/* * Copyright 2000-2021 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; } 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); 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(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 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) { BNerr(BN_F_BN_MOD_SQRT, 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)) { BNerr(BN_F_BN_MOD_SQRT, 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; }