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BN_sqrt

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This commit is contained in:
Bodo Möller 2000-11-30 00:18:19 +00:00
parent 06676624fc
commit cd2eebfd64
8 changed files with 438 additions and 21 deletions
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4
crypto/bn/Makefile.ssl
View File

@ -37,12 +37,12 @@ APPS=
LIB=$(TOP)/libcrypto.a
LIBSRC= bn_add.c bn_div.c bn_exp.c bn_lib.c bn_ctx.c bn_mul.c bn_mod.c \
bn_print.c bn_rand.c bn_shift.c bn_word.c bn_blind.c \
bn_kron.c bn_gcd.c bn_prime.c bn_err.c bn_sqr.c bn_asm.c \
bn_kron.c bn_sqrt.c bn_gcd.c bn_prime.c bn_err.c bn_sqr.c bn_asm.c \
bn_recp.c bn_mont.c bn_mpi.c bn_exp2.c
LIBOBJ= bn_add.o bn_div.o bn_exp.o bn_lib.o bn_ctx.o bn_mul.o bn_mod.o \
bn_print.o bn_rand.o bn_shift.o bn_word.o bn_blind.o \
bn_kron.o bn_gcd.o bn_prime.o bn_err.o bn_sqr.o $(BN_ASM) \
bn_kron.o bn_sqrt.o bn_gcd.o bn_prime.o bn_err.o bn_sqr.o $(BN_ASM) \
bn_recp.o bn_mont.o bn_mpi.o bn_exp2.o
SRC= $(LIBSRC)

9
crypto/bn/bn.h
View File

@ -238,7 +238,7 @@ typedef struct bignum_st
} BIGNUM;
/* Used for temp variables */
#define BN_CTX_NUM 16
#define BN_CTX_NUM 20
#define BN_CTX_NUM_POS 12
typedef struct bignum_ctx
{
@ -357,6 +357,7 @@ int BN_mod_sub(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *m, BN_
int BN_mod_sub_quick(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *m);
int BN_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
const BIGNUM *m, BN_CTX *ctx);
int BN_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);
int BN_mod_lshift1(BIGNUM *r, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);
int BN_mod_lshift1_quick(BIGNUM *r, const BIGNUM *a, const BIGNUM *m);
int BN_mod_lshift(BIGNUM *r, const BIGNUM *a, int n, const BIGNUM *m, BN_CTX *ctx);
@ -414,6 +415,8 @@ int BN_gcd(BIGNUM *r,const BIGNUM *a,const BIGNUM *b,BN_CTX *ctx);
int BN_kronecker(const BIGNUM *a,const BIGNUM *b,BN_CTX *ctx); /* returns -2 for error */
BIGNUM *BN_mod_inverse(BIGNUM *ret,
const BIGNUM *a, const BIGNUM *n,BN_CTX *ctx);
BIGNUM *BN_mod_sqrt(BIGNUM *ret,
const BIGNUM *a, const BIGNUM *n,BN_CTX *ctx);
BIGNUM *BN_generate_prime(BIGNUM *ret,int bits,int safe,
const BIGNUM *add, const BIGNUM *rem,
void (*callback)(int,int,void *),void *cb_arg);
@ -517,6 +520,7 @@ void bn_dump1(FILE *o, const char *a, const BN_ULONG *b,int n);
#define BN_F_BN_MOD_INVERSE 110
#define BN_F_BN_MOD_LSHIFT_QUICK 119
#define BN_F_BN_MOD_MUL_RECIPROCAL 111
#define BN_F_BN_MOD_SQRT 121
#define BN_F_BN_MPI2BN 112
#define BN_F_BN_NEW 113
#define BN_F_BN_RAND 114
@ -531,8 +535,11 @@ void bn_dump1(FILE *o, const char *a, const BN_ULONG *b,int n);
#define BN_R_EXPAND_ON_STATIC_BIGNUM_DATA 105
#define BN_R_INPUT_NOT_REDUCED 110
#define BN_R_INVALID_LENGTH 106
#define BN_R_NOT_A_SQUARE 111
#define BN_R_NOT_INITIALIZED 107
#define BN_R_NO_INVERSE 108
#define BN_R_P_IS_NOT_PRIME 112
#define BN_R_TOO_MANY_ITERATIONS 113
#define BN_R_TOO_MANY_TEMPORARY_VARIABLES 109
#ifdef __cplusplus

