/src/openssl34/crypto/bn/bn_gcd.c

Source
/*
 * Copyright 1995-2024 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"
#include "internal/constant_time.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 pow2_numbits, pow2_numbits_temp, pow2_condition_mask, pow2_flag;
    int i, j, top, rlen, glen, m, delta = 1, cond = 0, pow2_shifts, 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 */
    pow2_flag = 1;
    pow2_shifts = 0;
    pow2_numbits = 0;
    for (i = 0; i < r->dmax && i < g->dmax; i++) {
        pow2_numbits_temp = r->d[i] | g->d[i];
        pow2_condition_mask = constant_time_is_zero_bn(pow2_flag);
        pow2_flag &= constant_time_is_zero_bn(pow2_numbits_temp);
        pow2_shifts += pow2_flag;
        pow2_numbits = constant_time_select_bn(pow2_condition_mask,
            pow2_numbits, pow2_numbits_temp);
    }
    pow2_numbits = ~pow2_numbits;
    pow2_shifts *= BN_BITS2;
    pow2_flag = 1;
    for (j = 0; j < BN_BITS2; j++) {
        pow2_flag &= pow2_numbits;
        pow2_shifts += pow2_flag;
        pow2_numbits >>= 1;
    }

    /* subtract shared powers of two; shifts >= 1 */
    if (!BN_rshift(r, r, pow2_shifts)
        || !BN_rshift(g, g, pow2_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, pow2_shifts)
        || !BN_rshift1(r, r))
        goto err;

    ret = 1;

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

Coverage Report

Created: 2026-04-01 06:39

Line	Count	Source
1		/*
2		* Copyright 1995-2024 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		#include "internal/constant_time.h"
13
14		/*
15		* bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
16		* not contain branches that may leak sensitive information.
17		*
18		* This is a static function, we ensure all callers in this file pass valid
19		* arguments: all passed pointers here are non-NULL.
20		*/
21		static ossl_inline BIGNUM bn_mod_inverse_no_branch(BIGNUM in,
22		const BIGNUM a, const BIGNUM n,
23		BN_CTX ctx, int pnoinv)
24	43.5k	{
25	43.5k	BIGNUM A, B, X, Y, M, D, T, R = NULL;
26	43.5k	BIGNUM *ret = NULL;
27	43.5k	int sign;
28
29	43.5k	bn_check_top(a);
30	43.5k	bn_check_top(n);
31
32	43.5k	BN_CTX_start(ctx);
33	43.5k	A = BN_CTX_get(ctx);
34	43.5k	B = BN_CTX_get(ctx);
35	43.5k	X = BN_CTX_get(ctx);
36	43.5k	D = BN_CTX_get(ctx);
37	43.5k	M = BN_CTX_get(ctx);
38	43.5k	Y = BN_CTX_get(ctx);
39	43.5k	T = BN_CTX_get(ctx);
40	43.5k	if (T == NULL)
41	0	goto err;
42
43	43.5k	if (in == NULL)
44	0	R = BN_new();
45	43.5k	else
46	43.5k	R = in;
47	43.5k	if (R == NULL)
48	0	goto err;
49
50	43.5k	if (!BN_one(X))
51	0	goto err;
52	43.5k	BN_zero(Y);
53	43.5k	if (BN_copy(B, a) == NULL)
54	0	goto err;
55	43.5k	if (BN_copy(A, n) == NULL)
56	0	goto err;
57	43.5k	A->neg = 0;
58
59	43.5k	if (B->neg \|\| (BN_ucmp(B, A) >= 0)) {
60		/*
61		* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
62		* BN_div_no_branch will be called eventually.
63		*/
64	29.1k	{
65	29.1k	BIGNUM local_B;
66	29.1k	bn_init(&local_B);
67	29.1k	BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
68	29.1k	if (!BN_nnmod(B, &local_B, A, ctx))
69	0	goto err;
70		/* Ensure local_B goes out of scope before any further use of B */
71	29.1k	}
72	29.1k	}
73	43.5k	sign = -1;
74		/*-
75		* From B = a mod \|n\|, A = \|n\| it follows that
76		*
77		* 0 <= B < A,
78		* -signXa == B (mod \|n\|),
79		* signYa == A (mod \|n\|).
80		*/
81
82	17.7M	while (!BN_is_zero(B)) {
83	17.6M	BIGNUM *tmp;
84
85		/*-
86		* 0 < B < A,
87		* () -signX*a == B (mod \|n\|),
88		* signYa == A (mod \|n\|)
89		*/
90
91		/*
92		* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
93		* BN_div_no_branch will be called eventually.
