/* 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 #include #include "internal.h" BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { // Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm // (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory", // algorithm 1.5.1). |p| is assumed to be a 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 || !BN_set_word(ret, BN_is_bit_set(a, 0))) { if (ret != in) { BN_free(ret); } return NULL; } return ret; } OPENSSL_PUT_ERROR(BN, 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 || !BN_set_word(ret, BN_is_one(a))) { if (ret != in) { BN_free(ret); } return NULL; } 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) || !BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) { 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, // , // November 1992.) // t := 2*a if (!bn_mod_lshift1_consttime(t, A, p, ctx)) { goto end; } // b := (2*a)^((|p|-5)/8) if (!BN_rshift(q, p, 3)) { goto end; } q->neg = 0; if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) { 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) || !BN_sub_word(t, 1)) { goto end; } // x = a*b*t if (!BN_mod_mul(x, A, b, p, ctx) || !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_jacobi(y, q, ctx); // here 'q' is |p| if (r < -1) { goto end; } if (r == 0) { // m divides p OPENSSL_PUT_ERROR(BN, 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. OPENSSL_PUT_ERROR(BN, 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_mont(y, y, q, p, ctx, NULL)) { goto end; } if (BN_is_one(y)) { OPENSSL_PUT_ERROR(BN, 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_mont(x, A, t, p, ctx, NULL)) { 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) || !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) { OPENSSL_PUT_ERROR(BN, 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) || !BN_mod_mul(x, x, t, p, ctx) || !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)) { OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); err = 1; } } end: if (err) { if (ret != in) { BN_clear_free(ret); } ret = NULL; } BN_CTX_end(ctx); return ret; } int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) { BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2; int ok = 0, last_delta_valid = 0; if (in->neg) { OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER); return 0; } if (BN_is_zero(in)) { BN_zero(out_sqrt); return 1; } BN_CTX_start(ctx); if (out_sqrt == in) { estimate = BN_CTX_get(ctx); } else { estimate = out_sqrt; } tmp = BN_CTX_get(ctx); last_delta = BN_CTX_get(ctx); delta = BN_CTX_get(ctx); if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) { OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE); goto err; } // We estimate that the square root of an n-bit number is 2^{n/2}. if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) { goto err; } // This is Newton's method for finding a root of the equation |estimate|^2 - // |in| = 0. for (;;) { // |estimate| = 1/2 * (|estimate| + |in|/|estimate|) if (!BN_div(tmp, NULL, in, estimate, ctx) || !BN_add(tmp, tmp, estimate) || !BN_rshift1(estimate, tmp) || // |tmp| = |estimate|^2 !BN_sqr(tmp, estimate, ctx) || // |delta| = |in| - |tmp| !BN_sub(delta, in, tmp)) { OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB); goto err; } delta->neg = 0; // The difference between |in| and |estimate| squared is required to always // decrease. This ensures that the loop always terminates, but I don't have // a proof that it always finds the square root for a given square. if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) { break; } last_delta_valid = 1; tmp2 = last_delta; last_delta = delta; delta = tmp2; } if (BN_cmp(tmp, in) != 0) { OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); goto err; } ok = 1; err: if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) { ok = 0; } BN_CTX_end(ctx); return ok; }