/*
* Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
*
* Licensed under the OpenSSL license (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"
/* solves ax == 1 (mod n) */
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
const BIGNUM *a, const BIGNUM *n,
BN_CTX *ctx);
BIGNUM *BN_mod_inverse(BIGNUM *in,
const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
{
BIGNUM *rv;
int noinv;
rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
if (noinv)
BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
return rv;
}
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)) {
if (pnoinv != NULL)
*pnoinv = 1;
return NULL;
}
if (pnoinv != 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);
}
bn_check_top(a);
bn_check_top(n);
BN_CTX_start(ctx);
A = BN_CTX_get(ctx);
B = BN_CTX_get(ctx);
X = BN_CTX_get(ctx);
D = BN_CTX_get(ctx);
M = BN_CTX_get(ctx);
Y = BN_CTX_get(ctx);
T = BN_CTX_get(ctx);
if (T == NULL)
goto err;
if (in == NULL)
R = BN_new();
else
R = in;
if (R == NULL)
goto err;
BN_one(X);
BN_zero(Y);
if (BN_copy(B, a) == NULL)
goto err;
if (BN_copy(A, n) == NULL)
goto err;
A->neg = 0;
if (B->neg || (BN_ucmp(B, A) >= 0)) {
if (!BN_nnmod(B, B, A, ctx))
goto err;
}
sign = -1;
/*-
* From B = a mod |n|, A = |n| it follows that
*
* 0 <= B < A,
* -sign*X*a == B (mod |n|),
* sign*Y*a == A (mod |n|).
*/
if (BN_is_odd(n) && (BN_num_bits(n) <= 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 {
if (pnoinv)
*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;
}
/*
* BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
* not contain branches that may leak sensitive information.
*/
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
const BIGNUM *a, const BIGNUM *n,
BN_CTX *ctx)
{
BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
BIGNUM *ret = NULL;
int sign;
bn_check_top(a);
bn_check_top(n);
BN_CTX_start(ctx);
A = BN_CTX_get(ctx);
B = BN_CTX_get(ctx);
X = BN_CTX_get(ctx);
D = BN_CTX_get(ctx);
M = BN_CTX_get(ctx);
Y = BN_CTX_get(ctx);
T = BN_CTX_get(ctx);
if (T == NULL)
goto err;
if (in == NULL)
R = BN_new();
else
R = in;
if (R == NULL)
goto err;
BN_one(X);
BN_zero(Y);
if (BN_copy(B, a) == NULL)
goto err;
if (BN_copy(A, n) == NULL)
goto err;
A->neg = 0;
if (B->neg || (BN_ucmp(B, A) >= 0)) {
/*
* 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 {
BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH, BN_R_NO_INVERSE);
goto err;
}
ret = R;
err:
if ((ret == NULL) && (in == NULL))
BN_free(R);
BN_CTX_end(ctx);
bn_check_top(ret);
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 = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
/* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
& (~((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) */
& (~((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;
}