94		*/
95	17.6M	{
96	17.6M	BIGNUM local_A;
97	17.6M	bn_init(&local_A);
98	17.6M	BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
99
100		/* (D, M) := (A/B, A%B) ... */
101	17.6M	if (!BN_div(D, M, &local_A, B, ctx))
102	0	goto err;
103		/* Ensure local_A goes out of scope before any further use of A */
104	17.6M	}
105
106		/*-
107		* Now
108		* A = D*B + M;
109		* thus we have
110		* (*) signYa == DB + M (mod \|n\|).
111		*/
112
113	17.6M	tmp = A; /* keep the BIGNUM object, the value does not
114		* matter */
115
116		/* (A, B) := (B, A mod B) ... */
117	17.6M	A = B;
118	17.6M	B = M;
119		/* ... so we have 0 <= B < A again */
120
121		/*-
122		* Since the former M is now B and the former B is now A,
123		* (**) translates into
124		* signYa == D*A + B (mod \|n\|),
125		* i.e.
126		* signYa - D*A == B (mod \|n\|).
127		* Similarly, (*) translates into
128		* -signXa == A (mod \|n\|).
129		*
130		* Thus,
131		* signYa + DsignX*a == B (mod \|n\|),
132		* i.e.
133		* sign(Y + DX)*a == B (mod \|n\|).
134		*
135		* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
136		* -signXa == B (mod \|n\|),
137		* signYa == A (mod \|n\|).
138		* Note that X and Y stay non-negative all the time.
139		*/
140
141	17.6M	if (!BN_mul(tmp, D, X, ctx))
142	0	goto err;
143	17.6M	if (!BN_add(tmp, tmp, Y))
144	0	goto err;
145
146	17.6M	M = Y; /* keep the BIGNUM object, the value does not
147		* matter */
148	17.6M	Y = X;
149	17.6M	X = tmp;
150	17.6M	sign = -sign;
151	17.6M	}
152
153		/*-
154		* The while loop (Euclid's algorithm) ends when
155		* A == gcd(a,n);
156		* we have
157		* signYa == A (mod \|n\|),
158		* where Y is non-negative.
159		*/
160
161	43.5k	if (sign < 0) {
162	13.5k	if (!BN_sub(Y, n, Y))
163	0	goto err;
164	13.5k	}
165		/* Now Ya == A (mod \|n\|). /
166
167	43.5k	if (BN_is_one(A)) {
168		/* Ya == 1 (mod \|n\|) /
169	43.1k	if (!Y->neg && BN_ucmp(Y, n) < 0) {
170	43.1k	if (!BN_copy(R, Y))
171	0	goto err;
172	43.1k	} else {
173	0	if (!BN_nnmod(R, Y, n, ctx))
174	0	goto err;
175	0	}
176	43.1k	} else {
177	410	*pnoinv = 1;
178		/* caller sets the BN_R_NO_INVERSE error */
179	410	goto err;
180	410	}
181
182	43.1k	ret = R;
183	43.1k	*pnoinv = 0;
184
185	43.5k	err:
186	43.5k	if ((ret == NULL) && (in == NULL))
187	0	BN_free(R);
188	43.5k	BN_CTX_end(ctx);
189	43.5k	bn_check_top(ret);
190	43.5k	return ret;
191	43.1k	}
192
193		/*
194		* This is an internal function, we assume all callers pass valid arguments:
195		* all pointers passed here are assumed non-NULL.
