/src/openssl33/crypto/bn/bn_gcd.c

Source
/*
 * Copyright 1995-2020 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"

/*
 * bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
 * not contain branches that may leak sensitive information.
 *
 * This is a static function, we ensure all callers in this file pass valid
 * arguments: all passed pointers here are non-NULL.
 */
static ossl_inline BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in,
    const BIGNUM *a, const BIGNUM *n,
    BN_CTX *ctx, int *pnoinv)
{
    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;

    if (!BN_one(X))
        goto err;
    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)) {
        /*
         * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
         * BN_div_no_branch will be called eventually.
         */
        {
            BIGNUM local_B;
            bn_init(&local_B);
            BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
            if (!BN_nnmod(B, &local_B, A, ctx))
                goto err;
            /* Ensure local_B goes out of scope before any further use of B */
        }
    }
    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|).
     */

    while (!BN_is_zero(B)) {
        BIGNUM *tmp;

        /*-
         *      0 < B < A,
         * (*) -sign*X*a  ==  B   (mod |n|),
         *      sign*Y*a  ==  A   (mod |n|)
         */

        /*
         * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
         * BN_div_no_branch will be called eventually.
         */
        {
            BIGNUM local_A;
            bn_init(&local_A);
            BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);

            /* (D, M) := (A/B, A%B) ... */
            if (!BN_div(D, M, &local_A, B, ctx))
                goto err;
            /* Ensure local_A goes out of scope before any further use of A */
        }

        /*-
         * 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.
         */

        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 {
        *pnoinv = 1;
        /* caller sets the BN_R_NO_INVERSE error */
        goto err;
    }

    ret = R;
    *pnoinv = 0;

err:
    if ((ret == NULL) && (in == NULL))
        BN_free(R);
    BN_CTX_end(ctx);
    bn_check_top(ret);
    return ret;
}

/*
 * This is an internal function, we assume all callers pass valid arguments:
 * all pointers passed here are assumed non-NULL.
 */
BIGNUM *int_bn_mod_inverse(BIGNUM *in,
    const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
    int *pnoinv)
{
    BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
    BIGNUM *ret = NULL;
    int sign;

    /* This is invalid input so we don't worry about constant time here */
    if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
        *pnoinv = 1;
        return NULL;
    }

    *pnoinv = 0;

    if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
        || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
        return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
    }

    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;

    if (!BN_one(X))
        goto err;
    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) <= 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 systems, 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 {
        *pnoinv = 1;
        goto err;
    }
    ret = R;
err:
    if ((ret == NULL) && (in == NULL))
        BN_free(R);
    BN_CTX_end(ctx);
    bn_check_top(ret);
    return ret;
}

/* solves ax == 1 (mod n) */
BIGNUM *BN_mod_inverse(BIGNUM *in,
    const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
{
    BN_CTX *new_ctx = NULL;
    BIGNUM *rv;
    int noinv = 0;

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new_ex(NULL);
        if (ctx == NULL) {
            ERR_raise(ERR_LIB_BN, ERR_R_BN_LIB);
            return NULL;
        }
    }

    rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
    if (noinv)
        ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE);
    BN_CTX_free(new_ctx);
    return rv;
}

/*
 * The numbers a and b are coprime if the only positive integer that is a
 * divisor of both of them is 1.
 * i.e. gcd(a,b) = 1.
 *
 * Coprimes have the property: b has a multiplicative inverse modulo a
 * i.e there is some value x such that bx = 1 (mod a).
 *
 * Testing the modulo inverse is currently much faster than the constant
 * time version of BN_gcd().
 */
int BN_are_coprime(BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
{
    int ret = 0;
    BIGNUM *tmp;

    BN_CTX_start(ctx);
    tmp = BN_CTX_get(ctx);
    if (tmp == NULL)
        goto end;

    ERR_set_mark();
    BN_set_flags(a, BN_FLG_CONSTTIME);
    ret = (BN_mod_inverse(tmp, a, b, ctx) != NULL);
    /* Clear any errors (an error is returned if there is no inverse) */
    ERR_pop_to_mark();
end:
    BN_CTX_end(ctx);
    return ret;
}

