/* crypto/bn/bn_gcd.c */ /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com) * All rights reserved. * * This package is an SSL implementation written * by Eric Young (eay@cryptsoft.com). * The implementation was written so as to conform with Netscapes SSL. * * This library is free for commercial and non-commercial use as long as * the following conditions are aheared to. The following conditions * apply to all code found in this distribution, be it the RC4, RSA, * lhash, DES, etc., code; not just the SSL code. The SSL documentation * included with this distribution is covered by the same copyright terms * except that the holder is Tim Hudson (tjh@cryptsoft.com). * * Copyright remains Eric Young's, and as such any Copyright notices in * the code are not to be removed. * If this package is used in a product, Eric Young should be given attribution * as the author of the parts of the library used. * This can be in the form of a textual message at program startup or * in documentation (online or textual) provided with the package. * * 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 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 acknowledgement: * "This product includes cryptographic software written by * Eric Young (eay@cryptsoft.com)" * The word 'cryptographic' can be left out if the rouines from the library * being used are not cryptographic related :-). * 4. If you include any Windows specific code (or a derivative thereof) from * the apps directory (application code) you must include an acknowledgement: * "This product includes software written by Tim Hudson (tjh@cryptsoft.com)" * * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND * ANY EXPRESS 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 AUTHOR OR 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. * * The licence and distribution terms for any publically available version or * derivative of this code cannot be changed. i.e. this code cannot simply be * copied and put under another distribution licence * [including the GNU Public Licence.] */ /* ==================================================================== * Copyright (c) 1998-2001 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" static BIGNUM *euclid(BIGNUM *a, BIGNUM *b); int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) { BIGNUM *a,*b,*t; int ret=0; bn_check_top(in_a); bn_check_top(in_b); BN_CTX_start(ctx); a = BN_CTX_get(ctx); b = BN_CTX_get(ctx); if (a == NULL || b == NULL) goto err; if (BN_copy(a,in_a) == NULL) goto err; if (BN_copy(b,in_b) == NULL) goto err; a->neg = 0; b->neg = 0; if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; } t=euclid(a,b); if (t == NULL) goto err; if (BN_copy(r,t) == NULL) goto err; ret=1; err: BN_CTX_end(ctx); return(ret); } static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) { BIGNUM *t; int shifts=0; bn_check_top(a); bn_check_top(b); /* 0 <= b <= a */ while (!BN_is_zero(b)) { /* 0 < b <= a */ if (BN_is_odd(a)) { if (BN_is_odd(b)) { if (!BN_sub(a,a,b)) goto err; if (!BN_rshift1(a,a)) goto err; if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; } } else /* a odd - b even */ { if (!BN_rshift1(b,b)) goto err; if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; } } } else /* a is even */ { if (BN_is_odd(b)) { if (!BN_rshift1(a,a)) goto err; if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; } } else /* a even - b even */ { if (!BN_rshift1(a,a)) goto err; if (!BN_rshift1(b,b)) goto err; shifts++; } } /* 0 <= b <= a */ } if (shifts) { if (!BN_lshift(a,a,shifts)) goto err; } return(a); err: return(NULL); } /* solves ax == 1 (mod n) */ BIGNUM *BN_mod_inverse(BIGNUM *in, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; BIGNUM *ret=NULL; int sign; bn_check_top(a); bn_check_top(n); BN_CTX_start(ctx); A = BN_CTX_get(ctx); B = BN_CTX_get(ctx); X = BN_CTX_get(ctx); D = BN_CTX_get(ctx); M = BN_CTX_get(ctx); Y = BN_CTX_get(ctx); T = BN_CTX_get(ctx); if (T == NULL) goto err; if (in == NULL) R=BN_new(); else R=in; if (R == NULL) goto err; BN_one(X); BN_zero(Y); if (BN_copy(B,a) == NULL) goto err; if (BN_copy(A,n) == NULL) goto err; A->neg = 0; if (B->neg || (BN_ucmp(B, A) >= 0)) { if (!BN_nnmod(B, B, A, ctx)) goto err; } sign = -1; /* From B = a mod |n|, A = |n| it follows that * * 0 <= B < A, * -sign*X*a == B (mod |n|), * sign*Y*a == A (mod |n|). */ if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) { /* Binary inversion algorithm; requires odd modulus. * This is faster than the general algorithm if the modulus * is sufficiently small (about 400 .. 