/src/openssl30/crypto/bn/bn_gcd.c

Source
/*
 * Copyright 1995-2023 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_MALLOC_FAILURE);
            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;
}

/*-
 * 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: 2025-12-31 06:58

Line	Count	Source
1		/*
2		* Copyright 1995-2023 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	54.7k	{
24	54.7k	BIGNUM A, B, X, Y, M, D, T, R = NULL;
25	54.7k	BIGNUM *ret = NULL;
26	54.7k	int sign;
27
28	54.7k	bn_check_top(a);
29	54.7k	bn_check_top(n);
30
31	54.7k	BN_CTX_start(ctx);
32	54.7k	A = BN_CTX_get(ctx);
33	54.7k	B = BN_CTX_get(ctx);
34	54.7k	X = BN_CTX_get(ctx);
35	54.7k	D = BN_CTX_get(ctx);
36	54.7k	M = BN_CTX_get(ctx);
37	54.7k	Y = BN_CTX_get(ctx);
38	54.7k	T = BN_CTX_get(ctx);
39	54.7k	if (T == NULL)
40	0	goto err;
41
42	54.7k	if (in == NULL)
43	0	R = BN_new();
44	54.7k	else
45	54.7k	R = in;
46	54.7k	if (R == NULL)
47	0	goto err;
48
49	54.7k	if (!BN_one(X))
50	0	goto err;
51	54.7k	BN_zero(Y);
52	54.7k	if (BN_copy(B, a) == NULL)
53	0	goto err;
54	54.7k	if (BN_copy(A, n) == NULL)
55	0	goto err;
56	54.7k	A->neg = 0;
57
58	54.7k	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	36.6k	{
64	36.6k	BIGNUM local_B;
65	36.6k	bn_init(&local_B);
66	36.6k	BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
67	36.6k	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	36.6k	}
71	36.6k	}
72	54.7k	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	22.0M	while (!BN_is_zero(B)) {
82	21.9M	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	21.9M	{
95	21.9M	BIGNUM local_A;
96	21.9M	bn_init(&local_A);
97	21.9M	BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
98
99		/* (D, M) := (A/B, A%B) ... */
100	21.9M	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	21.9M	}
104
105		/*-
106		* Now
107		* A = D*B + M;
108		* thus we have
109		* (*) signYa == DB + M (mod \|n\|).
110		*/
111
112	21.9M	tmp = A; /* keep the BIGNUM object, the value does not
113		* matter */
114
115		/* (A, B) := (B, A mod B) ... */
116	21.9M	A = B;
117	21.9M	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	21.9M	if (!BN_mul(tmp, D, X, ctx))
141	0	goto err;
142	21.9M	if (!BN_add(tmp, tmp, Y))
143	0	goto err;
144
145	21.9M	M = Y; /* keep the BIGNUM object, the value does not
146		* matter */
147	21.9M	Y = X;
148	21.9M	X = tmp;
149	21.9M	sign = -sign;
150	21.9M	}
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	54.7k	if (sign < 0) {
161	17.2k	if (!BN_sub(Y, n, Y))
162	0	goto err;
163	17.2k	}
164		/* Now Ya == A (mod \|n\|). /
165
166	54.7k	if (BN_is_one(A)) {
167		/* Ya == 1 (mod \|n\|) /
168	54.3k	if (!Y->neg && BN_ucmp(Y, n) < 0) {
169	54.3k	if (!BN_copy(R, Y))
170	0	goto err;
171	54.3k	} else {
172	0	if (!BN_nnmod(R, Y, n, ctx))
173	0	goto err;
174	0	}
175	54.3k	} else {
176	464	*pnoinv = 1;
177		/* caller sets the BN_R_NO_INVERSE error */
178	464	goto err;
179	464	}
180
181	54.3k	ret = R;
182	54.3k	*pnoinv = 0;
183
184	54.7k	err:
185	54.7k	if ((ret == NULL) && (in == NULL))
186	0	BN_free(R);
187	54.7k	BN_CTX_end(ctx);
188	54.7k	bn_check_top(ret);
189	54.7k	return ret;
190	54.3k	}
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	1.04M	{
200	1.04M	BIGNUM A, B, X, Y, M, D, T, R = NULL;
201	1.04M	BIGNUM *ret = NULL;
202	1.04M	int sign;
203
204		/* This is invalid input so we don't worry about constant time here */
205	1.04M	if (BN_abs_is_word(n, 1) \|\| BN_is_zero(n)) {
206	570	*pnoinv = 1;
207	570	return NULL;
208	570	}
209
210	1.04M	*pnoinv = 0;
211
212	1.04M	if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
213	1.04M	\|\| (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
214	54.7k	return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
