/* crypto/bn/bn_exp.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) 1998-2000 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" #define TABLE_SIZE 32 /* this one works - simple but works */ int BN_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { int i,bits,ret=0; BIGNUM *v,*rr; BN_CTX_start(ctx); if ((r == a) || (r == p)) rr = BN_CTX_get(ctx); else rr = r; if ((v = BN_CTX_get(ctx)) == NULL) goto err; if (BN_copy(v,a) == NULL) goto err; bits=BN_num_bits(p); if (BN_is_odd(p)) { if (BN_copy(rr,a) == NULL) goto err; } else { if (!BN_one(rr)) goto err; } for (i=1; i<bits; i++) { if (!BN_sqr(v,v,ctx)) goto err; if (BN_is_bit_set(p,i)) { if (!BN_mul(rr,rr,v,ctx)) goto err; } } ret=1; err: if (r != rr) BN_copy(r,rr); BN_CTX_end(ctx); return(ret); } int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m, BN_CTX *ctx) { int ret; bn_check_top(a); bn_check_top(p); bn_check_top(m); /* For even modulus m = 2^k*m_odd, it might make sense to compute * a^p mod m_odd and a^p mod 2^k separately (with Montgomery * exponentiation for the odd part), using appropriate exponent * reductions, and combine the results using the CRT. * * For now, we use Montgomery only if the modulus is odd; otherwise, * exponentiation using the reciprocal-based quick remaindering * algorithm is used. * * (Timing obtained with expspeed.c [computations a^p mod m * where a, p, m are of the same length: 256, 512, 1024, 2048, * 4096, 8192 bits], compared to the running time of the * standard algorithm: * * BN_mod_exp_mont 33 .. 40 % [AMD K6-2, Linux, debug configuration] * 55 .. 77 % [UltraSparc processor, but * debug-solaris-sparcv8-gcc conf.] * * BN_mod_exp_recp 50 .. 70 % [AMD K6-2, Linux, debug configuration] * 62 .. 118 % [UltraSparc, debug-solaris-sparcv8-gcc] * * On the Sparc, BN_mod_exp_recp was faster than BN_mod_exp_mont * at 2048 and more bits, but at 512 and 1024 bits, it was * slower even than the standard algorithm! * * "Real" timings [linux-elf, solaris-sparcv9-gcc configurations] * should be obtained when the new Montgomery reduction code * has been integrated into OpenSSL.) */ #define MONT_MUL_MOD #define MONT_EXP_WORD #define RECP_MUL_MOD #ifdef MONT_MUL_MOD /* I have finally been able to take out this pre-condition of * the top bit being set. It was caused by an error in BN_div * with negatives. There was also another problem when for a^b%m * a >= m. eay 07-May-97 */ /* if ((m->d[m->top-1]&BN_TBIT) && BN_is_odd(m)) */ if (BN_is_odd(m)) { # ifdef MONT_EXP_WORD if (a->top == 1 && !a->neg) { BN_ULONG A = a->d[0]; ret=BN_mod_exp_mont_word(r,A,p,m,ctx,NULL); } else # endif ret=BN_mod_exp_mont(r,a,p,m,ctx,NULL); } else #endif #ifdef RECP_MUL_MOD { ret=BN_mod_exp_recp(r,a,p,m,ctx); } #else { ret=BN_mod_exp_simple(r,a,p,m,ctx); } #endif return(ret); } int BN_mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m, BN_CTX *ctx) { int i,j,bits,ret=0,wstart,wend,window,wvalue; int start=1,ts=0; BIGNUM *aa; BIGNUM val[TABLE_SIZE]; BN_RECP_CTX recp; bits=BN_num_bits(p); if (bits == 0) { ret = BN_one(r); return ret; } BN_CTX_start(ctx); if ((aa = BN_CTX_get(ctx)) == NULL) goto err; BN_RECP_CTX_init(&recp); if (m->neg) { /* ignore sign of 'm' */ if (!BN_copy(aa, m)) goto err; aa->neg = 0; if (BN_RECP_CTX_set(&recp,aa,ctx) <= 0) goto err; } else { if (BN_RECP_CTX_set(&recp,m,ctx) <= 0) goto err; } BN_init(&(val[0])); ts=1; if (!BN_nnmod(&(val[0]),a,m,ctx)) goto err; /* 1 */ if (BN_is_zero(&(val[0]))) { ret = BN_zero(r); goto err; } window = BN_window_bits_for_exponent_size(bits); if (window > 1) { if (!BN_mod_mul_reciprocal(aa,&(val[0]),&(val[0]),&recp,ctx)) goto err; /* 2 */ j=1<<(window-1); for (i=1; i<j; i++) { BN_init(&val[i]); if (!BN_mod_mul_reciprocal(&(val[i]),&(val[i-1]),aa,&recp,ctx)) goto err; } ts=i; } start=1; /* This is used to avoid multiplication etc * when there is only the value '1' in the * buffer. */ wvalue=0; /* The 'value' of the window */ wstart=bits-1; /* The top bit of the window */ wend=0; /* The bottom bit of the window */ if (!BN_one(r)) goto err; for (;;) { if (BN_is_bit_set(p,wstart) == 0) { if (!start) if (!BN_mod_mul_reciprocal(r,r,r,&recp,ctx)) goto err; if (wstart == 0) break; wstart--; continue; } /* We now have wstart on a 'set' bit, we now need to work out * how bit a window to do. To do this we need to scan * forward until the last set bit before the end of the * window */ j=wstart; wvalue=1; wend=0; for (i=1; i<window; i++) { if (wstart-i < 0) break; if (BN_is_bit_set(p,wstart-i)) { wvalue<<=(i-wend); wvalue|=1; wend=i; } } /* wend is the size of the current window */ j=wend+1; /* add the 'bytes above' */ if (!start) for (i=0; i<j; i++) { if (!BN_mod_mul_reciprocal(r,r,r,&recp,ctx)) goto err; } /* wvalue will be an odd number < 2^window */ if (!BN_mod_mul_reciprocal(r,r,&(val[wvalue>>1]),&recp,ctx)) goto err; /* move the 'window' down further */ wstart-=wend+1; wvalue=0; start=0; if (wstart < 0) break; } ret=1; err: BN_CTX_end(ctx); for (i=0; i<ts; i++) BN_clear_free(&(val[i])); BN_RECP_CTX_free(&recp); return(ret); } int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont) { int i,j,bits,ret=0,wstart,wend,window,wvalue; int start=1,ts=0; BIGNUM *d,*r; const BIGNUM *aa; BIGNUM val[TABLE_SIZE]; BN_MONT_CTX *mont=NULL; bn_check_top(a); bn_check_top(p); bn_check_top(m); if (!(m->d[0] & 1)) { BNerr(BN_F_BN_MOD_EXP_MONT,BN_R_CALLED_WITH_EVEN_MODULUS); return(0); } bits=BN_num_bits(p); if (bits == 0) { ret = BN_one(rr); return ret; } BN_CTX_start(ctx); d = BN_CTX_get(ctx); r = BN_CTX_get(ctx); if (d == NULL || r == NULL) goto err; /* If this is not done, things will break in the montgomery * part */ if (in_mont != NULL) mont=in_mont; else { if ((mont=BN_MONT_CTX_new()) == NULL) goto err; if (!BN_MONT_CTX_set(mont,m,ctx)) goto err; } BN_init(&val[0]); ts=1; if (a->neg || BN_ucmp(a,m) >= 0) { if (!BN_nnmod(&(val[0]),a,m,ctx)) goto err; aa= &(val[0]); } else aa=a; if (BN_is_zero(aa)) { ret = BN_zero(rr); goto err; } if (!BN_to_montgomery(&(val[0]),aa,mont,ctx)) goto err; /* 1 */ window = BN_window_bits_for_exponent_size(bits); if (window > 1) { if (!BN_mod_mul_montgomery(d,&(val[0]),&(val[0]),mont,ctx)) goto err; /* 2 */ j=1<<(window-1); for (i=1; i<j; i++) { BN_init(&(val[i])); if (!BN_mod_mul_montgomery(&(val[i]),&(val[i-1]),d,mont,ctx)) goto err; } ts=i; } start=1; /* This is used to avoid multiplication etc * when there is only the value '1' in the * buffer. */ wvalue=0; /* The 'value' of the window */ wstart=bits-1; /* The top bit of the window */ wend=0; /* The bottom bit of the window */ if (!BN_to_montgomery(r,BN_value_one(),mont,ctx)) goto err; for (;;) { if (BN_is_bit_set(p,wstart) == 0) { if (!start) { if (!BN_mod_mul_montgomery(r,r,r,mont,ctx)) goto err; } if (wstart == 0) break; wstart--; continue; } /* We now have wstart on a 'set' bit, we now need to work out * how bit a window to do. To do this we need to scan * forward until the last set bit before the end of the * window */ j=wstart; wvalue=1; wend=0; for (i=1; i<window; i++) { if (wstart-i < 0) break; if (BN_is_bit_set(p,wstart-i)) { wvalue<<=(i-wend); wvalue|=1; wend=i; } } /* wend is the size of the current window */ j=wend+1; /* add the 'bytes above' */ if (!start) for (i=0; i<j; i++) { if (!BN_mod_mul_montgomery(r,r,r,mont,ctx)) goto err; } /* wvalue will be an odd number < 2^window */ if (!BN_mod_mul_montgomery(r,r,&(val[wvalue>>1]),mont,ctx)) goto err; /* move the 'window' down further */ wstart-=wend+1; wvalue=0; start=0; if (wstart < 0) break; } if (!BN_from_montgomery(rr,r,mont,ctx)) goto err; ret=1; err: if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont); BN_CTX_end(ctx); for (i=0; i<ts; i++) BN_clear_free(&(val[i])); return(ret); } int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p, const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont) { BN_MONT_CTX *mont = NULL; int b, bits, ret=0; int r_is_one; BN_ULONG w, next_w; BIGNUM *d, *r, *t; BIGNUM *swap_tmp; #define BN_MOD_MUL_WORD(r, w, m) \ (BN_mul_word(r, (w)) && \ (/* BN_ucmp(r, (m)) < 0 ? 1 :*/ \ (BN_mod(t, r, m, ctx) && (swap_tmp = r, r = t, t = swap_tmp, 1)))) /* BN_MOD_MUL_WORD is only used with 'w' large, * so the BN_ucmp test is probably more overhead * than always using BN_mod (which uses BN_copy if * a similar test returns true). */ /* We can use BN_mod and do not need BN_nnmod because our * accumulator is never negative (the result of BN_mod does * not depend on the sign of the modulus). */ #define BN_TO_MONTGOMERY_WORD(r, w, mont) \ (BN_set_word(r, (w)) && BN_to_montgomery(r, r, (mont), ctx)) bn_check_top(p); bn_check_top(m); if (m->top == 0 || !(m->d[0] & 1)) { BNerr(BN_F_BN_MOD_EXP_MONT_WORD,BN_R_CALLED_WITH_EVEN_MODULUS); return(0); } if (m->top == 1) a %= m->d[0]; /* make sure that 'a' is reduced */ bits = BN_num_bits(p); if (bits == 0) { ret = BN_one(rr); return ret; } if (a == 0) { ret = BN_zero(rr); return ret; } BN_CTX_start(ctx); d = BN_CTX_get(ctx); r = BN_CTX_get(ctx); t = BN_CTX_get(ctx); if (d == NULL || r == NULL || t == NULL) goto err; if (in_mont != NULL) mont=in_mont; else { if ((mont = BN_MONT_CTX_new()) == NULL) goto err; if (!BN_MONT_CTX_set(mont, m, ctx)) goto err; } r_is_one = 1; /* except for Montgomery factor */ /* bits-1 >= 0 */ /* The result is accumulated in the product r*w. */ w = a; /* bit 