Calculate inverse in |BN_MONT_CTX_set| in constant time w.r.t. modulus.

Simplify the calculation of the Montgomery constants in
|BN_MONT_CTX_set|, making the inversion constant-time. It should also
be faster by avoiding any use of the |BIGNUM| API in favor of using
only 64-bit arithmetic.

Now it's obvious how it works. /s

Change-Id: I59a1e1c3631f426fbeabd0c752e0de44bcb5fd75
Reviewed-on: https://boringssl-review.googlesource.com/9031
Reviewed-by: Adam Langley <agl@google.com>
Commit-Queue: Adam Langley <agl@google.com>
CQ-Verified: CQ bot account: commit-bot@chromium.org <commit-bot@chromium.org>

crypto/bn/CMakeLists.txt

@@ -57,6 +57,7 @@ add_library(
   gcd.c
   kronecker.c
   montgomery.c
+  montgomery_inv.c
   mul.c
   prime.c
   random.c

crypto/bn/internal.h

@@ -156,6 +156,7 @@ BIGNUM *bn_expand(BIGNUM *bn, size_t bits);
 #define BN_MASK2l	(0xffffffffUL)
 #define BN_MASK2h	(0xffffffff00000000UL)
 #define BN_MASK2h1	(0xffffffff80000000UL)
+#define BN_MONT_CTX_N0_LIMBS 1
 #define BN_TBIT		(0x8000000000000000UL)
 #define BN_DEC_CONV	(10000000000000000000UL)
 #define BN_DEC_NUM	19
@@ -171,6 +172,12 @@ BIGNUM *bn_expand(BIGNUM *bn, size_t bits);
 #define BN_MASK2l	(0xffffUL)
 #define BN_MASK2h1	(0xffff8000UL)
 #define BN_MASK2h	(0xffff0000UL)
+/* On some 32-bit platforms, Montgomery multiplication is done using 64-bit
+ * arithmetic with SIMD instructions. On such platforms, |BN_MONT_CTX::n0|
+ * needs to be two words long. Only certain 32-bit platforms actually make use
+ * of n0[1] and shorter R value would suffice for the others. However,
+ * currently only the assembly files know which is which. */
+#define BN_MONT_CTX_N0_LIMBS 2
 #define BN_TBIT		(0x80000000UL)
 #define BN_DEC_CONV	(1000000000UL)
 #define BN_DEC_NUM	9
@@ -192,7 +199,6 @@ BIGNUM *bn_expand(BIGNUM *bn, size_t bits);
 #define Hw(t) (((BN_ULONG)((t)>>BN_BITS2))&BN_MASK2)
 #endif
 
-
 /* bn_set_words sets |bn| to the value encoded in the |num| words in |words|,
  * least significant word first. */
 int bn_set_words(BIGNUM *bn, const BN_ULONG *words, size_t num);
@@ -221,6 +227,8 @@ int bn_cmp_part_words(const BN_ULONG *a, const BN_ULONG *b, int cl, int dl);
 int bn_mul_mont(BN_ULONG *rp, const BN_ULONG *ap, const BN_ULONG *bp,
                 const BN_ULONG *np, const BN_ULONG *n0, int num);
 
+uint64_t bn_mont_n0(const BIGNUM *n);
+
 #if defined(OPENSSL_X86_64) && defined(_MSC_VER)
 #define BN_UMULT_LOHI(low, high, a, b) ((low) = _umul128((a), (b), &(high)))
 #endif

crypto/bn/montgomery.c

@@ -162,131 +162,61 @@ BN_MONT_CTX *BN_MONT_CTX_copy(BN_MONT_CTX *to, const BN_MONT_CTX *from) {
   return to;
 }
 
