bn_gcd.c

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/* crypto/bn/bn_gcd.c */
/* Copyright (C) 1995-1998 Eric Young ([email protected])
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young ([email protected]).
 * 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 ([email protected]).
 *
 * 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 ([email protected])"
 *    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 ([email protected])"
 *
 * 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
 *    [email protected].
 *
 * 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
 * ([email protected]).  This product includes software written by Tim
 * Hudson ([email protected]).
 *
 */

#if defined( INC_ALL )
  #include "bn_lcl.h"
#else
  #include "bn/bn_lcl.h"
#endif /* Compiler-specific includes */

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);
    bn_check_top(r);
    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;
        }
    bn_check_top(a);
    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);
    if (ret)
        bn_check_top(ret);
    return(ret);
    }