215	54.7k	}
216
217	990k	bn_check_top(a);
218	990k	bn_check_top(n);
219
220	990k	BN_CTX_start(ctx);
221	990k	A = BN_CTX_get(ctx);
222	990k	B = BN_CTX_get(ctx);
223	990k	X = BN_CTX_get(ctx);
224	990k	D = BN_CTX_get(ctx);
225	990k	M = BN_CTX_get(ctx);
226	990k	Y = BN_CTX_get(ctx);
227	990k	T = BN_CTX_get(ctx);
228	990k	if (T == NULL)
229	0	goto err;
230
231	990k	if (in == NULL)
232	0	R = BN_new();
233	990k	else
234	990k	R = in;
235	990k	if (R == NULL)
236	0	goto err;
237
238	990k	if (!BN_one(X))
239	0	goto err;
240	990k	BN_zero(Y);
241	990k	if (BN_copy(B, a) == NULL)
242	0	goto err;
243	990k	if (BN_copy(A, n) == NULL)
244	0	goto err;
245	990k	A->neg = 0;
246	990k	if (B->neg \|\| (BN_ucmp(B, A) >= 0)) {
247	987k	if (!BN_nnmod(B, B, A, ctx))
248	0	goto err;
249	987k	}
250	990k	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	990k	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	981k	int shift;
267
268	60.4M	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	59.4M	shift = 0;
282	82.8M	while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
283	23.3M	shift++;
284
285	23.3M	if (BN_is_odd(X)) {
286	11.8M	if (!BN_uadd(X, X, n))
287	0	goto err;
288	11.8M	}
289		/*
290		* now X is even, so we can easily divide it by two
291		*/
292	23.3M	if (!BN_rshift1(X, X))
293	0	goto err;
294	23.3M	}
295	59.4M	if (shift > 0) {
296	22.0M	if (!BN_rshift(B, B, shift))
297	0	goto err;
298	22.0M	}
299
300		/*
301		* Same for A and Y. Afterwards, (2) still holds.
302		*/
303	59.4M	shift = 0;
304	97.0M	while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
305	37.5M	shift++;
306
307	37.5M	if (BN_is_odd(Y)) {
308	24.0M	if (!BN_uadd(Y, Y, n))
309	0	goto err;
310	24.0M	}
311		/* now Y is even */
312	37.5M	if (!BN_rshift1(Y, Y))
313	0	goto err;
314	37.5M	}
315	59.4M	if (shift > 0) {
316	36.8M	if (!BN_rshift(A, A, shift))
317	0	goto err;
318	36.8M	}
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	59.4M	if (BN_ucmp(B, A) >= 0) {
333		/* -sign(X + Y)a == B - A (mod \|n\|) */
334	22.6M	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	22.6M	if (!BN_usub(B, B, A))
341	0	goto err;
342	36.8M	} else {
343		/* sign(X + Y)a == A - B (mod \|n\|) */
344	36.8M	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	36.8M	if (!BN_usub(A, A, B))
350	0	goto err;
351	36.8M	}
352	59.4M	}
353	981k	} else {
354		/* general inversion algorithm */
355
356	319k	while (!BN_is_zero(B)) {
357	311k	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	311k	if (BN_num_bits(A) == BN_num_bits(B)) {
367	68.1k	if (!BN_one(D))
368	0	goto err;
369	68.1k	if (!BN_sub(M, A, B))
370	0	goto err;
371	243k	} else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
372		/* A/B is 1, 2, or 3 */
373	108k	if (!BN_lshift1(T, B))
374	0	goto err;
375	108k	if (BN_ucmp(A, T) < 0) {
376		/* A < 2B, so D=1 /
377	61.7k	if (!BN_one(D))
378	0	goto err;
379	61.7k	if (!BN_sub(M, A, B))
380	0	goto err;
381	61.7k	} else {
382		/* A >= 2B, so D=2 or D=3 /
383	46.8k	if (!BN_sub(M, A, T))
384	0	goto err;
385	46.8k	if (!BN_add(D, T, B))
386	0	goto err; /* use D (:= 3B) as temp /
387	46.8k	if (BN_ucmp(A, D) < 0) {
388		/* A < 3B, so D=2 /
389	38.6k	if (!BN_set_word(D, 2))
390	0	goto err;
391		/*
392		* M (= A - 2*B) already has the correct value
393		*/
394	38.6k	} else {
395		/* only D=3 remains */
396	8.18k	if (!BN_set_word(D, 3))
397	0	goto err;
398		/*
399		* currently M = A - 2B, but we need M = A - 3B
400		*/
401	8.18k	if (!BN_sub(M, M, B))
402	0	goto err;
403	8.18k	}
404	46.8k	}
405	134k	} else {
406	134k	if (!BN_div(D, M, A, B, ctx))
407	0	goto err;
408	134k	}
409
410		/*-
411		* Now
412		* A = D*B + M;
413		* thus we have
414		* (*) signYa == DB + M (mod \|n\|).