/*-
 * This function is based on the constant-time GCD work by Bernstein and Yang:
 * https://eprint.iacr.org/2019/266
 * Generalized fast GCD function to allow even inputs.
 * The algorithm first finds the shared powers of 2 between
 * the inputs, and removes them, reducing at least one of the
 * inputs to an odd value. Then it proceeds to calculate the GCD.
 * Before returning the resulting GCD, we take care of adding
 * back the powers of two removed at the beginning.
 * Note 1: we assume the bit length of both inputs is public information,
 * since access to top potentially leaks this information.
 */
int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
{
    BIGNUM *g, *temp = NULL;
    BN_ULONG mask = 0;
    int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;

    /* Note 2: zero input corner cases are not constant-time since they are
     * handled immediately. An attacker can run an attack under this
     * assumption without the need of side-channel information. */
    if (BN_is_zero(in_b)) {
        ret = BN_copy(r, in_a) != NULL;
        r->neg = 0;
        return ret;
    }
    if (BN_is_zero(in_a)) {
        ret = BN_copy(r, in_b) != NULL;
        r->neg = 0;
        return ret;
    }

    bn_check_top(in_a);
    bn_check_top(in_b);

    BN_CTX_start(ctx);
    temp = BN_CTX_get(ctx);
    g = BN_CTX_get(ctx);

    /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
    if (g == NULL
        || !BN_lshift1(g, in_b)
        || !BN_lshift1(r, in_a))
        goto err;

    /* find shared powers of two, i.e. "shifts" >= 1 */
    for (i = 0; i < r->dmax && i < g->dmax; i++) {
        mask = ~(r->d[i] | g->d[i]);
        for (j = 0; j < BN_BITS2; j++) {
            bit &= mask;
            shifts += bit;
            mask >>= 1;
        }
    }

    /* subtract shared powers of two; shifts >= 1 */
    if (!BN_rshift(r, r, shifts)
        || !BN_rshift(g, g, shifts))
        goto err;

    /* expand to biggest nword, with room for a possible extra word */
    top = 1 + ((r->top >= g->top) ? r->top : g->top);
    if (bn_wexpand(r, top) == NULL
        || bn_wexpand(g, top) == NULL
        || bn_wexpand(temp, top) == NULL)
        goto err;

    /* re arrange inputs s.t. r is odd */
    BN_consttime_swap((~r->d[0]) & 1, r, g, top);

    /* compute the number of iterations */
    rlen = BN_num_bits(r);
    glen = BN_num_bits(g);
    m = 4 + 3 * ((rlen >= glen) ? rlen : glen);

    for (i = 0; i < m; i++) {
        /* conditionally flip signs if delta is positive and g is odd */
        cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
            /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
            & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1)));
        delta = (-cond & -delta) | ((cond - 1) & delta);
        r->neg ^= cond;
        /* swap */
        BN_consttime_swap(cond, r, g, top);

        /* elimination step */
        delta++;
        if (!BN_add(temp, g, r))
            goto err;
        BN_consttime_swap(g->d[0] & 1 /* g is odd */
                /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
                & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))),
            g, temp, top);
        if (!BN_rshift1(g, g))
            goto err;
    }

    /* remove possible negative sign */
    r->neg = 0;
    /* add powers of 2 removed, then correct the artificial shift */
    if (!BN_lshift(r, r, shifts)
        || !BN_rshift1(r, r))
        goto err;

    ret = 1;

