diff options
Diffstat (limited to 'ics-openvpn-stripped/main/openssl/crypto/bn/bn_sqrt.c')
-rw-r--r-- | ics-openvpn-stripped/main/openssl/crypto/bn/bn_sqrt.c | 393 |
1 files changed, 393 insertions, 0 deletions
diff --git a/ics-openvpn-stripped/main/openssl/crypto/bn/bn_sqrt.c b/ics-openvpn-stripped/main/openssl/crypto/bn/bn_sqrt.c new file mode 100644 index 00000000..6beaf9e5 --- /dev/null +++ b/ics-openvpn-stripped/main/openssl/crypto/bn/bn_sqrt.c @@ -0,0 +1,393 @@ +/* crypto/bn/bn_sqrt.c */ +/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> + * 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 *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; + } |