196		*/
197		BIGNUM int_bn_mod_inverse(BIGNUM in,
198		const BIGNUM a, const BIGNUM n, BN_CTX *ctx,
199		int *pnoinv)
200	951k	{
201	951k	BIGNUM A, B, X, Y, M, D, T, R = NULL;
202	951k	BIGNUM *ret = NULL;
203	951k	int sign;
204
205		/* This is invalid input so we don't worry about constant time here */
206	951k	if (BN_abs_is_word(n, 1) \|\| BN_is_zero(n)) {
207	419	*pnoinv = 1;
208	419	return NULL;
209	419	}
210
211	950k	*pnoinv = 0;
212
213	950k	if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
214	949k	\|\| (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
215	43.5k	return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
216	43.5k	}
217
218	907k	bn_check_top(a);
219	907k	bn_check_top(n);
220
221	907k	BN_CTX_start(ctx);
222	907k	A = BN_CTX_get(ctx);
223	907k	B = BN_CTX_get(ctx);
224	907k	X = BN_CTX_get(ctx);
225	907k	D = BN_CTX_get(ctx);
226	907k	M = BN_CTX_get(ctx);
227	907k	Y = BN_CTX_get(ctx);
228	907k	T = BN_CTX_get(ctx);
229	907k	if (T == NULL)
230	0	goto err;
231
232	907k	if (in == NULL)
233	0	R = BN_new();
234	907k	else
235	907k	R = in;
236	907k	if (R == NULL)
237	0	goto err;
238
239	907k	if (!BN_one(X))
240	0	goto err;
241	907k	BN_zero(Y);
242	907k	if (BN_copy(B, a) == NULL)
243	0	goto err;
244	907k	if (BN_copy(A, n) == NULL)
245	0	goto err;
246	907k	A->neg = 0;
247	907k	if (B->neg \|\| (BN_ucmp(B, A) >= 0)) {
248	904k	if (!BN_nnmod(B, B, A, ctx))
249	0	goto err;
250	904k	}
251	907k	sign = -1;
252		/*-
253		* From B = a mod \|n\|, A = \|n\| it follows that
254		*
255		* 0 <= B < A,
256		* -signXa == B (mod \|n\|),
257		* signYa == A (mod \|n\|).
258		*/
259
260	907k	if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
261		/*
262		* Binary inversion algorithm; requires odd modulus. This is faster
263		* than the general algorithm if the modulus is sufficiently small
264		* (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
265		* systems)
266		*/
267	899k	int shift;
268
269	55.3M	while (!BN_is_zero(B)) {
270		/*-
271		* 0 < B < \|n\|,
272		* 0 < A <= \|n\|,
273		* (1) -signXa == B (mod \|n\|),
274		* (2) signYa == A (mod \|n\|)
275		*/
276
277		/*
278		* Now divide B by the maximum possible power of two in the
279		* integers, and divide X by the same value mod \|n\|. When we're
280		* done, (1) still holds.
281		*/
282	54.4M	shift = 0;
283	75.4M	while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
284	21.0M	shift++;
285
286	21.0M	if (BN_is_odd(X)) {
287	10.6M	if (!BN_uadd(X, X, n))
288	0	goto err;
289	10.6M	}
290		/*
291		* now X is even, so we can easily divide it by two
292		*/
293	21.0M	if (!BN_rshift1(X, X))
294	0	goto err;
295	21.0M	}
296	54.4M	if (shift > 0) {
297	19.8M	if (!BN_rshift(B, B, shift))
298	0	goto err;
299	19.8M	}
300
301		/*
302		* Same for A and Y. Afterwards, (2) still holds.
303		*/
304	54.4M	shift = 0;
305	89.2M	while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
306	34.7M	shift++;
307
308	34.7M	if (BN_is_odd(Y)) {
309	22.3M	if (!BN_uadd(Y, Y, n))
310	0	goto err;
311	22.3M	}
312		/* now Y is even */
313	34.7M	if (!BN_rshift1(Y, Y))
314	0	goto err;
315	34.7M	}
316	54.4M	if (shift > 0) {
317	34.0M	if (!BN_rshift(A, A, shift))
318	0	goto err;
319	34.0M	}
320
321		/*-
322		* We still have (1) and (2).
323		* Both A and B are odd.
324		* The following computations ensure that
325		*
326		* 0 <= B < \|n\|,
327		* 0 < A < \|n\|,
328		* (1) -signXa == B (mod \|n\|),
329		* (2) signYa == A (mod \|n\|),
330		*
331		* and that either A or B is even in the next iteration.