-int BN_MONT_CTX_set(BN_MONT_CTX *mont, const BIGNUM *mod, BN_CTX *ctx) {
-  int ret = 0;
-  BIGNUM *Ri, *R;
-  BIGNUM tmod;
-  BN_ULONG buf[2];
+OPENSSL_COMPILE_ASSERT(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2,
+                       BN_MONT_CTX_N0_LIMBS_VALUE_INVALID);
+OPENSSL_COMPILE_ASSERT(sizeof(BN_ULONG) * BN_MONT_CTX_N0_LIMBS ==
+                       sizeof(uint64_t), BN_MONT_CTX_set_64_bit_mismatch);
 
+int BN_MONT_CTX_set(BN_MONT_CTX *mont, const BIGNUM *mod, BN_CTX *ctx) {
   if (BN_is_zero(mod)) {
     OPENSSL_PUT_ERROR(BN, BN_R_DIV_BY_ZERO);
     return 0;
   }
-
-  BN_CTX_start(ctx);
-  Ri = BN_CTX_get(ctx);
-  if (Ri == NULL) {
-    goto err;
-  }
-  R = &mont->RR; /* grab RR as a temp */
-  if (!BN_copy(&mont->N, mod)) {
-    goto err; /* Set N */
-  }
-  mont->N.neg = 0;
-
-  BN_init(&tmod);
-  tmod.d = buf;
-  tmod.dmax = 2;
-  tmod.neg = 0;
-
-#if defined(OPENSSL_BN_ASM_MONT) && (BN_BITS2 <= 32)
-  /* Only certain BN_BITS2<=32 platforms actually make use of
-   * n0[1], and we could use the #else case (with a shorter R
-   * value) for the others.  However, currently only the assembler
-   * files do know which is which. */
-
-  BN_zero(R);
-  if (!BN_set_bit(R, 2 * BN_BITS2)) {
-    goto err;
-  }
-
-  tmod.top = 0;
-  if ((buf[0] = mod->d[0])) {
-    tmod.top = 1;
+  if (!BN_is_odd(mod)) {
+    OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
+    return 0;
   }
-  if ((buf[1] = mod->top > 1 ? mod->d[1] : 0)) {
-    tmod.top = 2;
+  if (BN_is_negative(mod)) {
+    OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
+    return 0;
   }
 
-  if (BN_mod_inverse(Ri, R, &tmod, ctx) == NULL) {
-    goto err;
-  }
-  if (!BN_lshift(Ri, Ri, 2 * BN_BITS2)) {
-    goto err; /* R*Ri */
+  /* Save the modulus. */
+  if (!BN_copy(&mont->N, mod)) {
+    OPENSSL_PUT_ERROR(BN, ERR_R_INTERNAL_ERROR);
+    return 0;
   }
-  if (!BN_is_zero(Ri)) {
-    if (!BN_sub_word(Ri, 1)) {
-      goto err;
-    }
-  } else {
-    /* if N mod word size == 1 */
-    if (bn_expand(Ri, (int)sizeof(BN_ULONG) * 2) == NULL) {
-      goto err;
-    }
-    /* Ri-- (mod double word size) */
-    Ri->neg = 0;
-    Ri->d[0] = BN_MASK2;
-    Ri->d[1] = BN_MASK2;
-    Ri->top = 2;
+  if (BN_get_flags(mod, BN_FLG_CONSTTIME)) {
+    BN_set_flags(&mont->N, BN_FLG_CONSTTIME);
   }
 