err:
    BN_CTX_end(ctx);
    bn_check_top(r);
    return ret;
}

Coverage Report

Created: 2026-02-14 07:20

Line	Count	Source
1		/*
2		* Copyright 1995-2020 The OpenSSL Project Authors. All Rights Reserved.
3		*
4		* Licensed under the Apache License 2.0 (the "License"). You may not use
5		* this file except in compliance with the License. You can obtain a copy
6		* in the file LICENSE in the source distribution or at
7		* https://www.openssl.org/source/license.html
8		*/
9
10		#include "internal/cryptlib.h"
11		#include "bn_local.h"
12
13		/*
14		* bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
15		* not contain branches that may leak sensitive information.
16		*
17		* This is a static function, we ensure all callers in this file pass valid
18		* arguments: all passed pointers here are non-NULL.
19		*/
20		static ossl_inline BIGNUM bn_mod_inverse_no_branch(BIGNUM in,
21		const BIGNUM a, const BIGNUM n,
22		BN_CTX ctx, int pnoinv)
23	51.6k	{
24	51.6k	BIGNUM A, B, X, Y, M, D, T, R = NULL;
25	51.6k	BIGNUM *ret = NULL;
26	51.6k	int sign;
27
28	51.6k	bn_check_top(a);
29	51.6k	bn_check_top(n);
30
31	51.6k	BN_CTX_start(ctx);
32	51.6k	A = BN_CTX_get(ctx);
33	51.6k	B = BN_CTX_get(ctx);
34	51.6k	X = BN_CTX_get(ctx);
35	51.6k	D = BN_CTX_get(ctx);
36	51.6k	M = BN_CTX_get(ctx);
37	51.6k	Y = BN_CTX_get(ctx);
38	51.6k	T = BN_CTX_get(ctx);
39	51.6k	if (T == NULL)
40	0	goto err;
41
42	51.6k	if (in == NULL)
43	0	R = BN_new();
44	51.6k	else
45	51.6k	R = in;
46	51.6k	if (R == NULL)
47	0	goto err;
48
49	51.6k	if (!BN_one(X))
50	0	goto err;
51	51.6k	BN_zero(Y);
52	51.6k	if (BN_copy(B, a) == NULL)
53	0	goto err;
54	51.6k	if (BN_copy(A, n) == NULL)
55	0	goto err;
56	51.6k	A->neg = 0;
57
58	51.6k	if (B->neg \|\| (BN_ucmp(B, A) >= 0)) {
59		/*
60		* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
61		* BN_div_no_branch will be called eventually.
62		*/
63	34.5k	{
64	34.5k	BIGNUM local_B;
65	34.5k	bn_init(&local_B);
66	34.5k	BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
67	34.5k	if (!BN_nnmod(B, &local_B, A, ctx))
68	0	goto err;
69		/* Ensure local_B goes out of scope before any further use of B */
70	34.5k	}
71	34.5k	}
72	51.6k	sign = -1;
73		/*-
74		* From B = a mod \|n\|, A = \|n\| it follows that
75		*
76		* 0 <= B < A,
77		* -signXa == B (mod \|n\|),
78		* signYa == A (mod \|n\|).
79		*/
80
81	20.9M	while (!BN_is_zero(B)) {
82	20.8M	BIGNUM *tmp;
83
84		/*-
85		* 0 < B < A,
86		* () -signX*a == B (mod \|n\|),
87		* signYa == A (mod \|n\|)
88		*/
89
90		/*
91		* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
92		* BN_div_no_branch will be called eventually.
93		*/
94	20.8M	{
95	20.8M	BIGNUM local_A;
96	20.8M	bn_init(&local_A);
97	20.8M	BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
98
99		/* (D, M) := (A/B, A%B) ... */
100	20.8M	if (!BN_div(D, M, &local_A, B, ctx))
101	0	goto err;
102		/* Ensure local_A goes out of scope before any further use of A */
103	20.8M	}
104
105		/*-
106		* Now
107		* A = D*B + M;
108		* thus we have
109		* (*) signYa == DB + M (mod \|n\|).
110		*/
111