'bits-1' of 'p' is always set */ for (b = bits-2; b >= 0; b--) { /* First, square r*w. */ next_w = w*w; if ((next_w/w) != w) /* overflow */ { if (r_is_one) { if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err; r_is_one = 0; } else { if (!BN_MOD_MUL_WORD(r, w, m)) goto err; } next_w = 1; } w = next_w; if (!r_is_one) { if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) goto err; } /* Second, multiply r*w by 'a' if exponent bit is set. */ if (BN_is_bit_set(p, b)) { next_w = w*a; if ((next_w/a) != w) /* overflow */ { if (r_is_one) { if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err; r_is_one = 0; } else { if (!BN_MOD_MUL_WORD(r, w, m)) goto err; } next_w = a; } w = next_w; } } /* Finally, set r:=r*w. */ if (w != 1) { if (r_is_one) { if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err; r_is_one = 0; } else { if (!BN_MOD_MUL_WORD(r, w, m)) goto err; } } if (r_is_one) /* can happen only if a == 1*/ { if (!BN_one(rr)) goto err; } else { if (!BN_from_montgomery(rr, r, mont, ctx)) goto err; } ret = 1; err: if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont); BN_CTX_end(ctx); return(ret); } /* The old fallback, simple version :-) */ int BN_mod_exp_simple(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m, BN_CTX *ctx) { int i,j,bits,ret=0,wstart,wend,window,wvalue,ts=0; int start=1; BIGNUM *d; BIGNUM val[TABLE_SIZE]; bits=BN_num_bits(p); if (bits == 0) { ret = BN_one(r); return ret; } BN_CTX_start(ctx); if ((d = BN_CTX_get(ctx)) == NULL) goto err; BN_init(&(val[0])); ts=1; if (!BN_nnmod(&(val[0]),a,m,ctx)) goto err; /* 1 */ if (BN_is_zero(&(val[0]))) { ret = BN_zero(r); goto err; } window = BN_window_bits_for_exponent_size(bits); if (window > 1) { if (!BN_mod_mul(d,&(val[0]),&(val[0]),m,ctx)) goto err; /* 2 */ j=1<<(window-1); for (i=1; i<j; i++) { BN_init(&(val[i])); if (!BN_mod_mul(&(val[i]),&(val[i-1]),d,m,ctx)) goto err; } ts=i; } start=1; /* This is used to avoid multiplication etc * when there is only the value '1' in the * buffer. */ wvalue=0; /* The 'value' of the window */ wstart=bits-1; /* The top bit of the window */ wend=0; /* The bottom bit of the window */ if (!BN_one(r)) goto err; for (;;) { if (BN_is_bit_set(p,wstart) == 0) { if (!start) if (!BN_mod_mul(r,r,r,m,ctx)) goto err; if (wstart == 0) break; wstart--; continue; } /* We now have wstart on a 'set' bit, we now need to work out * how bit a window to do. To do this we need to scan * forward until the last set bit before the end of the * window */ j=wstart; wvalue=1; wend=0; for (i=1; i<window; i++) { if (wstart-i < 0) break; if (BN_is_bit_set(p,wstart-i)) { wvalue<<=(i-wend); wvalue|=1; wend=i; } } /* wend is the size of the current window */ j=wend+1; /* add the 'bytes above' */ if (!start) for (i=0; i<j; i++) { if (!BN_mod_mul(r,r,r,m,ctx)) goto err; } /* wvalue will be an odd number < 2^window */ if (!BN_mod_mul(r,r,&(val[wvalue>>1]),m,ctx)) goto err; /* move the 'window' down further */ wstart-=wend+1; wvalue=0; start=0; if (wstart < 0) break; } ret=1; err: BN_CTX_end(ctx); for (i=0; i<ts; i++) BN_clear_free(&(val[i])); return(ret); }