int ec_GFp_simple_set_compressed_coordinates(const EC_GROUP *group, EC_POINT *point, const BIGNUM *x_, int y_bit, BN_CTX *ctx) { BN_CTX *new_ctx = NULL; BIGNUM *tmp1, *tmp2, *x, *y; int ret = 0; ERR_clear_error(); if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) { return 0; } } y_bit = (y_bit != 0); BN_CTX_start(ctx); tmp1 = BN_CTX_get(ctx); tmp2 = BN_CTX_get(ctx); x = BN_CTX_get(ctx); y = BN_CTX_get(ctx); if (y == NULL) { goto err; } /* Recover y. We have a Weierstrass equation * y^2 = x^3 + a*x + b, * so y is one of the square roots of x^3 + a*x + b. */ /* tmp1 := x^3 */ if (!BN_nnmod(x, x_, &group->field, ctx)) { goto err; } if (group->meth->field_decode == 0) { /* field_{sqr,mul} work on standard representation */ if (!group->meth->field_sqr(group, tmp2, x_, ctx) || !group->meth->field_mul(group, tmp1, tmp2, x_, ctx)) { goto err; } } else { if (!BN_mod_sqr(tmp2, x_, &group->field, ctx) || !BN_mod_mul(tmp1, tmp2, x_, &group->field, ctx)) { goto err; } } /* tmp1 := tmp1 + a*x */ if (group->a_is_minus3) { if (!BN_mod_lshift1_quick(tmp2, x, &group->field) || !BN_mod_add_quick(tmp2, tmp2, x, &group->field) || !BN_mod_sub_quick(tmp1, tmp1, tmp2, &group->field)) { goto err; } } else { if (group->meth->field_decode) { if (!group->meth->field_decode(group, tmp2, &group->a, ctx) || !BN_mod_mul(tmp2, tmp2, x, &group->field, ctx)) { goto err; } } else { /* field_mul works on standard representation */ if (!group->meth->field_mul(group, tmp2, &group->a, x, ctx)) { goto err; } } if (!BN_mod_add_quick(tmp1, tmp1, tmp2, &group->field)) { goto err; } } /* tmp1 := tmp1 + b */ if (group->meth->field_decode) { if (!group->meth->field_decode(group, tmp2, &group->b, ctx) || !BN_mod_add_quick(tmp1, tmp1, tmp2, &group->field)) { goto err; } } else { if (!BN_mod_add_quick(tmp1, tmp1, &group->b, &group->field)) { goto err; } } if (!BN_mod_sqrt(y, tmp1, &group->field, ctx)) { unsigned long err = ERR_peek_last_error(); if (ERR_GET_LIB(err) == ERR_LIB_BN && ERR_GET_REASON(err) == BN_R_NOT_A_SQUARE) { ERR_clear_error(); OPENSSL_PUT_ERROR(EC, ec_GFp_simple_set_compressed_coordinates, EC_R_INVALID_COMPRESSED_POINT); } else { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_set_compressed_coordinates, ERR_R_BN_LIB); } goto err; } if (y_bit != BN_is_odd(y)) { if (BN_is_zero(y)) { int kron; kron = BN_kronecker(x, &group->field, ctx); if (kron == -2) { goto err; } if (kron == 1) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_set_compressed_coordinates, EC_R_INVALID_COMPRESSION_BIT); } else { /* BN_mod_sqrt() should have cought this error (not a square) */ OPENSSL_PUT_ERROR(EC, ec_GFp_simple_set_compressed_coordinates, EC_R_INVALID_COMPRESSED_POINT); } goto err; } if (!BN_usub(y, &group->field, y)) { goto err; } } if (y_bit != BN_is_odd(y)) { OPENSSL_PUT_ERROR(EC, ec_GFp_simple_set_compressed_coordinates, ERR_R_INTERNAL_ERROR); goto err; } if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx)) goto err; ret = 1; err: BN_CTX_end(ctx); if (new_ctx != NULL) BN_CTX_free(new_ctx); return ret; }
