diff --git a/sshbn.c b/sshbn.c index 65c0629e..5620bb02 100644 --- a/sshbn.c +++ b/sshbn.c @@ -203,7 +203,7 @@ static void internal_sub(const BignumInt *a, const BignumInt *b, * Result is returned in the first 2*len words of c. */ #define KARATSUBA_THRESHOLD 50 -static void internal_mul(BignumInt *a, BignumInt *b, +static void internal_mul(const BignumInt *a, const BignumInt *b, BignumInt *c, int len) { int i, j; @@ -348,6 +348,167 @@ static void internal_mul(BignumInt *a, BignumInt *b, } } +/* + * Variant form of internal_mul used for the initial step of + * Montgomery reduction. Only bothers outputting 'len' words + * (everything above that is thrown away). + */ +static void internal_mul_low(const BignumInt *a, const BignumInt *b, + BignumInt *c, int len) +{ + int i, j; + BignumDblInt t; + + if (len > KARATSUBA_THRESHOLD) { + + /* + * Karatsuba-aware version of internal_mul_low. As before, we + * express each input value as a shifted combination of two + * halves: + * + * a = a_1 D + a_0 + * b = b_1 D + b_0 + * + * Then the full product is, as before, + * + * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 + * + * Provided we choose D on the large side (so that a_0 and b_0 + * are _at least_ as long as a_1 and b_1), we don't need the + * topmost term at all, and we only need half of the middle + * term. So there's no point in doing the proper Karatsuba + * optimisation which computes the middle term using the top + * one, because we'd take as long computing the top one as + * just computing the middle one directly. + * + * So instead, we do a much more obvious thing: we call the + * fully optimised internal_mul to compute a_0 b_0, and we + * recursively call ourself to compute the _bottom halves_ of + * a_1 b_0 and a_0 b_1, each of which we add into the result + * in the obvious way. + * + * In other words, there's no actual Karatsuba _optimisation_ + * in this function; the only benefit in doing it this way is + * that we call internal_mul proper for a large part of the + * work, and _that_ can optimise its operation. + */ + + int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ + BignumInt *scratch; + + /* + * Allocate scratch space for the various bits and pieces + * we're going to be adding together. We need botlen*2 words + * for a_0 b_0 (though we may end up throwing away its topmost + * word), and toplen words for each of a_1 b_0 and a_0 b_1. + * That adds up to exactly 2*len. + */ + scratch = snewn(len*2, BignumInt); + + /* a_0 b_0 */ + internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen); + + /* a_1 b_0 */ + internal_mul_low(a, b + len - toplen, scratch + toplen, toplen); + + /* a_0 b_1 */ + internal_mul_low(a + len - toplen, b, scratch, toplen); + + /* Copy the bottom half of the big coefficient into place */ + for (j = 0; j < botlen; j++) + c[toplen + j] = scratch[2*toplen + botlen + j]; + + /* Add the two small coefficients, throwing away the returned carry */ + internal_add(scratch, scratch + toplen, scratch, toplen); + + /* And add that to the large coefficient, leaving the result in c. */ + internal_add(scratch, scratch + 2*toplen + botlen - toplen, + c, toplen); + + /* Free scratch. */ + for (j = 0; j < len*2; j++) + scratch[j] = 0; + sfree(scratch); + + } else { + + for (j = 0; j < len; j++) + c[j] = 0; + + for (i = len - 1; i >= 0; i--) { + t = 0; + for (j = len - 1; j >= len - i - 1; j--) { + t += MUL_WORD(a[i], (BignumDblInt) b[j]); + t += (BignumDblInt) c[i + j + 1 - len]; + c[i + j + 1 - len] = (BignumInt) t; + t = t >> BIGNUM_INT_BITS; + } + } + + } +} + +/* + * Montgomery reduction. Expects x to be a big-endian array of 2*len + * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len * + * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array + * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <= + * x' < n. + * + * 'n' and 'mninv' should be big-endian arrays of 'len' BignumInts + * each, containing respectively n and the multiplicative inverse of + * -n mod r. + * + * 'tmp' is an array of at least '3*len' BignumInts used as scratch + * space. + */ +static void monty_reduce(BignumInt *x, const BignumInt *n, + const BignumInt *mninv, BignumInt *tmp, int len) +{ + int i; + BignumInt carry; + + /* + * Multiply x by (-n)^{-1} mod r. This gives us a value m such + * that mn is congruent to -x mod r. Hence, mn+x is an exact + * multiple of r, and is also (obviously) congruent to x mod n. + */ + internal_mul_low(x + len, mninv, tmp, len); + + /* + * Compute t = (mn+x)/r in ordinary, non-modular, integer + * arithmetic. By construction this is exact, and is congruent mod + * n to x * r^{-1}, i.e. the answer we want. + * + * The following multiply leaves that answer in the _most_ + * significant half of the 'x' array, so then we must shift it + * down. + */ + internal_mul(tmp, n, tmp+len, len); + carry = internal_add(x, tmp+len, x, 2*len); + for (i = 0; i < len; i++) + x[len + i] = x[i], x[i] = 0; + + /* + * Reduce t mod n. This doesn't require a full-on division by n, + * but merely a test and single optional subtraction, since we can + * show that 0 <= t < 2n. + * + * Proof: + * + we computed m mod r, so 0 <= m < r. + * + so 0 <= mn < rn, obviously + * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn + * + yielding 0 <= (mn+x)/r < 2n as required. + */ + if (!carry) { + for (i = 0; i < len; i++) + if (x[len + i] != n[i]) + break; + } + if (carry || i >= len || x[len + i] > n[i]) + internal_sub(x+len, n, x+len, len); +} + static void internal_add_shifted(BignumInt *number, unsigned n, int shift) { @@ -469,14 +630,14 @@ static void internal_mod(BignumInt *a, int alen, } /* - * Compute (base ^ exp) % mod. + * Compute (base ^ exp) % mod. Uses the Montgomery multiplication + * technique. */ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) { - BignumInt *a, *b, *n, *m; - int mshift; - int mlen, i, j; - Bignum base, result; + BignumInt *a, *b, *x, *n, *mninv, *tmp; + int len, i, j; + Bignum base, base2, r, rn, inv, result; /* * The most significant word of mod needs to be non-zero. It @@ -490,37 +651,64 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) */ base = bigmod(base_in, mod); - /* Allocate m of size mlen, copy mod to m */ - /* We use big endian internally */ - mlen = mod[0]; - m = snewn(mlen, BignumInt); - for (j = 0; j < mlen; j++) - m[j] = mod[mod[0] - j]; + /* + * mod had better be odd, or we can't do Montgomery multiplication + * using a power of two at all. + */ + assert(mod[1] & 1); - /* Shift m left to make msb bit set */ - for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++) - if ((m[0] << mshift) & BIGNUM_TOP_BIT) - break; - if (mshift) { - for (i = 0; i < mlen - 1; i++) - m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); - m[mlen - 1] = m[mlen - 1] << mshift; - } + /* + * Compute the inverse of n mod r, for monty_reduce. (In fact we + * want the inverse of _minus_ n mod r, but we'll sort that out + * below.) + */ + len = mod[0]; + r = bn_power_2(BIGNUM_INT_BITS * len); + inv = modinv(mod, r); - /* Allocate n of size mlen, copy base to n */ - n = snewn(mlen, BignumInt); - i = mlen - base[0]; - for (j = 0; j < i; j++) - n[j] = 0; - for (j = 0; j < (int)base[0]; j++) - n[i + j] = base[base[0] - j]; + /* + * Multiply the base by r mod n, to get it into Montgomery + * representation. + */ + base2 = modmul(base, r, mod); + freebn(base); + base = base2; - /* Allocate a and b of size 2*mlen. Set a = 1 */ - a = snewn(2 * mlen, BignumInt); - b = snewn(2 * mlen, BignumInt); - for (i = 0; i < 2 * mlen; i++) - a[i] = 0; - a[2 * mlen - 1] = 1; + rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */ + + freebn(r); /* won't need this any more */ + + /* + * Set up internal arrays of the right lengths, in big-endian + * format, containing the base, the modulus, and the modulus's + * inverse. + */ + n = snewn(len, BignumInt); + for (j = 0; j < len; j++) + n[len - 1 - j] = mod[j + 1]; + + mninv = snewn(len, BignumInt); + for (j = 0; j < len; j++) + mninv[len - 1 - j] = (j < inv[0] ? inv[j + 1] : 0); + freebn(inv); /* we don't need this copy of it any more */ + /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */ + x = snewn(len, BignumInt); + for (j = 0; j < len; j++) + x[j] = 0; + internal_sub(x, mninv, mninv, len); + + /* x = snewn(len, BignumInt); */ /* already done above */ + for (j = 0; j < len; j++) + x[len - 1 - j] = (j < base[0] ? base[j + 1] : 0); + freebn(base); /* we don't need this copy of it any more */ + + a = snewn(2*len, BignumInt); + b = snewn(2*len, BignumInt); + for (j = 0; j < len; j++) + a[2*len - 1 - j] = (j < rn[0] ? rn[j + 1] : 0); + freebn(rn); + + tmp = snewn(3*len, BignumInt); /* Skip leading zero bits of exp. */ i = 0; @@ -536,11 +724,11 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) /* Main computation */ while (i < (int)exp[0]) { while (j >= 0) { - internal_mul(a + mlen, a + mlen, b, mlen); - internal_mod(b, mlen * 2, m, mlen, NULL, 0); + internal_mul(a + len, a + len, b, len); + monty_reduce(b, n, mninv, tmp, len); if ((exp[exp[0] - i] & (1 << j)) != 0) { - internal_mul(b + mlen, n, a, mlen); - internal_mod(a, mlen * 2, m, mlen, NULL, 0); + internal_mul(b + len, x, a, len); + monty_reduce(a, n, mninv, tmp, len); } else { BignumInt *t; t = a; @@ -553,38 +741,38 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) j = BIGNUM_INT_BITS-1; } - /* Fixup result in case the modulus was shifted */ - if (mshift) { - for (i = mlen - 1; i < 2 * mlen - 1; i++) - a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); - a[2 * mlen - 1] = a[2 * mlen - 1] << mshift; - internal_mod(a, mlen * 2, m, mlen, NULL, 0); - for (i = 2 * mlen - 1; i >= mlen; i--) - a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); - } + /* + * Final monty_reduce to get back from the adjusted Montgomery + * representation. + */ + monty_reduce(a, n, mninv, tmp, len); /* Copy result to buffer */ result = newbn(mod[0]); - for (i = 0; i < mlen; i++) - result[result[0] - i] = a[i + mlen]; + for (i = 0; i < len; i++) + result[result[0] - i] = a[i + len]; while (result[0] > 1 && result[result[0]] == 0) result[0]--; /* Free temporary arrays */ - for (i = 0; i < 2 * mlen; i++) + for (i = 0; i < 3 * len; i++) + tmp[i] = 0; + sfree(tmp); + for (i = 0; i < 2 * len; i++) a[i] = 0; sfree(a); - for (i = 0; i < 2 * mlen; i++) + for (i = 0; i < 2 * len; i++) b[i] = 0; sfree(b); - for (i = 0; i < mlen; i++) - m[i] = 0; - sfree(m); - for (i = 0; i < mlen; i++) + for (i = 0; i < len; i++) + mninv[i] = 0; + sfree(mninv); + for (i = 0; i < len; i++) n[i] = 0; sfree(n); - - freebn(base); + for (i = 0; i < len; i++) + x[i] = 0; + sfree(x); return result; }