332		*/
333	54.4M	if (BN_ucmp(B, A) >= 0) {
334		/* -sign(X + Y)a == B - A (mod \|n\|) */
335	20.3M	if (!BN_uadd(X, X, Y))
336	0	goto err;
337		/*
338		* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
339		* actually makes the algorithm slower
340		*/
341	20.3M	if (!BN_usub(B, B, A))
342	0	goto err;
343	34.0M	} else {
344		/* sign(X + Y)a == A - B (mod \|n\|) */
345	34.0M	if (!BN_uadd(Y, Y, X))
346	0	goto err;
347		/*
348		* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
349		*/
350	34.0M	if (!BN_usub(A, A, B))
351	0	goto err;
352	34.0M	}
353	54.4M	}
354	899k	} else {
355		/* general inversion algorithm */
356
357	280k	while (!BN_is_zero(B)) {
358	272k	BIGNUM *tmp;
359
360		/*-
361		* 0 < B < A,
362		* () -signX*a == B (mod \|n\|),
363		* signYa == A (mod \|n\|)
364		*/
365
366		/* (D, M) := (A/B, A%B) ... */
367	272k	if (BN_num_bits(A) == BN_num_bits(B)) {
368	60.1k	if (!BN_one(D))
369	0	goto err;
370	60.1k	if (!BN_sub(M, A, B))
371	0	goto err;
372	212k	} else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
373		/* A/B is 1, 2, or 3 */
374	95.9k	if (!BN_lshift1(T, B))
375	0	goto err;
376	95.9k	if (BN_ucmp(A, T) < 0) {
377		/* A < 2B, so D=1 /
378	54.6k	if (!BN_one(D))
379	0	goto err;
380	54.6k	if (!BN_sub(M, A, B))
381	0	goto err;
382	54.6k	} else {
383		/* A >= 2B, so D=2 or D=3 /
384	41.2k	if (!BN_sub(M, A, T))
385	0	goto err;
386	41.2k	if (!BN_add(D, T, B))
387	0	goto err; /* use D (:= 3B) as temp /
388	41.2k	if (BN_ucmp(A, D) < 0) {
389		/* A < 3B, so D=2 /
390	34.0k	if (!BN_set_word(D, 2))
391	0	goto err;
392		/*
393		* M (= A - 2*B) already has the correct value
394		*/
395	34.0k	} else {
396		/* only D=3 remains */
397	7.26k	if (!BN_set_word(D, 3))
398	0	goto err;
399		/*
400		* currently M = A - 2B, but we need M = A - 3B
401		*/
402	7.26k	if (!BN_sub(M, M, B))
403	0	goto err;
404	7.26k	}
405	41.2k	}
406	116k	} else {
407	116k	if (!BN_div(D, M, A, B, ctx))
408	0	goto err;
409	116k	}
410
411		/*-
412		* Now
413		* A = D*B + M;
414		* thus we have
415		* (*) signYa == DB + M (mod \|n\|).
416		*/
417
418	272k	tmp = A; /* keep the BIGNUM object, the value does not matter */
419
420		/* (A, B) := (B, A mod B) ... */
421	272k	A = B;
422	272k	B = M;
423		/* ... so we have 0 <= B < A again */
424
425		/*-
426		* Since the former M is now B and the former B is now A,
427		* (**) translates into
428		* signYa == D*A + B (mod \|n\|),
429		* i.e.
430		* signYa - D*A == B (mod \|n\|).
431		* Similarly, (*) translates into
432		* -signXa == A (mod \|n\|).
433		*
434		* Thus,
435		* signYa + DsignX*a == B (mod \|n\|),
436		* i.e.
437		* sign(Y + DX)*a == B (mod \|n\|).
438		*
439		* So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
440		* -signXa == B (mod \|n\|),
441		* signYa == A (mod \|n\|).
442		* Note that X and Y stay non-negative all the time.
443		*/
444
445		/*
446		* most of the time D is very small, so we can optimize tmp := D*X+Y
447		*/
448	272k	if (BN_is_one(D)) {
449	114k	if (!BN_add(tmp, X, Y))
450	0	goto err;
451	158k	} else {
452	158k	if (BN_is_word(D, 2)) {
453	48.3k	if (!BN_lshift1(tmp, X))
454	0	goto err;
455	109k	} else if (BN_is_word(D, 4)) {
456	15.9k	if (!BN_lshift(tmp, X, 2))
457	0	goto err;
458	93.8k	} else if (D->top == 1) {
459	93.6k	if (!BN_copy(tmp, X))
460	0	goto err;
461	93.6k	if (!BN_mul_word(tmp, D->d[0]))
462	0	goto err;
463	93.6k	} else {
464	187	if (!BN_mul(tmp, D, X, ctx))
465	0	goto err;
466	187	}
467	158k	if (!BN_add(tmp, tmp, Y))
468	0	goto err;
469	158k	}
470
471	272k	M = Y; /* keep the BIGNUM object, the value does not matter */
472	272k	Y = X;
473	272k	X = tmp;
474	272k	sign = -sign;
475	272k	}
476	7.10k	}
477
478		/*-
479		* The while loop (Euclid's algorithm) ends when
480		* A == gcd(a,n);
481		* we have
482		* signYa == A (mod \|n\|),
483		* where Y is non-negative.
