From cd2eebfd6497ac1ca4d450960ba843f14643a9b2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Bodo=20M=C3=B6ller?= Date: Thu, 30 Nov 2000 00:18:19 +0000 Subject: BN_sqrt --- crypto/bn/Makefile.ssl | 4 +- crypto/bn/bn.h | 9 +- crypto/bn/bn_err.c | 4 + crypto/bn/bn_exp.c | 39 +++++-- crypto/bn/bn_exp2.c | 10 +- crypto/bn/bn_kron.c | 2 - crypto/bn/bn_sqrt.c | 309 ++++++++++++++++++++++++++++++++++++++++++++++++- crypto/bn/bntest.c | 82 ++++++++++++- 8 files changed, 438 insertions(+), 21 deletions(-) diff --git a/crypto/bn/Makefile.ssl b/crypto/bn/Makefile.ssl index b32fd97c41..962cda9bae 100644 --- a/crypto/bn/Makefile.ssl +++ b/crypto/bn/Makefile.ssl @@ -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) diff --git a/crypto/bn/bn.h b/crypto/bn/bn.h index c81f3de8be..b55ac95642 100644 --- a/crypto/bn/bn.h +++ b/crypto/bn/bn.h @@ -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 diff --git a/crypto/bn/bn_err.c b/crypto/bn/bn_err.c index 75a3458b11..afb9320322 100644 --- a/crypto/bn/bn_err.c +++ b/crypto/bn/bn_err.c @@ -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} }; diff --git a/crypto/bn/bn_exp.c b/crypto/bn/bn_exp.c index 35ab56efc0..51c8282593 100644 --- a/crypto/bn/bn_exp.c +++ b/crypto/bn/bn_exp.c @@ -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); diff --git a/crypto/bn/bn_exp2.c b/crypto/bn/bn_exp2.c index 70c4d83a79..56f1c959bd 100644 --- a/crypto/bn/bn_exp2.c +++ b/crypto/bn/bn_exp2.c @@ -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); diff --git a/crypto/bn/bn_kron.c b/crypto/bn/bn_kron.c index 0dd8a194cb..49f75594ae 100644 --- a/crypto/bn/bn_kron.c +++ b/crypto/bn/bn_kron.c @@ -1,5 +1,3 @@ -/* totally untested */ - /* crypto/bn/bn_kron.c */ /* ==================================================================== * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. diff --git a/crypto/bn/bn_sqrt.c b/crypto/bn/bn_sqrt.c index dd6d86a43d..5176772e4e 100644 --- a/crypto/bn/bn_sqrt.c +++ b/crypto/bn/bn_sqrt.c @@ -1 +1,308 @@ -XXX +/* crypto/bn/bn_mod.c */ +/* Written by Lenka Fibikova + * 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; + } diff --git a/crypto/bn/bntest.c b/crypto/bn/bntest.c index a664f3b91a..8f5b72c7f2 100644 --- a/crypto/bn/bntest.c +++ b/crypto/bn/bntest.c @@ -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; -- cgit v1.2.3