int test_kron(BIO *bp, BN_CTX *ctx) { BIGNUM *a,*b,*r,*t; int i; int legendre, kronecker; int ret = 0; a = BN_new(); b = BN_new(); r = BN_new(); t = BN_new(); if (a == NULL || b == NULL || r == NULL || t == NULL) goto err; /* We test BN_kronecker(a, b, ctx) just for b odd (Jacobi symbol). * In this case we know that if b is prime, then BN_kronecker(a, b, ctx) * is congruent to $a^{(b-1)/2}$, modulo $b$ (Legendre symbol). * So we generate a random prime b and compare these values * for a number of random a's. (That is, we run the Solovay-Strassen * primality test to confirm that b is prime, except that we * don't want to test whether b is prime but whether BN_kronecker * works.) */ if (!BN_generate_prime(b, 512, 0, NULL, NULL, genprime_cb, NULL)) goto err; b->neg = rand_neg(); putc('\n', stderr); for (i = 0; i < num0; i++) { if (!BN_bntest_rand(a, 512, 0, 0)) goto err; a->neg = rand_neg(); /* t := (|b|-1)/2 (note that b is odd) */ if (!BN_copy(t, b)) goto err; t->neg = 0; if (!BN_sub_word(t, 1)) goto err; if (!BN_rshift1(t, t)) goto err; /* r := a^t mod b */ b->neg=0; if (!BN_mod_exp_recp(r, a, t, b, ctx)) goto err; b->neg=1; if (BN_is_word(r, 1)) legendre = 1; else if (BN_is_zero(r)) legendre = 0; else { if (!BN_add_word(r, 1)) goto err; if (0 != BN_ucmp(r, b)) { fprintf(stderr, "Legendre symbol computation failed\n"); goto err; } legendre = -1; } kronecker = BN_kronecker(a, b, ctx); if (kronecker < -1) goto err; /* we actually need BN_kronecker(a, |b|) */ if (a->neg && b->neg) kronecker = -kronecker; if (legendre != kronecker) { fprintf(stderr, "legendre != kronecker; a = "); BN_print_fp(stderr, a); fprintf(stderr, ", b = "); BN_print_fp(stderr, b); fprintf(stderr, "\n"); goto err; } putc('.', stderr); fflush(stderr); } putc('\n', stderr); fflush(stderr); ret = 1; err: if (a != NULL) BN_free(a); if (b != NULL) BN_free(b); if (r != NULL) BN_free(r); if (t != NULL) BN_free(t); return ret; }
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 *A, *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))) { if (ret != in) BN_free(ret); return NULL; } bn_check_top(ret); return ret; } BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); return(NULL); } if (BN_is_zero(a) || BN_is_one(a)) { if (ret == NULL) ret = BN_new(); if (ret == NULL) goto end; if (!BN_set_word(ret, BN_is_one(a))) { if (ret != in) BN_free(ret); return NULL; } bn_check_top(ret); return ret; } BN_CTX_start(ctx); A = BN_CTX_get(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; /* A = a mod p */ if (!BN_nnmod(A, a, p, ctx)) goto end; /* now write |p| - 1 as 2^e*q where q is odd */ e = 1; while (!BN_is_bit_set(p, e)) e++; /* we'll set q later (if needed) */ 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. (|p|-3)/4 + 1. */ if (!BN_rshift(q, p, 2)) goto end; q->neg = 0; if (!BN_add_word(q, 1)) goto end; if (!BN_mod_exp(ret, A, q, p, ctx)) goto end; err = 0; goto vrfy; } if (e == 2) { /* |p| == 5 (mod 8) * * In this case 2 is always a non-square since * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. * So if a really is a square, then 2*a is a non-square. * Thus for * b := (2*a)^((|p|-5)/8), * i := (2*a)*b^2 * we have * i^2 = (2*a)^((1 + (|p|-5)/4)*2) * = (2*a)^((p-1)/2) * = -1; * so if we set * x := a*b*(i-1), * then * x^2 = a^2 * b^2 * (i^2 - 2*i + 1) * = a^2 * b^2 * (-2*i) * = a*(-i)*(2*a*b^2) * = a*(-i)*i * = a. * * (This is due to A.O.L. Atkin, * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, * November 1992.) */ /* t := 2*a */ if (!BN_mod_lshift1_quick(t, A, p)) goto end; /* b := (2*a)^((|p|-5)/8) */ if (!BN_rshift(q, p, 3)) goto end; q->neg = 0; if (!BN_mod_exp(b, t, q, p, ctx)) goto end; /* y := b^2 */ if (!BN_mod_sqr(y, b, p, ctx)) goto end; /* t := (2*a)*b^2 - 1*/ if (!BN_mod_mul(t, t, y, p, ctx)) goto end; if (!BN_sub_word(t, 1)) goto end; /* x = a*b*t */ if (!BN_mod_mul(x, A, b, p, ctx)) goto end; if (!BN_mod_mul(x, x, t, p, ctx)) goto end; if (!BN_copy(ret, x)) goto end; err = 0; goto vrfy; } /* e > 2, so we really