112	20.8M	tmp = A; /* keep the BIGNUM object, the value does not
113		* matter */
114
115		/* (A, B) := (B, A mod B) ... */
116	20.8M	A = B;
117	20.8M	B = M;
118		/* ... so we have 0 <= B < A again */
119
120		/*-
121		* Since the former M is now B and the former B is now A,
122		* (**) translates into
123		* signYa == D*A + B (mod \|n\|),
124		* i.e.
125		* signYa - D*A == B (mod \|n\|).
126		* Similarly, (*) translates into
127		* -signXa == A (mod \|n\|).
128		*
129		* Thus,
130		* signYa + DsignX*a == B (mod \|n\|),
131		* i.e.
132		* sign(Y + DX)*a == B (mod \|n\|).
133		*
134		* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
135		* -signXa == B (mod \|n\|),
136		* signYa == A (mod \|n\|).
137		* Note that X and Y stay non-negative all the time.
138		*/
139
140	20.8M	if (!BN_mul(tmp, D, X, ctx))
141	0	goto err;
142	20.8M	if (!BN_add(tmp, tmp, Y))
143	0	goto err;
144
145	20.8M	M = Y; /* keep the BIGNUM object, the value does not
146		* matter */
147	20.8M	Y = X;
148	20.8M	X = tmp;
149	20.8M	sign = -sign;
150	20.8M	}
151
152		/*-
153		* The while loop (Euclid's algorithm) ends when
154		* A == gcd(a,n);
155		* we have
156		* signYa == A (mod \|n\|),
157		* where Y is non-negative.
158		*/
159
160	51.6k	if (sign < 0) {
161	15.5k	if (!BN_sub(Y, n, Y))
162	0	goto err;
163	15.5k	}
164		/* Now Ya == A (mod \|n\|). /
165
166	51.6k	if (BN_is_one(A)) {
167		/* Ya == 1 (mod \|n\|) /
168	51.2k	if (!Y->neg && BN_ucmp(Y, n) < 0) {
169	51.2k	if (!BN_copy(R, Y))
170	0	goto err;
171	51.2k	} else {
172	0	if (!BN_nnmod(R, Y, n, ctx))
173	0	goto err;
174	0	}
175	51.2k	} else {
176	470	*pnoinv = 1;
177		/* caller sets the BN_R_NO_INVERSE error */
178	470	goto err;
179	470	}
180
181	51.2k	ret = R;
182	51.2k	*pnoinv = 0;
183
184	51.6k	err:
185	51.6k	if ((ret == NULL) && (in == NULL))
186	0	BN_free(R);
187	51.6k	BN_CTX_end(ctx);
188	51.6k	bn_check_top(ret);
189	51.6k	return ret;
190	51.2k	}
191
192		/*
193		* This is an internal function, we assume all callers pass valid arguments:
194		* all pointers passed here are assumed non-NULL.
195		*/
196		BIGNUM int_bn_mod_inverse(BIGNUM in,
197		const BIGNUM a, const BIGNUM n, BN_CTX *ctx,
198		int *pnoinv)
199	1.09M	{
200	1.09M	BIGNUM A, B, X, Y, M, D, T, R = NULL;
201	1.09M	BIGNUM *ret = NULL;
202	1.09M	int sign;
203
204		/* This is invalid input so we don't worry about constant time here */
205	1.09M	if (BN_abs_is_word(n, 1) \|\| BN_is_zero(n)) {
206	521	*pnoinv = 1;
207	521	return NULL;
208	521	}
209
210	1.09M	*pnoinv = 0;
211
212	1.09M	if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
213	1.09M	\|\| (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
214	51.6k	return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
215	51.6k	}
216
217	1.04M	bn_check_top(a);
218	1.04M	bn_check_top(n);
219
220	1.04M	BN_CTX_start(ctx);
221	1.04M	A = BN_CTX_get(ctx);
222	1.04M	B = BN_CTX_get(ctx);
223	1.04M	X = BN_CTX_get(ctx);
224	1.04M	D = BN_CTX_get(ctx);
225	1.04M	M = BN_CTX_get(ctx);
226	1.04M	Y = BN_CTX_get(ctx);
227	1.04M	T = BN_CTX_get(ctx);
228	1.04M	if (T == NULL)
229	0	goto err;
230
231	1.04M	if (in == NULL)