484		*/
485
486	907k	if (sign < 0) {
487	903k	if (!BN_sub(Y, n, Y))
488	0	goto err;
489	903k	}
490		/* Now Ya == A (mod \|n\|). /
491
492	907k	if (BN_is_one(A)) {
493		/* Ya == 1 (mod \|n\|) /
494	901k	if (!Y->neg && BN_ucmp(Y, n) < 0) {
495	224k	if (!BN_copy(R, Y))
496	0	goto err;
497	677k	} else {
498	677k	if (!BN_nnmod(R, Y, n, ctx))
499	0	goto err;
500	677k	}
501	901k	} else {
502	5.77k	*pnoinv = 1;
503	5.77k	goto err;
504	5.77k	}
505	901k	ret = R;
506	907k	err:
507	907k	if ((ret == NULL) && (in == NULL))
508	0	BN_free(R);
509	907k	BN_CTX_end(ctx);
510	907k	bn_check_top(ret);
511	907k	return ret;
512	901k	}
513
514		/* solves ax == 1 (mod n) */
515		BIGNUM BN_mod_inverse(BIGNUM in,
516		const BIGNUM a, const BIGNUM n, BN_CTX *ctx)
517	1.13M	{
518	1.13M	BN_CTX *new_ctx = NULL;
519	1.13M	BIGNUM *rv;
520	1.13M	int noinv = 0;
521
522	1.13M	if (ctx == NULL) {
523	0	ctx = new_ctx = BN_CTX_new_ex(NULL);
524	0	if (ctx == NULL) {
525	0	ERR_raise(ERR_LIB_BN, ERR_R_BN_LIB);
526	0	return NULL;
527	0	}
528	0	}
529
530	1.13M	rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
531	1.13M	if (noinv)
532	1.13M	ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE);
533	1.13M	BN_CTX_free(new_ctx);
534	1.13M	return rv;
535	1.13M	}
536
537		/*
538		* The numbers a and b are coprime if the only positive integer that is a
539		* divisor of both of them is 1.
540		* i.e. gcd(a,b) = 1.
541		*
542		* Coprimes have the property: b has a multiplicative inverse modulo a
543		* i.e there is some value x such that bx = 1 (mod a).
544		*
545		* Testing the modulo inverse is currently much faster than the constant
546		* time version of BN_gcd().
547		*/
548		int BN_are_coprime(BIGNUM a, const BIGNUM b, BN_CTX *ctx)
549	0	{
550	0	int ret = 0;
551	0	BIGNUM *tmp;
552
553	0	BN_CTX_start(ctx);
554	0	tmp = BN_CTX_get(ctx);
555	0	if (tmp == NULL)
556	0	goto end;
557
558	0	ERR_set_mark();
559	0	BN_set_flags(a, BN_FLG_CONSTTIME);
560	0	ret = (BN_mod_inverse(tmp, a, b, ctx) != NULL);
561		/* Clear any errors (an error is returned if there is no inverse) */
562	0	ERR_pop_to_mark();
563	0	end:
564	0	BN_CTX_end(ctx);
565	0	return ret;
566	0	}
567
568		/*-
569		* This function is based on the constant-time GCD work by Bernstein and Yang:
570		* https://eprint.iacr.org/2019/266
571		* Generalized fast GCD function to allow even inputs.
572		* The algorithm first finds the shared powers of 2 between
573		* the inputs, and removes them, reducing at least one of the
574		* inputs to an odd value. Then it proceeds to calculate the GCD.
575		* Before returning the resulting GCD, we take care of adding
576		* back the powers of two removed at the beginning.
577		* Note 1: we assume the bit length of both inputs is public information,
578		* since access to top potentially leaks this information.