4
crypto/bn/bn_err.c
View File

@ -83,6 +83,7 @@ static ERR_STRING_DATA BN_str_functs[]=
{ERR_PACK(0,BN_F_BN_MOD_INVERSE,0), "BN_mod_inverse"},
{ERR_PACK(0,BN_F_BN_MOD_LSHIFT_QUICK,0), "BN_mod_lshift_quick"},
{ERR_PACK(0,BN_F_BN_MOD_MUL_RECIPROCAL,0), "BN_mod_mul_reciprocal"},
{ERR_PACK(0,BN_F_BN_MOD_SQRT,0), "BN_mod_sqrt"},
{ERR_PACK(0,BN_F_BN_MPI2BN,0), "BN_mpi2bn"},
{ERR_PACK(0,BN_F_BN_NEW,0), "BN_new"},
{ERR_PACK(0,BN_F_BN_RAND,0), "BN_rand"},
@ -100,8 +101,11 @@ static ERR_STRING_DATA BN_str_reasons[]=
{BN_R_EXPAND_ON_STATIC_BIGNUM_DATA ,"expand on static bignum data"},
{BN_R_INPUT_NOT_REDUCED ,"input not reduced"},
{BN_R_INVALID_LENGTH ,"invalid length"},
{BN_R_NOT_A_SQUARE ,"not a square"},
{BN_R_NOT_INITIALIZED ,"not initialized"},
{BN_R_NO_INVERSE ,"no inverse"},
{BN_R_P_IS_NOT_PRIME ,"p is not prime"},
{BN_R_TOO_MANY_ITERATIONS ,"too many iterations"},
{BN_R_TOO_MANY_TEMPORARY_VARIABLES ,"too many temporary variables"},
{0,NULL}
};

39
crypto/bn/bn_exp.c
View File

@ -205,6 +205,8 @@ int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
if (a->top == 1 && !a->neg)
{
BN_ULONG A = a->d[0];
if (m->top == 1)
A %= m->d[0]; /* make sure that A is reduced */
ret=BN_mod_exp_mont_word(r,A,p,m,ctx,NULL);
}
else
@ -235,8 +237,13 @@ int BN_mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
if (bits == 0)
{
BN_one(r);
return(1);
ret = BN_one(r);
return ret;
}
if (BN_is_zero(a))
{
ret = BN_zero(r);
return ret;
}
BN_CTX_start(ctx);
@ -355,8 +362,13 @@ int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
bits=BN_num_bits(p);
if (bits == 0)
{
BN_one(rr);
return(1);
ret = BN_one(rr);
return ret;
}
if (BN_is_zero(a))
{
ret = BN_zero(rr);
return ret;
}
BN_CTX_start(ctx);
d = BN_CTX_get(ctx);
@ -500,9 +512,15 @@ int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p,
bits = BN_num_bits(p);
if (bits == 0)
{
BN_one(rr);
return(1);
ret = BN_one(rr);
return ret;
}
if (a == 0)
{
ret = BN_zero(rr);
return ret;
}
BN_CTX_start(ctx);
d = BN_CTX_get(ctx);
r = BN_CTX_get(ctx);
@ -611,8 +629,13 @@ int BN_mod_exp_simple(BIGNUM *r,
if (bits == 0)
{
BN_one(r);
return(1);
ret = BN_one(r);
return ret;
}
if (BN_is_zero(a))
{
ret = BN_one(r);
return ret;
}
BN_CTX_start(ctx);

10
crypto/bn/bn_exp2.c
View File

@ -141,9 +141,15 @@ int BN_mod_exp2_mont(BIGNUM *rr, const BIGNUM *a1, const BIGNUM *p1,
bits2=BN_num_bits(p2);
if ((bits1 == 0) && (bits2 == 0))
{
BN_one(rr);
return(1);
ret = BN_one(rr);
return ret;
}
if (BN_is_zero(a1) || BN_is_zero(a2))
{
ret = BN_zero(rr);
return ret;
}
bits=(bits1 > bits2)?bits1:bits2;
BN_CTX_start(ctx);

2
crypto/bn/bn_kron.c
View File

@ -1,5 +1,3 @@
/* totally untested */
/* crypto/bn/bn_kron.c */
/* ====================================================================
* Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved.