232	0	R = BN_new();
233	1.04M	else
234	1.04M	R = in;
235	1.04M	if (R == NULL)
236	0	goto err;
237
238	1.04M	if (!BN_one(X))
239	0	goto err;
240	1.04M	BN_zero(Y);
241	1.04M	if (BN_copy(B, a) == NULL)
242	0	goto err;
243	1.04M	if (BN_copy(A, n) == NULL)
244	0	goto err;
245	1.04M	A->neg = 0;
246	1.04M	if (B->neg \|\| (BN_ucmp(B, A) >= 0)) {
247	1.03M	if (!BN_nnmod(B, B, A, ctx))
248	0	goto err;
249	1.03M	}
250	1.04M	sign = -1;
251		/*-
252		* From B = a mod \|n\|, A = \|n\| it follows that
253		*
254		* 0 <= B < A,
255		* -signXa == B (mod \|n\|),
256		* signYa == A (mod \|n\|).
257		*/
258
259	1.04M	if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
260		/*
261		* Binary inversion algorithm; requires odd modulus. This is faster
262		* than the general algorithm if the modulus is sufficiently small
263		* (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
264		* systems)
265		*/
266	1.03M	int shift;
267
268	63.6M	while (!BN_is_zero(B)) {
269		/*-
270		* 0 < B < \|n\|,
271		* 0 < A <= \|n\|,
272		* (1) -signXa == B (mod \|n\|),
273		* (2) signYa == A (mod \|n\|)
274		*/
275
276		/*
277		* Now divide B by the maximum possible power of two in the
278		* integers, and divide X by the same value mod \|n\|. When we're
279		* done, (1) still holds.
280		*/
281	62.6M	shift = 0;
282	87.4M	while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
283	24.7M	shift++;
284
285	24.7M	if (BN_is_odd(X)) {
286	12.6M	if (!BN_uadd(X, X, n))
287	0	goto err;
288	12.6M	}
289		/*
290		* now X is even, so we can easily divide it by two
291		*/
292	24.7M	if (!BN_rshift1(X, X))
293	0	goto err;
294	24.7M	}
295	62.6M	if (shift > 0) {
296	23.4M	if (!BN_rshift(B, B, shift))
297	0	goto err;
298	23.4M	}
299
300		/*
301		* Same for A and Y. Afterwards, (2) still holds.
302		*/
303	62.6M	shift = 0;
304	101M	while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
305	39.3M	shift++;
306
307	39.3M	if (BN_is_odd(Y)) {
308	24.8M	if (!BN_uadd(Y, Y, n))
309	0	goto err;
310	24.8M	}
311		/* now Y is even */
312	39.3M	if (!BN_rshift1(Y, Y))
313	0	goto err;
314	39.3M	}
315	62.6M	if (shift > 0) {
316	38.5M	if (!BN_rshift(A, A, shift))
317	0	goto err;
318	38.5M	}
319
320		/*-
321		* We still have (1) and (2).
322		* Both A and B are odd.
323		* The following computations ensure that
324		*
325		* 0 <= B < \|n\|,
326		* 0 < A < \|n\|,
327		* (1) -signXa == B (mod \|n\|),
328		* (2) signYa == A (mod \|n\|),
329		*
330		* and that either A or B is even in the next iteration.
331		*/
332	62.6M	if (BN_ucmp(B, A) >= 0) {
333		/* -sign(X + Y)a == B - A (mod \|n\|) */
334	24.0M	if (!BN_uadd(X, X, Y))
335	0	goto err;
336		/*
337		* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
338		* actually makes the algorithm slower
339		*/
340	24.0M	if (!BN_usub(B, B, A))
341	0	goto err;
342	38.5M	} else {
343		/* sign(X + Y)a == A - B (mod \|n\|) */
344	38.5M	if (!BN_uadd(Y, Y, X))
345	0	goto err;
346		/*
347		* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
348		*/
349	38.5M	if (!BN_usub(A, A, B))
350	0	goto err;
351	38.5M	}
352	62.6M	}
353	1.03M	} else {