500 bits on 32-bit * sytems, but much more on 64-bit systems) */ int shift; while (!BN_is_zero(B)) { /* * 0 < B < |n|, * 0 < A <= |n|, * (1) -sign*X*a == B (mod |n|), * (2) sign*Y*a == A (mod |n|) */ /* Now divide B by the maximum possible power of two in the integers, * and divide X by the same value mod |n|. * When we're done, (1) still holds. */ shift = 0; while (!BN_is_bit_set(B, shift)) /* note that 0 < B */ { shift++; if (BN_is_odd(X)) { if (!BN_uadd(X, X, n)) goto err; } /* now X is even, so we can easily divide it by two */ if (!BN_rshift1(X, X)) goto err; } if (shift > 0) { if (!BN_rshift(B, B, shift)) goto err; } /* Same for A and Y. Afterwards, (2) still holds. */ shift = 0; while (!BN_is_bit_set(A, shift)) /* note that 0 < A */ { shift++; if (BN_is_odd(Y)) { if (!BN_uadd(Y, Y, n)) goto err; } /* now Y is even */ if (!BN_rshift1(Y, Y)) goto err; } if (shift > 0) { if (!BN_rshift(A, A, shift)) goto err; } /* We still have (1) and (2). * Both A and B are odd. * The following computations ensure that * * 0 <= B < |n|, * 0 < A < |n|, * (1) -sign*X*a == B (mod |n|), * (2) sign*Y*a == A (mod |n|), * * and that either A or B is even in the next iteration. */ if (BN_ucmp(B, A) >= 0) { /* -sign*(X + Y)*a == B - A (mod |n|) */ if (!BN_uadd(X, X, Y)) goto err; /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that * actually makes the algorithm slower */ if (!BN_usub(B, B, A)) goto err; } else { /* sign*(X + Y)*a == A - B (mod |n|) */ if (!BN_uadd(Y, Y, X)) goto err; /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */ if (!BN_usub(A, A, B)) goto err; } } } else { /* general inversion algorithm */ while (!BN_is_zero(B)) { BIGNUM *tmp; /* * 0 < B < A, * (*) -sign*X*a == B (mod |n|), * sign*Y*a == A (mod |n|) */ /* (D, M) := (A/B, A%B) ... */ if (BN_num_bits(A) == BN_num_bits(B)) { if (!BN_one(D)) goto err; if (!BN_sub(M,A,B)) goto err; } else if (BN_num_bits(A) == BN_num_bits(B) + 1) { /* A/B is 1, 2, or 3 */ if (!BN_lshift1(T,B)) goto err; if (BN_ucmp(A,T) < 0) { /* A < 2*B, so D=1 */ if (!BN_one(D)) goto err; if (!BN_sub(M,A,B)) goto err; } else { /* A >= 2*B, so D=2 or D=3 */ if (!BN_sub(M,A,T)) goto err; if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */ if (BN_ucmp(A,D) < 0) { /* A < 3*B, so D=2 */ if (!BN_set_word(D,2)) goto err; /* M (= A - 2*B) already has the correct value */ } else { /* only D=3 remains */ if (!BN_set_word(D,3)) goto err; /* currently M = A - 2*B, but we need M = A - 3*B */ if (!BN_sub(M,M,B)) goto err; } } } else { if (!BN_div(D,M,A,B,ctx)) goto err; } /* Now * A = D*B + M; * thus we have * (**) sign*Y*a == D*B + M (mod |n|). */ tmp=A; /* keep the BIGNUM object, the value does not matter */ /* (A, B) := (B, A mod B) ... */ A=B; B=M; /* ... so we have 0 <= B < A again */ /* Since the former M is now B and the former B is now A, * (**) translates into * sign*Y*a == D*A + B (mod |n|), * i.e. * sign*Y*a - D*A == B (mod |n|). * Similarly, (*) translates into * -sign*X*a == A (mod |n|). * * Thus, * sign*Y*a + D*sign*X*a == B (mod |n|), * i.e. * sign*(Y + D*X)*a == B (mod |n|). * * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at * -sign*X*a == B (mod |n|), * sign*Y*a == A (mod |n|). * Note that X and Y stay non-negative all the time. */ /* most of the time D is very small, so we can optimize tmp := D*X+Y */ if (BN_is_one(D)) { if (!BN_add(tmp,X,Y)) goto err; } else { if (BN_is_word(D,2)) { if (!BN_lshift1(tmp,X)) goto err; } else if (BN_is_word(D,4)) { if (!BN_lshift(tmp,X,2)) goto err; } else if (D->top == 1) { if (!BN_copy(tmp,X)) goto err; if (!BN_mul_word(tmp,D->d[0])) goto err; } else { if (!BN_mul(tmp,D,X,ctx)) goto err; } if (!BN_add(tmp,tmp,Y)) goto err; } M=Y; /* keep the BIGNUM object, the value does not matter */ Y=X; X=tmp; sign = -sign; } } /* * The while loop (Euclid's algorithm) ends when * A == gcd(a,n); * we have * sign*Y*a == A (mod |n|), * where Y is non-negative. */ if (sign < 0) { if (!BN_sub(Y,n,Y)) goto err; } /* Now Y*a == A (mod |n|). */ if (BN_is_one(A)) { /* Y*a == 1 (mod |n|) */ if (!Y->neg && BN_ucmp(Y,n) < 0) { if (!BN_copy(R,Y)) goto err; } else { if (!BN_nnmod(R,Y,n,ctx)) goto err; } } else { BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE); goto err; } ret=R; err: if ((ret == NULL) && (in == NULL)) BN_free(R); BN_CTX_end(ctx); return(ret); }