_printv("\n");
ret = 0;
}
return (ret);
}
/***************************************************************************
* Boilerplate

256
t/t_mpi_gcd.c Normal file
View file

@ -0,0 +1,256 @@
/*
* Copyright (c) 2017 Dag-Erling Smørgrav
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. The name of the author may not be used to endorse or promote
* products derived from this software without specific prior written
* permission.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include "cryb/impl.h"
#include <sys/types.h>
#include <stdint.h>
#include <string.h>
#include <unistd.h>
#include <cryb/endian.h>
#include <cryb/mpi.h>
#include <cryb/test.h>
#include "t_mpi.h"
/***************************************************************************
* Test cases
*/
static struct t_gcd_case {
const char *desc;
uint8_t a[16];
size_t amsb;
int aneg:1;
uint8_t b[16];
size_t bmsb;
int bneg:1;
uint8_t e[16];
size_t emsb;
int eneg:1;
} t_gcd_cases[] = {
{
"gcd(0, 0) == 0",
{ }, 0, 0,
{ }, 0, 0,
{ }, 0, 0,
},
{
"gcd(0x20140901, 0x20140901) == 0x20140901",
{ 0x20, 0x14, 0x09, 0x01 }, 30, 0,
{ 0x20, 0x14, 0x09, 0x01 }, 30, 0,
{ 0x20, 0x14, 0x09, 0x01 }, 30, 0,
},
/* exercise both sides of the 0 check */
{
"gcd(0x20140901, 0) == 0",
{ 0x20, 0x14, 0x09, 0x01 }, 30, 0,
{ }, 0, 0,
{ }, 0, 0,
},
{
"gcd(0, 0x20140901) == 0",
{ }, 0, 0,
{ 0x20, 0x14, 0x09, 0x01 }, 30, 0,
{ }, 0, 0,
},
/* exercise both sides of the parity reduction */
{
"gcd(512, 2048) == 512",
{ 0x02, 0x00 }, 10, 0,
{ 0x08, 0x00 }, 12, 0,
{ 0x02, 0x00 }, 10, 0,
},
{
"gcd(2048, 512) == 512",
{ 0x08, 0x00 }, 12, 0,
{ 0x02, 0x00 }, 10, 0,
{ 0x02, 0x00 }, 10, 0,
},
/* HAC 14.55 */
{
"gcd(1764, 868) == 28",
{ 0x06, 0xe4 }, 11, 0,
{ 0x03, 0x64 }, 10, 0,
{ 0x1c }, 5, 0,
},
/* HAC 14.60 (small coprimes) */
{
"gcd(768454923, 542167814) == 1",
{ 0x2d, 0xcd, 0xb1, 0x0b }, 30, 0,
{ 0x20, 0x50, 0xd3, 0x06 }, 30, 0,
{ 0x01 }, 1, 0,
},
/* largish primes */
{
"gcd(83464029052373107673, 52563826518442013507) == 1",
{ 0x04, 0x86, 0x4b, 0xcc, 0x36, 0x35, 0x95, 0x03, 0xd9 }, 59, 0,
{ 0x02, 0xd9, 0x78, 0x39, 0xb9, 0x32, 0xfe, 0x73, 0x43 }, 58, 0,
{ 0x01 }, 1, 0,
},
/*
* Two examples of a pathological case for Stein's algorithm:
* gcd(k * n, k) where k and n are prime and n >> k. The code
* goes into a loop which makes the absolute minimum amount of
* progress per iteration:
*
* gcd(k * n, k) = gcd(k * (n - 1) / 2, k)
* = gcd(k * ((n - 1) / 2 - 1) / 2, k)
* = ...
*
* for approximately log2(k * n) iterations.
*/
{
"gcd(2^61-1, 1) == 1",
{ 0x1f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff }, 61, 0,
{ 0x01 }, 1, 0,
{ 0x01 }, 1, 0,
},
{
"gcd((2^61-1)*3, 3) == 3",
{ 0x5f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfd }, 63, 0,
{ 0x03 }, 2, 0,
{ 0x03 }, 2, 0,
},
};
/***************************************************************************
* Test functions
*/
static int
t_mpi_gcd_tc(char **desc CRYB_UNUSED, void *arg)
{
struct t_gcd_case *tc = arg;
cryb_mpi a = CRYB_MPI_ZERO, b = CRYB_MPI_ZERO, e = CRYB_MPI_ZERO;
cryb_mpi x = CRYB_MPI_ZERO;
int ret = 1;
mpi_load(&a, tc->a, (tc->amsb + 7) / 8);
a.neg = tc->aneg;
mpi_load(&b, tc->b, (tc->bmsb + 7) / 8);
b.neg = tc->bneg;
mpi_load(&e, tc->e, (tc->emsb + 7) / 8);
e.neg = tc->eneg;
ret &= t_compare_i(0, mpi_gcd_abs(&x, &a, &b));
ret &= t_compare_mpi(&e, &x);
mpi_destroy(&a);
mpi_destroy(&b);
mpi_destroy(&e);
mpi_destroy(&x);
return (ret);
}
static int
t_mpi_gcd_ident_ab(char **desc CRYB_UNUSED, void *arg CRYB_UNUSED)
{
cryb_mpi a = CRYB_MPI_ZERO, x = CRYB_MPI_ZERO;
int ret = 1;
mpi_set(&a, 0x20140901);
ret &= t_compare_i(0, mpi_gcd_abs(&x, &a, &a));
ret &= t_compare_mpi_u32(0x20140901, &x);
mpi_destroy(&a);
mpi_destroy(&x);
return (ret);
}
static int
t_mpi_gcd_ident_xa(char **desc CRYB_UNUSED, void *arg CRYB_UNUSED)
{
cryb_mpi a = CRYB_MPI_ZERO, b = CRYB_MPI_ZERO;
int ret = 1;
mpi_set(&a, 0x20140901);
mpi_set(&b, 0x20140901);
ret &= t_compare_i(0, mpi_gcd_abs(&a, &a, &b));
ret &= t_compare_mpi_u32(0x20140901, &a);
mpi_destroy(&a);
mpi_destroy(&b);
return (ret);
}
static int
t_mpi_gcd_ident_xb(char **desc CRYB_UNUSED, void *arg CRYB_UNUSED)
{
cryb_mpi a = CRYB_MPI_ZERO, b = CRYB_MPI_ZERO;
int ret = 1;
mpi_set(&a, 0x20140901);
mpi_set(&b, 0x20140901);
ret &= t_compare_i(0, mpi_gcd_abs(&b, &a, &b));
ret &= t_compare_mpi_u32(0x20140901, &b);
mpi_destroy(&a);
mpi_destroy(&b);
return (ret);
}
/***************************************************************************
* Boilerplate
*/
static int
t_prepare(int argc, char *argv[])
{
unsigned int i;
(void)argc;
(void)argv;
t_mpi_prepare();
for (i = 0; i < sizeof t_gcd_cases / sizeof t_gcd_cases[0]; ++i)
t_add_test(t_mpi_gcd_tc, &t_gcd_cases[i],
"%s", t_gcd_cases[i].desc);
t_add_test(t_mpi_gcd_ident_ab, NULL, "x = gcd(a, a)");
t_add_test(t_mpi_gcd_ident_xa, NULL, "a = gcd(a, b)");
t_add_test(t_mpi_gcd_ident_xb, NULL, "b = gcd(a, b)");
return (0);
}
static void
t_cleanup(void)
{
t_mpi_cleanup();
}
int
main(int argc, char *argv[])
{
t_main(t_prepare, t_cleanup, argc, argv);
}