309
crypto/bn/bn_sqrt.c
View File

@ -1 +1,308 @@
XXX
/* crypto/bn/bn_mod.c */
/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
* and Bodo Moeller for the OpenSSL project. */
/* ====================================================================
* Copyright (c) 1998-2000 The OpenSSL Project. 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. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* openssl-core@openssl.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED 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 OpenSSL PROJECT OR
* ITS 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.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com).
*
*/
#include "cryptlib.h"
#include "bn_lcl.h"
BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
/* Returns 'ret' such that
* ret^2 == a (mod p),
* using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
* in Algebraic Computational Number Theory", algorithm 1.5.1).
* 'p' must be prime!
*/
{
BIGNUM *ret = in;
int err = 1;
int r;
BIGNUM *b, *q, *t, *x, *y;
int e, i, j;
if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
{
if (BN_abs_is_word(p, 2))
{
if (ret == NULL)
ret = BN_new();
if (ret == NULL)
goto end;
if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
{
BN_free(ret);
return NULL;
}
return ret;
}
BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
return(NULL);
}
#if 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
r = BN_kronecker(a, p, ctx);
if (r < -1) return NULL;
if (r == -1)
{
BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
return(NULL);
}
#endif
BN_CTX_start(ctx);
b = BN_CTX_get(ctx);
q = BN_CTX_get(ctx);
t = BN_CTX_get(ctx);
x = BN_CTX_get(ctx);
y = BN_CTX_get(ctx);
if (y == NULL) goto end;
if (ret == NULL)
ret = BN_new();
if (ret == NULL) goto end;
/* now write |p| - 1 as 2^e*q where q is odd */
e = 1;
while (!BN_is_bit_set(p, e))
e++;
if (!BN_rshift(q, p, e)) goto end;
q->neg = 0;
if (e == 1)
{
/* The easy case: (p-1)/2 is odd, so 2 has an inverse
* modulo (p-1)/2, and square roots can be computed
* directly by modular exponentiation.
* We have
* 2 * (p+1)/4 == 1 (mod (p-1)/2),
* so we can use exponent (p+1)/4, i.e. (q+1)/2.
*/
if (!BN_add_word(q,1)) goto end;
if (!BN_rshift1(q,q)) goto end;
if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
err = 0;
goto end;
}
/* e > 1, so we really have to use the Tonelli/Shanks algorithm.
* First, find some y that is not a square. */
i = 1;
do
{
/* For efficiency, try small numbers first;
* if this fails, try random numbers.
*/
if (i < 20)
{
if (!BN_set_word(y, i)) goto end;
}
else
{
if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
if (BN_ucmp(y, p) >= 0)
{
if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
}
/* now 0 <= y < |p| */
if (BN_is_zero(y))
if (!BN_set_word(y, i)) goto end;
}
r = BN_kronecker(y, p, ctx);
if (r < -1) goto end;
if (r == 0)
{
/* m divides p */
BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
goto end;
}
}
while (r == 1 && i++ < 80);
if (r != -1)
{
/* Many rounds and still no non-square -- this is more likely
* a bug than just bad luck.
* Even if p is not prime, we should have found some y
* such that r == -1.
*/
BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
goto end;
}
/* Now that we have some non-square, we can find an element
* of order 2^e by computing its q'th power. */
if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
if (BN_is_one(y))
{
BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
goto end;
}
/* Now we know that (if p is indeed prime) there is an integer
* k, 0 <= k < 2^e, such that
*
* a^q * y^k == 1 (mod p).
*
* As a^q is a square and y is not, k must be even.
* q+1 is even, too, so there is an element
*
* X := a^((q+1)/2) * y^(k/2),
*
* and it satisfies
*
* X^2 = a^q * a * y^k
* = a,
*
* so it is the square root that we are looking for.
*/
/* t := (q-1)/2 (note that q is odd) */
if (!BN_rshift1(t, q)) goto end;
/* x := a^((q-1)/2) */
if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
{
if (!BN_nnmod(t, a, p, ctx)) goto end;
if (BN_is_zero(t))
{
/* special case: a == 0 (mod p) */
if (!BN_zero(ret)) goto end;
err = 0;
goto end;
}
else
if (!BN_one(x)) goto end;
}
else
{
if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
if (BN_is_zero(x))
{
/* special case: a == 0 (mod p) */
if (!BN_zero(ret)) goto end;
err = 0;
goto end;
}
}
/* b := a*x^2 (= a^q) */
if (!BN_mod_sqr(b, x, p, ctx)) goto end;
if (!BN_mod_mul(b, b, a, p, ctx)) goto end;
/* x := a*x (= a^((q+1)/2)) */
if (!BN_mod_mul(x, x, a, p, ctx)) goto end;
while (1)
{
/* Now b is a^q * y^k for some even k (0 <= k < 2^E
* where E refers to the original value of e, which we
* don't keep in a variable), and x is a^((q+1)/2) * y^(k/2).
*
* We have a*b = x^2,
* y^2^(e-1) = -1,
* b^2^(e-1) = 1.
*/
if (BN_is_one(b))
{
if (!BN_copy(ret, x)) goto end;
err = 0;
goto end;
}
/* find smallest i such that b^(2^i) = 1 */
i = 1;
if (!BN_mod_sqr(t, b, p, ctx)) goto end;
while (!BN_is_one(t))
{
i++;
if (i == e)
{
BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
goto end;
}
if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
}
/* t := y^2^(e - i - 1) */
if (!BN_copy(t, y)) goto end;
for (j = e - i - 1; j > 0; j--)
{
if (!BN_mod_sqr(t, t, p, ctx)) goto end;
}
if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
e = i;
}
end:
if (err)
{
if (ret != NULL && ret != in)
{
BN_clear_free(ret);
}
ret = NULL;
}
BN_CTX_end(ctx);
return ret;
}

