diff options
Diffstat (limited to 'src/crypto/fipsmodule/bn/gcd.c')
-rw-r--r-- | src/crypto/fipsmodule/bn/gcd.c | 492 |
1 files changed, 274 insertions, 218 deletions
diff --git a/src/crypto/fipsmodule/bn/gcd.c b/src/crypto/fipsmodule/bn/gcd.c index 850d4467..7868b407 100644 --- a/src/crypto/fipsmodule/bn/gcd.c +++ b/src/crypto/fipsmodule/bn/gcd.c @@ -114,121 +114,310 @@ #include "internal.h" -static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) { - BIGNUM *t; - int shifts = 0; - // 0 <= b <= a - while (!BN_is_zero(b)) { - // 0 < b <= a +static BN_ULONG word_is_odd_mask(BN_ULONG a) { return (BN_ULONG)0 - (a & 1); } - 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 - if (!BN_rshift1(b, b)) { - goto err; - } - if (BN_cmp(a, b) < 0) { - t = a; - a = b; - b = t; - } - } - } else { - // a is even - 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 - if (!BN_rshift1(a, a)) { - goto err; - } - if (!BN_rshift1(b, b)) { - goto err; - } - shifts++; - } - } - // 0 <= b <= a +static void maybe_rshift1_words(BN_ULONG *a, BN_ULONG mask, BN_ULONG *tmp, + size_t num) { + bn_rshift1_words(tmp, a, num); + bn_select_words(a, mask, tmp, a, num); +} + +static void maybe_rshift1_words_carry(BN_ULONG *a, BN_ULONG carry, + BN_ULONG mask, BN_ULONG *tmp, + size_t num) { + maybe_rshift1_words(a, mask, tmp, num); + if (num != 0) { + carry &= mask; + a[num - 1] |= carry << (BN_BITS2-1); + } +} + +static BN_ULONG maybe_add_words(BN_ULONG *a, BN_ULONG mask, const BN_ULONG *b, + BN_ULONG *tmp, size_t num) { + BN_ULONG carry = bn_add_words(tmp, a, b, num); + bn_select_words(a, mask, tmp, a, num); + return carry & mask; +} + +static int bn_gcd_consttime(BIGNUM *r, unsigned *out_shift, const BIGNUM *x, + const BIGNUM *y, BN_CTX *ctx) { + size_t width = x->width > y->width ? x->width : y->width; + if (width == 0) { + *out_shift = 0; + BN_zero(r); + return 1; } - if (shifts) { - if (!BN_lshift(a, a, shifts)) { - goto err; - } + // This is a constant-time implementation of Stein's algorithm (binary GCD). + int ret = 0; + BN_CTX_start(ctx); + BIGNUM *u = BN_CTX_get(ctx); + BIGNUM *v = BN_CTX_get(ctx); + BIGNUM *tmp = BN_CTX_get(ctx); + if (u == NULL || v == NULL || tmp == NULL || + !BN_copy(u, x) || + !BN_copy(v, y) || + !bn_resize_words(u, width) || + !bn_resize_words(v, width) || + !bn_resize_words(tmp, width)) { + goto err; + } + + // Each loop iteration halves at least one of |u| and |v|. Thus we need at + // most the combined bit width of inputs for at least one value to be zero. + unsigned x_bits = x->width * BN_BITS2, y_bits = y->width * BN_BITS2; + unsigned num_iters = x_bits + y_bits; + if (num_iters < x_bits) { + OPENSSL_PUT_ERROR(BN, BN_R_BIGNUM_TOO_LONG); + goto err; } - return a; + unsigned shift = 0; + for (unsigned i = 0; i < num_iters; i++) { + BN_ULONG both_odd = word_is_odd_mask(u->d[0]) & word_is_odd_mask(v->d[0]); + + // If both |u| and |v| are odd, subtract the smaller from the larger. + BN_ULONG u_less_than_v = + (BN_ULONG)0 - bn_sub_words(tmp->d, u->d, v->d, width); + bn_select_words(u->d, both_odd & ~u_less_than_v, tmp->d, u->d, width); + bn_sub_words(tmp->d, v->d, u->d, width); + bn_select_words(v->d, both_odd & u_less_than_v, tmp->d, v->d, width); + + // At least one of |u| and |v| is now even. + BN_ULONG u_is_odd = word_is_odd_mask(u->d[0]); + BN_ULONG v_is_odd = word_is_odd_mask(v->d[0]); + assert(!