354		/* general inversion algorithm */
355
356	306k	while (!BN_is_zero(B)) {
357	298k	BIGNUM *tmp;
358
359		/*-
360		* 0 < B < A,
361		* () -signX*a == B (mod \|n\|),
362		* signYa == A (mod \|n\|)
363		*/
364
365		/* (D, M) := (A/B, A%B) ... */
366	298k	if (BN_num_bits(A) == BN_num_bits(B)) {
367	64.9k	if (!BN_one(D))
368	0	goto err;
369	64.9k	if (!BN_sub(M, A, B))
370	0	goto err;
371	233k	} else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
372		/* A/B is 1, 2, or 3 */
373	105k	if (!BN_lshift1(T, B))
374	0	goto err;
375	105k	if (BN_ucmp(A, T) < 0) {
376		/* A < 2B, so D=1 /
377	59.4k	if (!BN_one(D))
378	0	goto err;
379	59.4k	if (!BN_sub(M, A, B))
380	0	goto err;
381	59.4k	} else {
382		/* A >= 2B, so D=2 or D=3 /
383	45.5k	if (!BN_sub(M, A, T))
384	0	goto err;
385	45.5k	if (!BN_add(D, T, B))
386	0	goto err; /* use D (:= 3B) as temp /
387	45.5k	if (BN_ucmp(A, D) < 0) {
388		/* A < 3B, so D=2 /
389	37.3k	if (!BN_set_word(D, 2))
390	0	goto err;
391		/*
392		* M (= A - 2*B) already has the correct value
393		*/
394	37.3k	} else {
395		/* only D=3 remains */
396	8.23k	if (!BN_set_word(D, 3))
397	0	goto err;
398		/*
399		* currently M = A - 2B, but we need M = A - 3B
400		*/
401	8.23k	if (!BN_sub(M, M, B))
402	0	goto err;
403	8.23k	}
404	45.5k	}
405	128k	} else {
406	128k	if (!BN_div(D, M, A, B, ctx))
407	0	goto err;
408	128k	}
409
410		/*-
411		* Now
412		* A = D*B + M;
413		* thus we have
414		* (*) signYa == DB + M (mod \|n\|).
415		*/
416
417	298k	tmp = A; /* keep the BIGNUM object, the value does not matter */
418
419		/* (A, B) := (B, A mod B) ... */
420	298k	A = B;
421	298k	B = M;
422		/* ... so we have 0 <= B < A again */
423
424		/*-
425		* Since the former M is now B and the former B is now A,
426		* (**) translates into
427		* signYa == D*A + B (mod \|n\|),
428		* i.e.
429		* signYa - D*A == B (mod \|n\|).
430		* Similarly, (*) translates into
431		* -signXa == A (mod \|n\|).
432		*
433		* Thus,
434		* signYa + DsignX*a == B (mod \|n\|),
435		* i.e.
436		* sign(Y + DX)*a == B (mod \|n\|).
437		*
438		* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
439		* -signXa == B (mod \|n\|),
440		* signYa == A (mod \|n\|).
441		* Note that X and Y stay non-negative all the time.
442		*/
443
444		/*
445		* most of the time D is very small, so we can optimize tmp := D*X+Y
446		*/
447	298k	if (BN_is_one(D)) {
448	124k	if (!BN_add(tmp, X, Y))
449	0	goto err;
450	173k	} else {
451	173k	if (BN_is_word(D, 2)) {
452	53.2k	if (!BN_lshift1(tmp, X))
453	0	goto err;
454	120k	} else if (BN_is_word(D, 4)) {
455	17.7k	if (!BN_lshift(tmp, X, 2))
456	0	goto err;
457	102k	} else if (D->top == 1) {
458	102k	if (!BN_copy(tmp, X))
459	0	goto err;
460	102k	if (!BN_mul_word(tmp, D->d[0]))
461	0	goto err;
462	102k	} else {
463	229	if (!BN_mul(tmp, D, X, ctx))
464	0	goto err;
465	229	}
466	173k	if (!BN_add(tmp, tmp, Y))
467	0	goto err;
468	173k	}
469
470	298k	M = Y; /* keep the BIGNUM object, the value does not matter */
471	298k	Y = X;
472	298k	X = tmp;
473	298k	sign = -sign;
474	298k	}
475	8.02k	}
476
477		/*-
478		* The while loop (Euclid's algorithm) ends when
