/*
- * Copyright 1995-2017 The OpenSSL Project Authors. All Rights Reserved.
+ * Copyright 1995-2018 The OpenSSL Project Authors. All Rights Reserved.
*
- * Licensed under the OpenSSL license (the "License"). You may not use
+ * 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_lcl.h"
+#include "bn_local.h"
-static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
-
-int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
+/*
+ * 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, *t;
- int ret = 0;
+ BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
+ BIGNUM *ret = NULL;
+ int sign;
- bn_check_top(in_a);
- bn_check_top(in_b);
+ bn_check_top(a);
+ bn_check_top(n);
BN_CTX_start(ctx);
- a = BN_CTX_get(ctx);
- b = BN_CTX_get(ctx);
- if (b == NULL)
+ 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_copy(a, in_a) == NULL)
+ BN_one(X);
+ BN_zero(Y);
+ if (BN_copy(B, a) == NULL)
goto err;
- if (BN_copy(b, in_b) == NULL)
+ if (BN_copy(A, n) == NULL)
goto err;
- a->neg = 0;
- b->neg = 0;
+ A->neg = 0;
- if (BN_cmp(a, b) < 0) {
- t = a;
- a = b;
- b = t;
+ 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 */
+ }
}
- t = euclid(a, b);
- if (t == NULL)
- 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_copy(r, t) == NULL)
- goto err;
- ret = 1;
- err:
- BN_CTX_end(ctx);
- bn_check_top(r);
- return ret;
-}
+ while (!BN_is_zero(B)) {
+ BIGNUM *tmp;
-static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
-{
- BIGNUM *t;
- int shifts = 0;
+ /*-
+ * 0 < B < A,
+ * (*) -sign*X*a == B (mod |n|),
+ * sign*Y*a == A (mod |n|)
+ */
- bn_check_top(a);
- bn_check_top(b);
+ /*
+ * 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);
- /* 0 <= b <= a */
- while (!BN_is_zero(b)) {
- /* 0 < b <= a */
+ /* (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 */
+ }
- if (BN_is_odd(a)) {
- if (BN_is_odd(b)) {
- if (!BN_sub(a, a, b))
- goto err;
- if (!BN_rshift1(a, a))
- goto err;
- if (BN_cmp(a, b) < 0) {
- t = a;
- a = b;
- b = t;
- }
- } else { /* a odd - b even */
+ /*-
+ * Now
+ * A = D*B + M;
+ * thus we have
+ * (**) sign*Y*a == D*B + M (mod |n|).
+ */
- if (!BN_rshift1(b, b))
- goto err;
- if (BN_cmp(a, b) < 0) {
- t = a;
- a = b;
- b = t;
- }
- }
- } else { /* a is even */
+ tmp = A; /* keep the BIGNUM object, the value does not
+ * matter */
- if (BN_is_odd(b)) {
- if (!BN_rshift1(a, a))
- goto err;
- if (BN_cmp(a, b) < 0) {
- t = a;
- a = b;
- b = t;
- }
- } else { /* a even - b even */
+ /* (A, B) := (B, A mod B) ... */
+ A = B;
+ B = M;
+ /* ... so we have 0 <= B < A again */
- if (!BN_rshift1(a, a))
- goto err;
- if (!BN_rshift1(b, b))
- goto err;
- shifts++;
- }
- }
- /* 0 <= b <= a */
+ /*-
+ * 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;
}
- if (shifts) {
- if (!BN_lshift(a, a, shifts))
+ /*-
+ * 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;
}
- bn_check_top(a);
- return a;
- err:
- return NULL;
-}
+ /* Now Y*a == A (mod |n|). */
-/* solves ax == 1 (mod n) */
-static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
- const BIGNUM *a, const BIGNUM *n,
- BN_CTX *ctx);
+ 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;
+ }
-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;
+ 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 *ret = NULL;
int sign;
- if (pnoinv)
- *pnoinv = 0;
+ /* 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);
+ return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
}
bn_check_top(a);
goto err;
}
} else {
- if (pnoinv)
- *pnoinv = 1;
+ *pnoinv = 1;
goto err;
}
ret = R;
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)
+/* solves ax == 1 (mod n) */
+BIGNUM *BN_mod_inverse(BIGNUM *in,
+ const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
{
- BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
- BIGNUM *ret = NULL;
- int sign;
-
- bn_check_top(a);
- bn_check_top(n);
-
- BN_CTX_start(ctx);
- A = BN_CTX_get(ctx);
- B = BN_CTX_get(ctx);
- X = BN_CTX_get(ctx);
- D = BN_CTX_get(ctx);
- M = BN_CTX_get(ctx);
- Y = BN_CTX_get(ctx);
- T = BN_CTX_get(ctx);
- if (T == NULL)
- goto err;
-
- if (in == NULL)
- R = BN_new();
- else
- R = in;
- if (R == NULL)
- goto err;
-
- BN_one(X);
- BN_zero(Y);
- if (BN_copy(B, a) == NULL)
- goto err;
- if (BN_copy(A, n) == NULL)
- goto err;
- A->neg = 0;
+ BN_CTX *new_ctx = NULL;
+ BIGNUM *rv;
+ int noinv = 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 */
+ if (ctx == NULL) {
+ ctx = new_ctx = BN_CTX_new_ex(NULL);
+ if (ctx == NULL) {
+ BNerr(BN_F_BN_MOD_INVERSE, ERR_R_MALLOC_FAILURE);
+ return NULL;
}
}
- 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;
+ rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
+ if (noinv)
+ BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
+ BN_CTX_free(new_ctx);
+ return rv;
+}
- /*-
- * 0 < B < A,
- * (*) -sign*X*a == B (mod |n|),
- * sign*Y*a == A (mod |n|)
- */
+/*-
+ * 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;
+ }
- /*
- * 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);
+ bn_check_top(in_a);
+ bn_check_top(in_b);
- /* (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 */
- }
+ BN_CTX_start(ctx);
+ temp = BN_CTX_get(ctx);
+ g = BN_CTX_get(ctx);
- /*-
- * Now
- * A = D*B + M;
- * thus we have
- * (**) sign*Y*a == D*B + M (mod |n|).
- */
+ /* 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;
- tmp = A; /* keep the BIGNUM object, the value does not
- * matter */
+ /* 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;
+ }
+ }
- /* (A, B) := (B, A mod B) ... */
- A = B;
- B = M;
- /* ... so we have 0 <= B < A again */
+ /* subtract shared powers of two; shifts >= 1 */
+ if (!BN_rshift(r, r, shifts)
+ || !BN_rshift(g, g, shifts))
+ goto err;
- /*-
- * 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.
- */
+ /* 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;
- if (!BN_mul(tmp, D, X, ctx))
+ /* 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;
- if (!BN_add(tmp, tmp, Y))
+ 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;
-
- 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.
- */
+ /* 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;
- if (sign < 0) {
- if (!BN_sub(Y, n, Y))
- goto err;
- }
- /* Now Y*a == A (mod |n|). */
+ ret = 1;
- 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);
+ bn_check_top(r);
return ret;
}
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