(u_is_odd & v_is_odd)); + + // If both are even, the final GCD gains a factor of two. + shift += 1 & (~u_is_odd & ~v_is_odd); + + // Halve any which are even. + maybe_rshift1_words(u->d, ~u_is_odd, tmp->d, width); + maybe_rshift1_words(v->d, ~v_is_odd, tmp->d, width); + } + + // One of |u| or |v| is zero at this point. The algorithm usually makes |u| + // zero, unless |y| was already zero on input. Fix this by combining the + // values. + assert(BN_is_zero(u) || BN_is_zero(v)); + for (size_t i = 0; i < width; i++) { + v->d[i] |= u->d[i]; + } + + *out_shift = shift; + ret = bn_set_words(r, v->d, width); err: - return NULL; + BN_CTX_end(ctx); + return ret; } -int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) { - BIGNUM *a, *b, *t; +int BN_gcd(BIGNUM *r, const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx) { + unsigned shift; + return bn_gcd_consttime(r, &shift, x, y, ctx) && + BN_lshift(r, r, shift); +} + +int bn_is_relatively_prime(int *out_relatively_prime, const BIGNUM *x, + const BIGNUM *y, BN_CTX *ctx) { int ret = 0; + BN_CTX_start(ctx); + unsigned shift; + BIGNUM *gcd = BN_CTX_get(ctx); + if (gcd == NULL || + !bn_gcd_consttime(gcd, &shift, x, y, ctx)) { + goto err; + } + + // Check that 2^|shift| * |gcd| is one. + if (gcd->width == 0) { + *out_relatively_prime = 0; + } else { + BN_ULONG mask = shift | (gcd->d[0] ^ 1); + for (int i = 1; i < gcd->width; i++) { + mask |= gcd->d[i]; + } + *out_relatively_prime = mask == 0; + } + ret = 1; + +err: + BN_CTX_end(ctx); + return ret; +} +int bn_lcm_consttime(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { BN_CTX_start(ctx); - a = BN_CTX_get(ctx); - b = BN_CTX_get(ctx); + unsigned shift; + BIGNUM *gcd = BN_CTX_get(ctx); + int ret = gcd != NULL && + bn_mul_consttime(r, a, b, ctx) && + bn_gcd_consttime(gcd, &shift, a, b, ctx) && + bn_div_consttime(r, NULL, r, gcd, ctx) && + bn_rshift_secret_shift(r, r, shift, ctx); + BN_CTX_end(ctx); + return ret; +} - if (a == NULL || b == NULL) { - goto err; +int bn_mod_inverse_consttime(BIGNUM *r, int *out_no_inverse, const BIGNUM *a, + const BIGNUM *n, BN_CTX *ctx) { + *out_no_inverse = 0; + if (BN_is_negative(a) || BN_ucmp(a, n) >= 0) { + OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED); + return 0; } - if (BN_copy(a, in_a) == NULL) { - goto err; + if (BN_is_zero(a)) { + if (BN_is_one(n)) { + BN_zero(r); + return 1; + } + *out_no_inverse = 1; + OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); + return 0; } - if (BN_copy(b, in_b) == NULL) { - goto err; + + // This is a constant-time implementation of the extended binary GCD + // algorithm. It is adapted from the Handbook of Applied Cryptography, section + // 14.4.3, algorithm 14.51, and modified to bound coefficients and avoid + // negative numbers. + // + // For more details and proof of correctness, see + // https://github.com/mit-plv/fiat-crypto/pull/333. In particular, see |step| + // and |mod_inverse_consttime| for the algorithm in Gallina and see + // |mod_inverse_consttime_spec| for the correctness result. + + if (!BN_is_odd(a) && !BN_is_odd(n)) { + *out_no_inverse = 1; + OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); + return 0; } - a->neg = 0; - b->neg = 0; + // This function exists to compute the RSA private exponent, where |a| is