479		* A == gcd(a,n);
480		* we have
481		* signYa == A (mod \|n\|),
482		* where Y is non-negative.
483		*/
484
485	1.04M	if (sign < 0) {
486	1.03M	if (!BN_sub(Y, n, Y))
487	0	goto err;
488	1.03M	}
489		/* Now Ya == A (mod \|n\|). /
490
491	1.04M	if (BN_is_one(A)) {
492		/* Ya == 1 (mod \|n\|) /
493	1.03M	if (!Y->neg && BN_ucmp(Y, n) < 0) {
494	240k	if (!BN_copy(R, Y))
495	0	goto err;
496	794k	} else {
497	794k	if (!BN_nnmod(R, Y, n, ctx))
498	0	goto err;
499	794k	}
500	1.03M	} else {
501	6.63k	*pnoinv = 1;
502	6.63k	goto err;
503	6.63k	}
504	1.03M	ret = R;
505	1.04M	err:
506	1.04M	if ((ret == NULL) && (in == NULL))
507	0	BN_free(R);
508	1.04M	BN_CTX_end(ctx);
509	1.04M	bn_check_top(ret);
510	1.04M	return ret;
511	1.03M	}
512
513		/* solves ax == 1 (mod n) */
514		BIGNUM BN_mod_inverse(BIGNUM in,
515		const BIGNUM a, const BIGNUM n, BN_CTX *ctx)
516	1.07M	{
517	1.07M	BN_CTX *new_ctx = NULL;
518	1.07M	BIGNUM *rv;
519	1.07M	int noinv = 0;
520
521	1.07M	if (ctx == NULL) {
522	0	ctx = new_ctx = BN_CTX_new_ex(NULL);
523	0	if (ctx == NULL) {
524	0	ERR_raise(ERR_LIB_BN, ERR_R_BN_LIB);
525	0	return NULL;
526	0	}
527	0	}
528
529	1.07M	rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
530	1.07M	if (noinv)
531	1.07M	ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE);
532	1.07M	BN_CTX_free(new_ctx);
533	1.07M	return rv;
534	1.07M	}
535
536		/*
537		* The numbers a and b are coprime if the only positive integer that is a
538		* divisor of both of them is 1.
539		* i.e. gcd(a,b) = 1.
540		*
541		* Coprimes have the property: b has a multiplicative inverse modulo a
542		* i.e there is some value x such that bx = 1 (mod a).
543		*
544		* Testing the modulo inverse is currently much faster than the constant
545		* time version of BN_gcd().
546		*/
547		int BN_are_coprime(BIGNUM a, const BIGNUM b, BN_CTX *ctx)
548	0	{
549	0	int ret = 0;
550	0	BIGNUM *tmp;
551
552	0	BN_CTX_start(ctx);
553	0	tmp = BN_CTX_get(ctx);
554	0	if (tmp == NULL)
555	0	goto end;
556
557	0	ERR_set_mark();
558	0	BN_set_flags(a, BN_FLG_CONSTTIME);
559	0	ret = (BN_mod_inverse(tmp, a, b, ctx) != NULL);
560		/* Clear any errors (an error is returned if there is no inverse) */
561	0	ERR_pop_to_mark();
562	0	end:
563	0	BN_CTX_end(ctx);
564	0	return ret;
565	0	}
566
567		/*-
568		* This function is based on the constant-time GCD work by Bernstein and Yang:
569		* https://eprint.iacr.org/2019/266
570		* Generalized fast GCD function to allow even inputs.
571		* The algorithm first finds the shared powers of 2 between
572		* the inputs, and removes them, reducing at least one of the
573		* inputs to an odd value. Then it proceeds to calculate the GCD.
574		* Before returning the resulting GCD, we take care of adding
575		* back the powers of two removed at the beginning.
576		* Note 1: we assume the bit length of both inputs is public information,
577		* since access to top potentially leaks this information.
578		*/
579		int BN_gcd(BIGNUM r, const BIGNUM in_a, const BIGNUM in_b, BN_CTX ctx)
580	1.44k	{
581	1.44k	BIGNUM g, temp = NULL;