579		*/
580		int BN_gcd(BIGNUM r, const BIGNUM in_a, const BIGNUM in_b, BN_CTX ctx)
581	5.99k	{
582	5.99k	BIGNUM g, temp = NULL;
583	5.99k	BN_ULONG pow2_numbits, pow2_numbits_temp, pow2_condition_mask, pow2_flag;
584	5.99k	int i, j, top, rlen, glen, m, delta = 1, cond = 0, pow2_shifts, ret = 0;
585
586		/* Note 2: zero input corner cases are not constant-time since they are
587		* handled immediately. An attacker can run an attack under this
588		* assumption without the need of side-channel information. */
589	5.99k	if (BN_is_zero(in_b)) {
590	43	ret = BN_copy(r, in_a) != NULL;
591	43	r->neg = 0;
592	43	return ret;
593	43	}
594	5.95k	if (BN_is_zero(in_a)) {
595	45	ret = BN_copy(r, in_b) != NULL;
596	45	r->neg = 0;
597	45	return ret;
598	45	}
599
600	5.91k	bn_check_top(in_a);
601	5.91k	bn_check_top(in_b);
602
603	5.91k	BN_CTX_start(ctx);
604	5.91k	temp = BN_CTX_get(ctx);
605	5.91k	g = BN_CTX_get(ctx);
606
607		/* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
608	5.91k	if (g == NULL
609	5.91k	\|\| !BN_lshift1(g, in_b)
610	5.91k	\|\| !BN_lshift1(r, in_a))
611	0	goto err;
612
613		/* find shared powers of two, i.e. "shifts" >= 1 */
614	5.91k	pow2_flag = 1;
615	5.91k	pow2_shifts = 0;
616	5.91k	pow2_numbits = 0;
617	150k	for (i = 0; i < r->dmax && i < g->dmax; i++) {
618	144k	pow2_numbits_temp = r->d[i] \| g->d[i];
619	144k	pow2_condition_mask = constant_time_is_zero_bn(pow2_flag);
620	144k	pow2_flag &= constant_time_is_zero_bn(pow2_numbits_temp);
621	144k	pow2_shifts += pow2_flag;
622	144k	pow2_numbits = constant_time_select_bn(pow2_condition_mask,
623	144k	pow2_numbits, pow2_numbits_temp);
624	144k	}
625	5.91k	pow2_numbits = ~pow2_numbits;
626	5.91k	pow2_shifts *= BN_BITS2;
627	5.91k	pow2_flag = 1;
628	384k	for (j = 0; j < BN_BITS2; j++) {
629	378k	pow2_flag &= pow2_numbits;
630	378k	pow2_shifts += pow2_flag;
631	378k	pow2_numbits >>= 1;
632	378k	}
633
634		/* subtract shared powers of two; shifts >= 1 */
635	5.91k	if (!BN_rshift(r, r, pow2_shifts)
636	5.91k	\|\| !BN_rshift(g, g, pow2_shifts))
637	0	goto err;
638
639		/* expand to biggest nword, with room for a possible extra word */
640	5.91k	top = 1 + ((r->top >= g->top) ? r->top : g->top);
641	5.91k	if (bn_wexpand(r, top) == NULL
642	5.91k	\|\| bn_wexpand(g, top) == NULL
643	5.91k	\|\| bn_wexpand(temp, top) == NULL)
644	0	goto err;
645
646		/* re arrange inputs s.t. r is odd */
647	5.91k	BN_consttime_swap((~r->d[0]) & 1, r, g, top);
648
649		/* compute the number of iterations */
650	5.91k	rlen = BN_num_bits(r);
651	5.91k	glen = BN_num_bits(g);
652	5.91k	m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
653
654	40.5M	for (i = 0; i < m; i++) {
655		/* conditionally flip signs if delta is positive and g is odd */
656	40.5M	cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
657		/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
658	40.5M	& (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1)));
659	40.5M	delta = (-cond & -delta) \| ((cond - 1) & delta);
660	40.5M	r->neg ^= cond;
661		/* swap */
662	40.5M	BN_consttime_swap(cond, r, g, top);
663
664		/* elimination step */
665	40.5M	delta++;
666	40.5M	if (!BN_add(temp, g, r))
667	0	goto err;
668	40.5M	BN_consttime_swap(g->d[0] & 1 /* g is odd */
669		/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
670	40.5M	& (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))),
671	40.5M	g, temp, top);
672	40.5M	if (!BN_rshift1(g, g))
673	0	goto err;
674	40.5M	}
675
676		/* remove possible negative sign */
677	5.91k	r->neg = 0;
678		/* add powers of 2 removed, then correct the artificial shift */
679	5.91k	if (!BN_lshift(r, r, pow2_shifts)
680	5.91k	\|\| !BN_rshift1(r, r))
681	0	goto err;
682
683	5.91k	ret = 1;
684
685	5.91k	err:
686	5.91k	BN_CTX_end(ctx);
687	5.91k	bn_check_top(r);
688	5.91k	return ret;
689	5.91k	}