415		*/
416
417	311k	tmp = A; /* keep the BIGNUM object, the value does not matter */
418
419		/* (A, B) := (B, A mod B) ... */
420	311k	A = B;
421	311k	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	311k	if (BN_is_one(D)) {
448	129k	if (!BN_add(tmp, X, Y))
449	0	goto err;
450	181k	} else {
451	181k	if (BN_is_word(D, 2)) {
452	55.2k	if (!BN_lshift1(tmp, X))
453	0	goto err;
454	126k	} else if (BN_is_word(D, 4)) {
455	18.4k	if (!BN_lshift(tmp, X, 2))
456	0	goto err;
457	107k	} else if (D->top == 1) {
458	107k	if (!BN_copy(tmp, X))
459	0	goto err;
460	107k	if (!BN_mul_word(tmp, D->d[0]))
461	0	goto err;
462	107k	} else {
463	216	if (!BN_mul(tmp, D, X, ctx))
464	0	goto err;
465	216	}
466	181k	if (!BN_add(tmp, tmp, Y))
467	0	goto err;
468	181k	}
469
470	311k	M = Y; /* keep the BIGNUM object, the value does not matter */
471	311k	Y = X;
472	311k	X = tmp;
473	311k	sign = -sign;
474	311k	}
475	8.46k	}
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	990k	if (sign < 0) {
486	986k	if (!BN_sub(Y, n, Y))
487	0	goto err;
488	986k	}
489		/* Now Ya == A (mod \|n\|). /
490
491	990k	if (BN_is_one(A)) {
492		/* Ya == 1 (mod \|n\|) /
493	983k	if (!Y->neg && BN_ucmp(Y, n) < 0) {
494	241k	if (!BN_copy(R, Y))
495	0	goto err;
496	741k	} else {
497	741k	if (!BN_nnmod(R, Y, n, ctx))
498	0	goto err;
499	741k	}
500	983k	} else {
501	6.95k	*pnoinv = 1;
502	6.95k	goto err;
503	6.95k	}
504	983k	ret = R;
505	990k	err:
506	990k	if ((ret == NULL) && (in == NULL))
507	0	BN_free(R);
508	990k	BN_CTX_end(ctx);
509	990k	bn_check_top(ret);
510	990k	return ret;
511	983k	}
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	1.02M	{
517	1.02M	BN_CTX *new_ctx = NULL;
518	1.02M	BIGNUM *rv;
519	1.02M	int noinv = 0;
520
521	1.02M	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_MALLOC_FAILURE);
525	0	return NULL;
526	0	}
527	0	}
528
529	1.02M	rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
530	1.02M	if (noinv)
531	1.02M	ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE);
532	1.02M	BN_CTX_free(new_ctx);
533	1.02M	return rv;
534	1.02M	}
535
536		/*-
537		* This function is based on the constant-time GCD work by Bernstein and Yang:
538		* https://eprint.iacr.org/2019/266
539		* Generalized fast GCD function to allow even inputs.
540		* The algorithm first finds the shared powers of 2 between
541		* the inputs, and removes them, reducing at least one of the
542		* inputs to an odd value. Then it proceeds to calculate the GCD.