82
crypto/bn/bntest.c
View File

@ -92,6 +92,7 @@ int test_mod_mul(BIO *bp,BN_CTX *ctx);
int test_mod_exp(BIO *bp,BN_CTX *ctx);
int test_exp(BIO *bp,BN_CTX *ctx);
int test_kron(BIO *bp,BN_CTX *ctx);
int test_sqrt(BIO *bp,BN_CTX *ctx);
int rand_neg(void);
static int results=0;
@ -233,6 +234,10 @@ int main(int argc, char *argv[])
if (!test_kron(out,ctx)) goto err;
BIO_flush(out);
message(out,"BN_mod_sqrt");
if (!test_sqrt(out,ctx)) goto err;
BIO_flush(out);
BN_CTX_free(ctx);
BIO_free(out);
@ -940,11 +945,6 @@ int test_kron(BIO *bp, BN_CTX *ctx)
if (!BN_generate_prime(b, 512, 0, NULL, NULL, genprime_cb, NULL)) goto err;
putc('\n', stderr);
if (1 != BN_is_prime(b, 10, NULL, ctx, NULL))
{
fprintf(stderr, "BN_is_prime failed\n");
goto err;
}
for (i = 0; i < num0; i++)
{
@ -998,6 +998,78 @@ int test_kron(BIO *bp, BN_CTX *ctx)
return ret;
}
int test_sqrt(BIO *bp, BN_CTX *ctx)
{
BIGNUM *a,*p,*r;
int i, j;
int ret = 0;
a = BN_new();
p = BN_new();
r = BN_new();
if (a == NULL || p == NULL || r == NULL) goto err;
for (i = 0; i < 16; i++)
{
if (i < 8)
{
unsigned primes[8] = { 2, 3, 7, 11, 13, 17, 19 };
if (!BN_set_word(p, primes[i])) goto err;
}
else
{
if (!BN_set_word(a, 32)) goto err;
if (!BN_set_word(r, 2*i + 1)) goto err;
if (!BN_generate_prime(p, 256, 0, a, r, genprime_cb, NULL)) goto err;
putc('\n', stderr);
}
for (j = 0; j < num2; j++)
{
/* construct 'a' such that it is a square modulo p,
* but in general not a proper square and not reduced modulo p */
if (!BN_rand(r, 256, 0, 3)) goto err;
if (!BN_nnmod(r, r, p, ctx)) goto err;
if (!BN_mod_sqr(r, r, p, ctx)) goto err;
if (!BN_rand(a, 256, 0, 3)) goto err;
if (!BN_nnmod(a, a, p, ctx)) goto err;
if (!BN_mod_sqr(a, a, p, ctx)) goto err;
if (!BN_mul(a, a, r, ctx)) goto err;
if (!BN_mod_sqrt(r, a, p, ctx)) goto err;
if (!BN_mod_sqr(r, r, p, ctx)) goto err;
if (!BN_nnmod(a, a, p, ctx)) goto err;
if (BN_cmp(a, r) != 0)
{
fprintf(stderr, "BN_mod_sqrt failed: a = ");
BN_print_fp(stderr, a);
fprintf(stderr, ", r = ");
BN_print_fp(stderr, r);
fprintf(stderr, ", p = ");
BN_print_fp(stderr, p);
fprintf(stderr, "\n");
goto err;
}
putc('.', stderr);
fflush(stderr);
}
putc('\n', stderr);
fflush(stderr);
}
ret = 1;
err:
if (a != NULL) BN_free(a);
if (p != NULL) BN_free(p);
if (r != NULL) BN_free(r);
return ret;
}
int test_lshift(BIO *bp,BN_CTX *ctx,BIGNUM *a_)
{
BIGNUM *a,*b,*c,*d;