582	1.44k	BN_ULONG mask = 0;
583	1.44k	int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
584
585		/* Note 2: zero input corner cases are not constant-time since they are
586		* handled immediately. An attacker can run an attack under this
587		* assumption without the need of side-channel information. */
588	1.44k	if (BN_is_zero(in_b)) {
589	5	ret = BN_copy(r, in_a) != NULL;
590	5	r->neg = 0;
591	5	return ret;
592	5	}
593	1.44k	if (BN_is_zero(in_a)) {
594	9	ret = BN_copy(r, in_b) != NULL;
595	9	r->neg = 0;
596	9	return ret;
597	9	}
598
599	1.43k	bn_check_top(in_a);
600	1.43k	bn_check_top(in_b);
601
602	1.43k	BN_CTX_start(ctx);
603	1.43k	temp = BN_CTX_get(ctx);
604	1.43k	g = BN_CTX_get(ctx);
605
606		/* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
607	1.43k	if (g == NULL
608	1.43k	\|\| !BN_lshift1(g, in_b)
609	1.43k	\|\| !BN_lshift1(r, in_a))
610	0	goto err;
611
612		/* find shared powers of two, i.e. "shifts" >= 1 */
613	38.5k	for (i = 0; i < r->dmax && i < g->dmax; i++) {
614	37.1k	mask = ~(r->d[i] \| g->d[i]);
615	2.41M	for (j = 0; j < BN_BITS2; j++) {
616	2.37M	bit &= mask;
617	2.37M	shifts += bit;
618	2.37M	mask >>= 1;
619	2.37M	}
620	37.1k	}
621
622		/* subtract shared powers of two; shifts >= 1 */
623	1.43k	if (!BN_rshift(r, r, shifts)
624	1.43k	\|\| !BN_rshift(g, g, shifts))
625	0	goto err;
626
627		/* expand to biggest nword, with room for a possible extra word */
628	1.43k	top = 1 + ((r->top >= g->top) ? r->top : g->top);
629	1.43k	if (bn_wexpand(r, top) == NULL
630	1.43k	\|\| bn_wexpand(g, top) == NULL
631	1.43k	\|\| bn_wexpand(temp, top) == NULL)
632	0	goto err;
633
634		/* re arrange inputs s.t. r is odd */
635	1.43k	BN_consttime_swap((~r->d[0]) & 1, r, g, top);
636
637		/* compute the number of iterations */
638	1.43k	rlen = BN_num_bits(r);
639	1.43k	glen = BN_num_bits(g);
640	1.43k	m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
641
642	11.7M	for (i = 0; i < m; i++) {
643		/* conditionally flip signs if delta is positive and g is odd */
644	11.7M	cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
645		/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
646	11.7M	& (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1)));
647	11.7M	delta = (-cond & -delta) \| ((cond - 1) & delta);
648	11.7M	r->neg ^= cond;
649		/* swap */
650	11.7M	BN_consttime_swap(cond, r, g, top);
651
652		/* elimination step */
653	11.7M	delta++;
654	11.7M	if (!BN_add(temp, g, r))
655	0	goto err;
656	11.7M	BN_consttime_swap(g->d[0] & 1 /* g is odd */
657		/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
658	11.7M	& (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))),
659	11.7M	g, temp, top);
660	11.7M	if (!BN_rshift1(g, g))
661	0	goto err;
662	11.7M	}
663
664		/* remove possible negative sign */
665	1.43k	r->neg = 0;
666		/* add powers of 2 removed, then correct the artificial shift */
667	1.43k	if (!BN_lshift(r, r, shifts)
668	1.43k	\|\| !BN_rshift1(r, r))
669	0	goto err;
670
671	1.43k	ret = 1;
672
673	1.43k	err:
674	1.43k	BN_CTX_end(ctx);
675	1.43k	bn_check_top(r);
676	1.43k	return ret;
677	1.43k	}