have to use the Tonelli/Shanks algorithm. * First, find some y that is not a square. */ if (!BN_copy(q, p)) goto end; /* use 'q' as temp */ q->neg = 0; i = 2; do { /* For efficiency, try small numbers first; * if this fails, try random numbers. */ if (i < 22) { 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, q, ctx); /* here 'q' is |p| */ 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 < 82); 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; } /* Here's our actual 'q': */ if (!BN_rshift(q, q, e)) 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) */ BN_zero(ret); 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) */ BN_zero(ret); 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 vrfy; } /* 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; } vrfy: if (!err) { /* verify the result -- the input might have been not a square * (test added in 0.9.8) */ if (!BN_mod_sqr(x, ret, p, ctx)) err = 1; if (!err && 0 != BN_cmp(x, A)) { BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); err = 1; } } end: if (err) { if (ret != NULL && ret != in) { BN_clear_free(ret); } ret = NULL; } BN_CTX_end(ctx); bn_check_top(ret); return ret; }
void do_mul_exp(BIGNUM *r, BIGNUM *a, BIGNUM *b, BIGNUM *c, BN_CTX *ctx) { int i,k; double tm; long num; num=BASENUM; for (i=NUM_START; i<NUM_SIZES; i++) { #ifdef C_PRIME # ifdef TEST_SQRT if (!BN_set_word(a, 64)) goto err; if (!BN_set_word(b, P_MOD_64)) goto err; # define ADD a # define REM b # else # define ADD NULL # define REM NULL # endif if (!BN_generate_prime(c,sizes[i],0,ADD,REM,genprime_cb,NULL)) goto err; putc('\n', stderr); fflush(stderr); #endif for (k=0; k<num; k++) { if (k%50 == 0) /* Average over num/50 different choices of random numbers. */ { if (!BN_pseudo_rand(a,sizes[i],1,0)) goto err; if (!BN_pseudo_rand(b,sizes[i],1,0)) goto err; #ifndef C_PRIME if (!BN_pseudo_rand(c,sizes[i],1,1)) goto err; #endif #ifdef TEST_SQRT if (!BN_mod_sqr(a,a,c,ctx)) goto err; if (!BN_mod_sqr(b,b,c,ctx)) goto err; #else if (!BN_nnmod(a,a,c,ctx)) goto err; if (!BN_nnmod(b,b,c,ctx)) goto err; #endif if (k == 0) Time_F(START); } #if defined(TEST_EXP) if (!BN_mod_exp(r,a,b,c,ctx)) goto err; #elif defined(TEST_MUL) { int i = 0; for (i = 0; i < 50; i++) if (!BN_mod_mul(r,a,b,c,ctx)) goto err; } #elif defined(TEST_SQR) { int i = 0; for (i = 0; i < 50; i++) { if (!BN_mod_sqr(r,a,c,ctx)) goto err; if (!BN_mod_sqr(r,b,c,ctx)) goto err; } } #elif defined(TEST_GCD) if (!BN_gcd(r,a,b,ctx)) goto err; if (!BN_gcd(r,b,c,ctx)) goto err; if (!BN_gcd(r,c,a,ctx)) goto err; #elif defined(TEST_KRON) if (-2 == BN_kronecker(a,b,ctx)) goto err; if (-2 == BN_kronecker(b,c,ctx)) goto err; if (-2 == BN_kronecker(c,a,ctx)) goto err; #elif defined(TEST_INV) if (!BN_mod_inverse(r,a,c,ctx)) goto err; if (!BN_mod_inverse(r,b,c,ctx)) goto err; #else /* TEST_SQRT */ if (!BN_mod_sqrt(r,a,c,ctx)) goto err; if (!BN_mod_sqrt(r,b,c,ctx)) goto err; #endif } tm=Time_F(STOP); printf( #if defined(TEST_EXP) "modexp %4d ^ %4d %% %4d" #elif defined(TEST_MUL) "50*modmul %4d %4d %4d" #elif defined(TEST_SQR) "100*modsqr %4d %4d %4d" #elif defined(TEST_GCD) "3*gcd %4d %4d %4d" #elif defined(TEST_KRON) "3*kronecker %4d %4d %4d" #elif defined(TEST_INV) "2*inv %4d %4d mod %4d" #else /* TEST_SQRT */ "2*sqrt [prime == %d (mod 64)] %4d %4d mod %4d" #endif " -> %8.3fms %5.1f (%ld)\n", #ifdef TEST_SQRT P_MOD_64, #endif sizes[i],sizes[i],sizes[i],tm*1000.0/num,tm*mul_c[i]/num, num); num/=7; if (num <= 0) num=1; } return; err: ERR_print_errors_fp(stderr); }