-  if (!BN_div(Ri, NULL, Ri, &tmod, ctx)) {
-    goto err;
-  }
-  /* Ni = (R*Ri-1)/N,
-   * keep only couple of least significant words: */
-  mont->n0[0] = (Ri->top > 0) ? Ri->d[0] : 0;
-  mont->n0[1] = (Ri->top > 1) ? Ri->d[1] : 0;
+  /* Find n0 such that n0 * N == -1 (mod r).
+   *
+   * Only certain BN_BITS2<=32 platforms actually make use of n0[1]. For the
+   * others, we could use a shorter R value and use faster |BN_ULONG|-based
+   * math instead of |uint64_t|-based math, which would be double-precision.
+   * However, currently only the assembler files know which is which. */
+  uint64_t n0 = bn_mont_n0(mod);
+  mont->n0[0] = (BN_ULONG)n0;
+#if BN_MONT_CTX_N0_LIMBS == 2
+  mont->n0[1] = (BN_ULONG)(n0 >> BN_BITS2);
 #else
-  BN_zero(R);
-  if (!BN_set_bit(R, BN_BITS2)) {
-    goto err; /* R */
-  }
-
-  buf[0] = mod->d[0]; /* tmod = N mod word size */
-  buf[1] = 0;
-  tmod.top = buf[0] != 0 ? 1 : 0;
-  /* Ri = R^-1 mod N*/
-  if (BN_mod_inverse(Ri, R, &tmod, ctx) == NULL) {
-    goto err;
-  }
-  if (!BN_lshift(Ri, Ri, BN_BITS2)) {
-    goto err; /* R*Ri */
-  }
-  if (!BN_is_zero(Ri)) {
-    if (!BN_sub_word(Ri, 1)) {
-      goto err;
-    }
-  } else {
-    /* if N mod word size == 1 */
-    if (!BN_set_word(Ri, BN_MASK2)) {
-      goto err; /* Ri-- (mod word size) */
-    }
-  }
-  if (!BN_div(Ri, NULL, Ri, &tmod, ctx)) {
-    goto err;
-  }
-  /* Ni = (R*Ri-1)/N,
-   * keep only least significant word: */
-  mont->n0[0] = (Ri->top > 0) ? Ri->d[0] : 0;
   mont->n0[1] = 0;
 #endif
 
-  /* RR = (2^ri)^2 == 2^(ri*2) == 1 << (ri*2), which has its (ri*2)th bit set. */
-  int ri = (BN_num_bits(mod) + (BN_BITS2 - 1)) / BN_BITS2 * BN_BITS2;
-  BN_zero(&(mont->RR));
-  if (!BN_set_bit(&(mont->RR), ri * 2)) {
-    goto err;
-  }
-  if (!BN_mod(&(mont->RR), &(mont->RR), &(mont->N), ctx)) {
-    goto err;
+  /* Save RR = R**2 (mod N). R is the smallest power of 2**BN_BITS such that R
+   * > mod. Even though the assembly on some 32-bit platforms works with 64-bit
+   * values, using |BN_BITS2| here, rather than |BN_MONT_CTX_N0_LIMBS *
+   * BN_BITS2|, is correct because because R^2 will still be a multiple of the
+   * latter as |BN_MONT_CTX_N0_LIMBS| is either one or two. */
+  unsigned lgBigR = (BN_num_bits(mod) + (BN_BITS2 - 1)) / BN_BITS2 * BN_BITS2;
+  BN_zero(&mont->RR);
+  if (!BN_set_bit(&mont->RR, lgBigR * 2) ||
+      !BN_mod(&mont->RR, &mont->RR, &mont->N, ctx)) {
+    return 0;
   }
 