543		* Before returning the resulting GCD, we take care of adding
544		* back the powers of two removed at the beginning.
545		* Note 1: we assume the bit length of both inputs is public information,
546		* since access to top potentially leaks this information.
547		*/
548		int BN_gcd(BIGNUM r, const BIGNUM in_a, const BIGNUM in_b, BN_CTX ctx)
549	1.50k	{
550	1.50k	BIGNUM g, temp = NULL;
551	1.50k	BN_ULONG mask = 0;
552	1.50k	int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
553
554		/* Note 2: zero input corner cases are not constant-time since they are
555		* handled immediately. An attacker can run an attack under this
556		* assumption without the need of side-channel information. */
557	1.50k	if (BN_is_zero(in_b)) {
558	4	ret = BN_copy(r, in_a) != NULL;
559	4	r->neg = 0;
560	4	return ret;
561	4	}
562	1.49k	if (BN_is_zero(in_a)) {
563	9	ret = BN_copy(r, in_b) != NULL;
564	9	r->neg = 0;
565	9	return ret;
566	9	}
567
568	1.49k	bn_check_top(in_a);
569	1.49k	bn_check_top(in_b);
570
571	1.49k	BN_CTX_start(ctx);
572	1.49k	temp = BN_CTX_get(ctx);
573	1.49k	g = BN_CTX_get(ctx);
574
575		/* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
576	1.49k	if (g == NULL
577	1.49k	\|\| !BN_lshift1(g, in_b)
578	1.49k	\|\| !BN_lshift1(r, in_a))
579	0	goto err;
580
581		/* find shared powers of two, i.e. "shifts" >= 1 */
582	40.6k	for (i = 0; i < r->dmax && i < g->dmax; i++) {
583	39.1k	mask = ~(r->d[i] \| g->d[i]);
584	2.54M	for (j = 0; j < BN_BITS2; j++) {
585	2.50M	bit &= mask;
586	2.50M	shifts += bit;
587	2.50M	mask >>= 1;
588	2.50M	}
589	39.1k	}
590
591		/* subtract shared powers of two; shifts >= 1 */
592	1.49k	if (!BN_rshift(r, r, shifts)
593	1.49k	\|\| !BN_rshift(g, g, shifts))
594	0	goto err;
595
596		/* expand to biggest nword, with room for a possible extra word */
597	1.49k	top = 1 + ((r->top >= g->top) ? r->top : g->top);
598	1.49k	if (bn_wexpand(r, top) == NULL
599	1.49k	\|\| bn_wexpand(g, top) == NULL
600	1.49k	\|\| bn_wexpand(temp, top) == NULL)
601	0	goto err;
602
603		/* re arrange inputs s.t. r is odd */
604	1.49k	BN_consttime_swap((~r->d[0]) & 1, r, g, top);
605
606		/* compute the number of iterations */
607	1.49k	rlen = BN_num_bits(r);
608	1.49k	glen = BN_num_bits(g);
609	1.49k	m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
610
611	11.6M	for (i = 0; i < m; i++) {
612		/* conditionally flip signs if delta is positive and g is odd */
613	11.6M	cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
614		/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
615	11.6M	& (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1)));
616	11.6M	delta = (-cond & -delta) \| ((cond - 1) & delta);
617	11.6M	r->neg ^= cond;
618		/* swap */
619	11.6M	BN_consttime_swap(cond, r, g, top);
620
621		/* elimination step */
622	11.6M	delta++;
623	11.6M	if (!BN_add(temp, g, r))
624	0	goto err;
625	11.6M	BN_consttime_swap(g->d[0] & 1 /* g is odd */
626		/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
627	11.6M	& (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))),
628	11.6M	g, temp, top);
629	11.6M	if (!BN_rshift1(g, g))
630	0	goto err;
631	11.6M	}
632
633		/* remove possible negative sign */
634	1.49k	r->neg = 0;
635		/* add powers of 2 removed, then correct the artificial shift */
636	1.49k	if (!BN_lshift(r, r, shifts)
637	1.49k	\|\| !BN_rshift1(r, r))
638	0	goto err;
639
640	1.49k	ret = 1;
641
642	1.49k	err:
643	1.49k	BN_CTX_end(ctx);
644	1.49k	bn_check_top(r);
645	1.49k	return ret;
646	1.49k	}