| /* |
| * Copyright 2000-2022 The OpenSSL Project Authors. All Rights Reserved. |
| * |
| * Licensed under the Apache License 2.0 (the "License"). You may not use |
| * this file except in compliance with the License. You can obtain a copy |
| * in the file LICENSE in the source distribution or at |
| * https://www.openssl.org/source/license.html |
| */ |
| |
| #include "internal/cryptlib.h" |
| #include "bn_local.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, otherwise an error or |
| * an incorrect "result" will be returned. |
| */ |
| { |
| BIGNUM *ret = in; |
| int err = 1; |
| int r; |
| BIGNUM *A, *b, *q, *t, *x, *y; |
| int e, i, j; |
| int used_ctx = 0; |
| |
| 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; |
| } |
| |
| ERR_raise(ERR_LIB_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) |
| 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); |
| used_ctx = 1; |
| 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, |
| * Subject: Square Roots and Cognate Matters modulo p=8n+5. |
| * URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026 |
| * 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_priv_rand_ex(y, BN_num_bits(p), 0, 0, 0, ctx)) |
| 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 */ |
| ERR_raise(ERR_LIB_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. |
| */ |
| ERR_raise(ERR_LIB_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(y, y, q, p, ctx)) |
| goto end; |
| if (BN_is_one(y)) { |
| ERR_raise(ERR_LIB_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(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 the smallest i, 0 < i < e, such that b^(2^i) = 1. */ |
| for (i = 1; i < e; i++) { |
| if (i == 1) { |
| if (!BN_mod_sqr(t, b, p, ctx)) |
| goto end; |
| |
| } else { |
| if (!BN_mod_mul(t, t, t, p, ctx)) |
| goto end; |
| } |
| if (BN_is_one(t)) |
| break; |
| } |
| /* If not found, a is not a square or p is not prime. */ |
| if (i >= e) { |
| ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE); |
| 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)) { |
| ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE); |
| err = 1; |
| } |
| } |
| |
| end: |
| if (err) { |
| if (ret != in) |
| BN_clear_free(ret); |
| ret = NULL; |
| } |
| if (used_ctx) |
| BN_CTX_end(ctx); |
| bn_check_top(ret); |
| return ret; |
| } |