-  ret = 1;
-
-err:
-  BN_CTX_end(ctx);
-  return ret;
+  return 1;
 }
 
 int BN_MONT_CTX_set_locked(BN_MONT_CTX **pmont, CRYPTO_MUTEX *lock,

crypto/bn/montgomery_inv.c

@@ -0,0 +1,158 @@
+/* Copyright 2016 Brian Smith.
+ *
+ * Permission to use, copy, modify, and/or distribute this software for any
+ * purpose with or without fee is hereby granted, provided that the above
+ * copyright notice and this permission notice appear in all copies.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+ * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+ * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+ * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+ * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
+ * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
+ * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
+
+#include <openssl/bn.h>
+
+#include <assert.h>
+
+#include "internal.h"
+#include "../internal.h"
+
+
+static uint64_t bn_neg_inv_mod_r_u64(uint64_t n);
+
+OPENSSL_COMPILE_ASSERT(BN_MONT_CTX_N0_LIMBS == 1 || BN_MONT_CTX_N0_LIMBS == 2,
+                       BN_MONT_CTX_N0_LIMBS_VALUE_INVALID);
+OPENSSL_COMPILE_ASSERT(sizeof(uint64_t) ==
+                       BN_MONT_CTX_N0_LIMBS * sizeof(BN_ULONG),
+                       BN_MONT_CTX_N0_LIMBS_DOES_NOT_MATCH_UINT64_T);
+
+/* LG_LITTLE_R is log_2(r). */
+#define LG_LITTLE_R (BN_MONT_CTX_N0_LIMBS * BN_BITS2)
+
+uint64_t bn_mont_n0(const BIGNUM *n) {
+  /* These conditions are checked by the caller, |BN_MONT_CTX_set|. */
+  assert(!BN_is_zero(n));
+  assert(!BN_is_negative(n));
+  assert(BN_is_odd(n));
+
+  /* r == 2**(BN_MONT_CTX_N0_LIMBS * BN_BITS2) and LG_LITTLE_R == lg(r). This
+   * ensures that we can do integer division by |r| by simply ignoring
+   * |BN_MONT_CTX_N0_LIMBS| limbs. Similarly, we can calculate values modulo
+   * |r| by just looking at the lowest |BN_MONT_CTX_N0_LIMBS| limbs. This is
+   * what makes Montgomery multiplication efficient.
+   *
+   * As shown in Algorithm 1 of "Fast Prime Field Elliptic Curve Cryptography
+   * with 256 Bit Primes" by Shay Gueron and Vlad Krasnov, in the loop of a
+   * multi-limb Montgomery multiplication of |a * b (mod n)|, given the
+   * unreduced product |t == a * b|, we repeatedly calculate:
+   *
+   *    t1 := t % r         |t1| is |t|'s lowest limb (see previous paragraph).
+   *    t2 := t1*n0*n
+   *    t3 := t + t2
+   *    t := t3 / r         copy all limbs of |t3| except the lowest to |t|.
+   *
+   * In the last step, it would only make sense to ignore the lowest limb of
+   * |t3| if it were zero. The middle steps ensure that this is the case:
+   *
+   *                            t3 ==  0 (mod r)
+   *                        t + t2 ==  0 (mod r)
+   *                   t + t1*n0*n ==  0 (mod r)
+   *                       t1*n0*n == -t (mod r)
+   *                        t*n0*n == -t (mod r)
+   *                          n0*n == -1 (mod r)
+   *                            n0 == -1/n (mod r)
+   *
+   * Thus, in each iteration of the loop, we multiply by the constant factor
+   * |n0|, the negative inverse of n (mod r). */
+
+  /* n_mod_r = n % r. As explained above, this is done by taking the lowest
+   * |BN_MONT_CTX_N0_LIMBS| limbs of |n|. */
+  uint64_t n_mod_r = n->d[0];
+#if BN_MONT_CTX_N0_LIMBS == 2
+  if (n->top > 1) {
+    n_mod_r |= (uint64_t)n->d[1] << BN_BITS2;
+  }
+#endif
+
+  return bn_neg_inv_mod_r_u64(n_mod_r);
+}
+
+/* bn_neg_inv_r_mod_n_u64 calculates the -1/n mod r; i.e. it calculates |v|