one + // word. We'll thus use |a_width| when available. + size_t n_width = n->width, a_width = a->width; + if (a_width > n_width) { + a_width = n_width; + } - if (BN_cmp(a, b) < 0) { - t = a; - a = b; - b = t; + int ret = 0; + BN_CTX_start(ctx); + BIGNUM *u = BN_CTX_get(ctx); + BIGNUM *v = BN_CTX_get(ctx); + BIGNUM *A = BN_CTX_get(ctx); + BIGNUM *B = BN_CTX_get(ctx); + BIGNUM *C = BN_CTX_get(ctx); + BIGNUM *D = BN_CTX_get(ctx); + BIGNUM *tmp = BN_CTX_get(ctx); + BIGNUM *tmp2 = BN_CTX_get(ctx); + if (u == NULL || v == NULL || A == NULL || B == NULL || C == NULL || + D == NULL || tmp == NULL || tmp2 == NULL || + !BN_copy(u, a) || + !BN_copy(v, n) || + !BN_one(A) || + !BN_one(D) || + // For convenience, size |u| and |v| equivalently. + !bn_resize_words(u, n_width) || + !bn_resize_words(v, n_width) || + // |A| and |C| are bounded by |m|. + !bn_resize_words(A, n_width) || + !bn_resize_words(C, n_width) || + // |B| and |D| are bounded by |a|. + !bn_resize_words(B, a_width) || + !bn_resize_words(D, a_width) || + // |tmp| and |tmp2| may be used at either size. + !bn_resize_words(tmp, n_width) || + !bn_resize_words(tmp2, n_width)) { + goto err; } - t = euclid(a, b); - if (t == NULL) { + + // Each loop iteration halves at least one of |u| and |v|. Thus we need at + // most the combined bit width of inputs for at least one value to be zero. + unsigned a_bits = a_width * BN_BITS2, n_bits = n_width * BN_BITS2; + unsigned num_iters = a_bits + n_bits; + if (num_iters < a_bits) { + OPENSSL_PUT_ERROR(BN, BN_R_BIGNUM_TOO_LONG); goto err; } - if (BN_copy(r, t) == NULL) { + // Before and after each loop iteration, the following hold: + // + // u = A*a - B*n + // v = D*n - C*a + // 0 < u <= a + // 0 <= v <= n + // 0 <= A < n + // 0 <= B <= a + // 0 <= C < n + // 0 <= D <= a + // + // After each loop iteration, u and v only get smaller, and at least one of + // them shrinks by at least a factor of two. + for (unsigned i = 0; i < num_iters; i++) { + BN_ULONG both_odd = word_is_odd_mask(u->d[0]) & word_is_odd_mask(v->d[0]); + + // If both |u| and |v| are odd, subtract the smaller from the larger. + BN_ULONG v_less_than_u = + (BN_ULONG)0 - bn_sub_words(tmp->d, v->d, u->d, n_width); + bn_select_words(v->d, both_odd & ~v_less_than_u, tmp->d, v->d, n_width); + bn_sub_words(tmp->d, u->d, v->d, n_width); + bn_select_words(u->d, both_odd & v_less_than_u, tmp->d, u->d, n_width); + + // If we updated one of the values, update the corresponding coefficient. + BN_ULONG carry = bn_add_words(tmp->d, A->d, C->d, n_width); + carry -= bn_sub_words(tmp2->d, tmp->d, n->d, n_width); + bn_select_words(tmp->d, carry, tmp->d, tmp2->d, n_width); + bn_select_words(A->d, both_odd & v_less_than_u, tmp->d, A->d, n_width); + bn_select_words(C->d, both_odd & ~v_less_than_u, tmp->d, C->d, n_width); + + bn_add_words(tmp->d, B->d, D->d, a_width); + bn_sub_words(tmp2->d, tmp->d, a->d, a_width); + bn_select_words(tmp->d, carry, tmp->d, tmp2->d, a_width); + bn_select_words(B->d, both_odd & v_less_than_u, tmp->d, B->d, a_width); + bn_select_words(D->d, both_odd & ~v_less_than_u, tmp->d, D->d, a_width); + + // Our loop invariants hold at this point. Additionally, exactly one of |u| + // and |v| is now even. + BN_ULONG u_is_even = ~word_is_odd_mask(u->d[0]); + BN_ULONG v_is_even = ~word_is_odd_mask(v->d[0]); + assert(u_is_even != v_is_even); + + // Halve the even one and adjust the corresponding coefficient. + maybe_rshift1_words(u->d, u_is_even, tmp->d, n_width); + BN_ULONG A_or_B_is_odd = + word_is_odd_mask(A->d[0]) | word_is_odd_mask(B->d[0]); + BN_ULONG A_carry = + maybe_add_words(A->d, A_or_B_is_odd & u_is_even, n->d, tmp->d, n_width); + BN_ULONG B_carry = + maybe_add_words(B->d, A_or_B_is_odd & u_is_even, a->d, tmp->d, a_width); + maybe_rshift1_words_carry(A->d, A_carry, u_is_even, tmp->d, n_width); + maybe_rshift1_words_carry(B->d, B_carry, u_is_even, tmp->d, a_width); + + maybe_rshift1_words(v->d, v_is_even, tmp->d, n_width); + BN_ULONG C_or_D_is_odd = + word_is_odd_mask(C->d[0]) | word_is_odd_mask(D->d[0]); + BN_ULONG C_carry = + maybe_add_words(C->d, C_or_D_is_odd & v_is_even, n->d, tmp->d, n_width); + BN_ULONG D_carry = + maybe_add_words(D->d, C_or_D_is_odd & v_is_even, a->d, tmp->d, a_width); + maybe_rshift1_words_carry(C->d, C_carry, v_is_even, tmp->d, n_width); + maybe_rshift1_words_carry(D->d, D_carry, v_is_even, tmp->d, a_width); + } + + assert(BN_is_zero(v)); + if (!BN_is_one(u)) { + *out_no_inverse = 1; + OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); goto err; } - ret = 1; + + ret = BN_copy(r, A) != NULL; err: BN_CTX_end(ctx); return ret; } -// solves ax == 1 (mod n) -static int bn_mod_inverse_general(BIGNUM *out, int *out_no_inverse, - const BIGNUM *a, const BIGNUM *n, - BN_CTX *ctx); - int BN_mod_inverse_odd(BIGNUM *out, int *out_no_inverse, const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) { *out_no_inverse = 0; @@ -423,7 +612,7 @@ BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n, int no_inverse; if (!BN_is_odd(n)) { - if (!bn_mod_inverse_general(out, &no_inverse, a, n, ctx)) { + if (!bn_mod_inverse_consttime(out, &no_inverse, a, n, ctx)) { goto err; } } else if (!BN_mod_inverse_odd(out, &no_inverse, a, n, ctx)) { @@ -469,139 +658,6 @@ err: return ret; } -// bn_mod_inverse_general is the general inversion algorithm that works for -// both even and odd |n|. It was specifically designed to contain fewer -// branches that may leak sensitive information; see "New Branch Prediction -// Vulnerabilities in OpenSSL and Necessary Software Countermeasures" by -// Onur Acıçmez, Shay Gueron, and Jean-Pierre Seifert. -static int bn_mod_inverse_general(BIGNUM *out, int *out_no_inverse, - const BIGNUM *a, const BIGNUM *n, - BN_CTX *ctx) { - BIGNUM *A, *B, *X, *Y, *M, *D, *T; - int ret = 0; - int sign; - - *out_no_inverse = 0; - - 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; - } - - BIGNUM *R = out; - - BN_zero(Y); - if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) { - goto err; - } - A->neg = 0; - - 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|) - - // (D, M) := (A/B, A%B) ... - 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. - - 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 (!BN_is_one(A)) { - *out_no_inverse = 1; - OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE); - goto err; - } - - // 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|). - - // 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; - } - } - - ret = 1; - -err: - BN_CTX_end(ctx); - return ret; -} - int bn_mod_inverse_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx, const BN_MONT_CTX *mont_p) { BN_CTX_start(ctx); |