+ * such that u*r - v*n == 1. |r| is the constant defined in |bn_mont_n0|. |n|
+ * must be odd.
+ *
+ * This is derived from |xbinGCD| in the "Montgomery Multiplication" chapter of
+ * "Hacker's Delight" by Henry S. Warren, Jr.:
+ * http://www.hackersdelight.org/MontgomeryMultiplication.pdf.
+ *
+ * This is inspired by Joppe W. Bos's "Constant Time Modular Inversion"
+ * (http://www.joppebos.com/files/CTInversion.pdf) so that the inversion is
+ * constant-time with respect to |n|. We assume uint64_t additions,
+ * subtractions, shifts, and bitwise operations are all constant time, which
+ * may be a large leap of faith on 32-bit targets. We avoid division and
+ * multiplication, which tend to be the most problematic in terms of timing
+ * leaks.
+ *
+ * Most GCD implementations return values such that |u*r + v*n == 1|, so the
+ * caller would have to negate the resultant |v| for the purpose of Montgomery
+ * multiplication. This implementation does the negation implicitly by doing
+ * the computations as a difference instead of a sum. */
+static uint64_t bn_neg_inv_mod_r_u64(uint64_t n) {
+  assert(n % 2 == 1);
+
+  /* alpha == 2**(lg r - 1) == r / 2. */
+  static const uint64_t alpha = UINT64_C(1) << (LG_LITTLE_R - 1);
+
+  const uint64_t beta = n;
+
+  uint64_t u = 1;
+  uint64_t v = 0;
+
+  /* The invariant maintained from here on is:
+   * 2**(lg r - i) == u*2*alpha - v*beta. */
+  for (size_t i = 0; i < LG_LITTLE_R; ++i) {
+#if BN_BITS2 == 64 && defined(BN_ULLONG)
+    assert((BN_ULLONG)(1) << (LG_LITTLE_R - i) ==
+           ((BN_ULLONG)u * 2 * alpha) - ((BN_ULLONG)v * beta));
+#endif
+
+    /* Delete a common factor of 2 in u and v if |u| is even. Otherwise, set
+     * |u = (u + beta) / 2| and |v = (v / 2) + alpha|. */
+
+    uint64_t u_is_odd = UINT64_C(0) - (u & 1); /* Either 0xff..ff or 0. */
+
+    /* The addition can overflow, so use Dietz's method for it.
+     *
+     * Dietz calculates (x+y)/2 by (x⊕y)>>1 + x&y. This is valid for all
+     * (unsigned) x and y, even when x+y overflows. Evidence for 32-bit values
+     * (embedded in 64 bits to so that overflow can be ignored):
+     *
+     * (declare-fun x () (_ BitVec 64))
+     * (declare-fun y () (_ BitVec 64))
+     * (assert (let (
+     *    (one (_ bv1 64))
+     *    (thirtyTwo (_ bv32 64)))
+     *    (and
+     *      (bvult x (bvshl one thirtyTwo))
+     *      (bvult y (bvshl one thirtyTwo))
+     *      (not (=
+     *        (bvadd (bvlshr (bvxor x y) one) (bvand x y))
+     *        (bvlshr (bvadd x y) one)))
+     * )))
+     * (check-sat) */
+    uint64_t beta_if_u_is_odd = beta & u_is_odd; /* Either |beta| or 0. */
+    u = ((u ^ beta_if_u_is_odd) >> 1) + (u & beta_if_u_is_odd);
+
+    uint64_t alpha_if_u_is_odd = alpha & u_is_odd; /* Either |alpha| or 0. */
+    v = (v >> 1) + alpha_if_u_is_odd;
+  }
+
+  /* The invariant now shows that u*r - v*n == 1 since r == 2 * alpha. */
+#if BN_BITS2 == 64 && defined(BN_ULLONG)
+  assert(1 == ((BN_ULLONG)u * 2 * alpha) - ((BN_ULLONG)v * beta));
+#endif
+
+  return v;
+}

crypto/rsa/rsa_impl.c

@@ -681,8 +681,7 @@ static int mod_exp(BIGNUM *r0, const BIGNUM *I, RSA *rsa, BN_CTX *ctx) {
     BIGNUM local_p, local_q;
     BIGNUM *p = NULL, *q = NULL;
 
-    /* Make sure BN_mod_inverse in Montgomery intialization uses the
-     * BN_FLG_CONSTTIME flag. */
+    /* Make sure BN_mod in Montgomery initialization uses BN_FLG_CONSTTIME. */
     BN_init(&local_p);
     p = &local_p;
     BN_with_flags(p, rsa->p, BN_FLG_CONSTTIME);