X-Git-Url: https://git.distorted.org.uk/u/mdw/putty/blobdiff_plain/32e51f76a4d1eb739b6ba581bc845880eef13b18..HEAD:/sshbn.c diff --git a/sshbn.c b/sshbn.c index c0ae5288..a5e0552f 100644 --- a/sshbn.c +++ b/sshbn.c @@ -6,6 +6,7 @@ #include #include #include +#include #include "misc.h" @@ -120,7 +121,11 @@ Bignum Zero = bnZero, One = bnOne; static Bignum newbn(int length) { - Bignum b = snewn(length + 1, BignumInt); + Bignum b; + + assert(length >= 0 && length < INT_MAX / BIGNUM_INT_BITS); + + b = snewn(length + 1, BignumInt); if (!b) abort(); /* FIXME */ memset(b, 0, (length + 1) * sizeof(*b)); @@ -148,41 +153,448 @@ void freebn(Bignum b) /* * Burn the evidence, just in case. */ - memset(b, 0, sizeof(b[0]) * (b[0] + 1)); + smemclr(b, sizeof(b[0]) * (b[0] + 1)); sfree(b); } Bignum bn_power_2(int n) { - Bignum ret = newbn(n / BIGNUM_INT_BITS + 1); + Bignum ret; + + assert(n >= 0); + + ret = newbn(n / BIGNUM_INT_BITS + 1); bignum_set_bit(ret, n, 1); return ret; } /* + * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all + * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried + * off the top. + */ +static BignumInt internal_add(const BignumInt *a, const BignumInt *b, + BignumInt *c, int len) +{ + int i; + BignumDblInt carry = 0; + + for (i = len-1; i >= 0; i--) { + carry += (BignumDblInt)a[i] + b[i]; + c[i] = (BignumInt)carry; + carry >>= BIGNUM_INT_BITS; + } + + return (BignumInt)carry; +} + +/* + * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are + * all big-endian arrays of 'len' BignumInts. Any borrow from the top + * is ignored. + */ +static void internal_sub(const BignumInt *a, const BignumInt *b, + BignumInt *c, int len) +{ + int i; + BignumDblInt carry = 1; + + for (i = len-1; i >= 0; i--) { + carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK); + c[i] = (BignumInt)carry; + carry >>= BIGNUM_INT_BITS; + } +} + +/* * Compute c = a * b. * Input is in the first len words of a and b. * Result is returned in the first 2*len words of c. + * + * 'scratch' must point to an array of BignumInt of size at least + * mul_compute_scratch(len). (This covers the needs of internal_mul + * and all its recursive calls to itself.) */ -static void internal_mul(BignumInt *a, BignumInt *b, - BignumInt *c, int len) +#define KARATSUBA_THRESHOLD 50 +static int mul_compute_scratch(int len) { - int i, j; - BignumDblInt t; - - for (j = 0; j < 2 * len; j++) - c[j] = 0; - - for (i = len - 1; i >= 0; i--) { - t = 0; - for (j = len - 1; j >= 0; j--) { - t += MUL_WORD(a[i], (BignumDblInt) b[j]); - t += (BignumDblInt) c[i + j + 1]; - c[i + j + 1] = (BignumInt) t; - t = t >> BIGNUM_INT_BITS; - } - c[i] = (BignumInt) t; + int ret = 0; + while (len > KARATSUBA_THRESHOLD) { + int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ + int midlen = botlen + 1; + ret += 4*midlen; + len = midlen; + } + return ret; +} +static void internal_mul(const BignumInt *a, const BignumInt *b, + BignumInt *c, int len, BignumInt *scratch) +{ + if (len > KARATSUBA_THRESHOLD) { + int i; + + /* + * Karatsuba divide-and-conquer algorithm. Cut each input in + * half, so that it's expressed as two big 'digits' in a giant + * base D: + * + * a = a_1 D + a_0 + * b = b_1 D + b_0 + * + * Then the product is of course + * + * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0 + * + * and we compute the three coefficients by recursively + * calling ourself to do half-length multiplications. + * + * The clever bit that makes this worth doing is that we only + * need _one_ half-length multiplication for the central + * coefficient rather than the two that it obviouly looks + * like, because we can use a single multiplication to compute + * + * (a_1 + a_0) (b_1 + b_0) = a_1 b_1 + a_1 b_0 + a_0 b_1 + a_0 b_0 + * + * and then we subtract the other two coefficients (a_1 b_1 + * and a_0 b_0) which we were computing anyway. + * + * Hence we get to multiply two numbers of length N in about + * three times as much work as it takes to multiply numbers of + * length N/2, which is obviously better than the four times + * as much work it would take if we just did a long + * conventional multiply. + */ + + int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ + int midlen = botlen + 1; + BignumDblInt carry; +#ifdef KARA_DEBUG + int i; +#endif + + /* + * The coefficients a_1 b_1 and a_0 b_0 just avoid overlapping + * in the output array, so we can compute them immediately in + * place. + */ + +#ifdef KARA_DEBUG + printf("a1,a0 = 0x"); + for (i = 0; i < len; i++) { + if (i == toplen) printf(", 0x"); + printf("%0*x", BIGNUM_INT_BITS/4, a[i]); + } + printf("\n"); + printf("b1,b0 = 0x"); + for (i = 0; i < len; i++) { + if (i == toplen) printf(", 0x"); + printf("%0*x", BIGNUM_INT_BITS/4, b[i]); + } + printf("\n"); +#endif + + /* a_1 b_1 */ + internal_mul(a, b, c, toplen, scratch); +#ifdef KARA_DEBUG + printf("a1b1 = 0x"); + for (i = 0; i < 2*toplen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, c[i]); + } + printf("\n"); +#endif + + /* a_0 b_0 */ + internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen, scratch); +#ifdef KARA_DEBUG + printf("a0b0 = 0x"); + for (i = 0; i < 2*botlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, c[2*toplen+i]); + } + printf("\n"); +#endif + + /* Zero padding. midlen exceeds toplen by at most 2, so just + * zero the first two words of each input and the rest will be + * copied over. */ + scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0; + + for (i = 0; i < toplen; i++) { + scratch[midlen - toplen + i] = a[i]; /* a_1 */ + scratch[2*midlen - toplen + i] = b[i]; /* b_1 */ + } + + /* compute a_1 + a_0 */ + scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen); +#ifdef KARA_DEBUG + printf("a1plusa0 = 0x"); + for (i = 0; i < midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); + } + printf("\n"); +#endif + /* compute b_1 + b_0 */ + scratch[midlen] = internal_add(scratch+midlen+1, b+toplen, + scratch+midlen+1, botlen); +#ifdef KARA_DEBUG + printf("b1plusb0 = 0x"); + for (i = 0; i < midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen+i]); + } + printf("\n"); +#endif + + /* + * Now we can do the third multiplication. + */ + internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen, + scratch + 4*midlen); +#ifdef KARA_DEBUG + printf("a1plusa0timesb1plusb0 = 0x"); + for (i = 0; i < 2*midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); + } + printf("\n"); +#endif + + /* + * Now we can reuse the first half of 'scratch' to compute the + * sum of the outer two coefficients, to subtract from that + * product to obtain the middle one. + */ + scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0; + for (i = 0; i < 2*toplen; i++) + scratch[2*midlen - 2*toplen + i] = c[i]; + scratch[1] = internal_add(scratch+2, c + 2*toplen, + scratch+2, 2*botlen); +#ifdef KARA_DEBUG + printf("a1b1plusa0b0 = 0x"); + for (i = 0; i < 2*midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[i]); + } + printf("\n"); +#endif + + internal_sub(scratch + 2*midlen, scratch, + scratch + 2*midlen, 2*midlen); +#ifdef KARA_DEBUG + printf("a1b0plusa0b1 = 0x"); + for (i = 0; i < 2*midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen+i]); + } + printf("\n"); +#endif + + /* + * And now all we need to do is to add that middle coefficient + * back into the output. We may have to propagate a carry + * further up the output, but we can be sure it won't + * propagate right the way off the top. + */ + carry = internal_add(c + 2*len - botlen - 2*midlen, + scratch + 2*midlen, + c + 2*len - botlen - 2*midlen, 2*midlen); + i = 2*len - botlen - 2*midlen - 1; + while (carry) { + assert(i >= 0); + carry += c[i]; + c[i] = (BignumInt)carry; + carry >>= BIGNUM_INT_BITS; + i--; + } +#ifdef KARA_DEBUG + printf("ab = 0x"); + for (i = 0; i < 2*len; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, c[i]); + } + printf("\n"); +#endif + + } else { + int i; + BignumInt carry; + BignumDblInt t; + const BignumInt *ap, *bp; + BignumInt *cp, *cps; + + /* + * Multiply in the ordinary O(N^2) way. + */ + + for (i = 0; i < 2 * len; i++) + c[i] = 0; + + for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) { + carry = 0; + for (cp = cps, bp = b + len; cp--, bp-- > b ;) { + t = (MUL_WORD(*ap, *bp) + carry) + *cp; + *cp = (BignumInt) t; + carry = (BignumInt)(t >> BIGNUM_INT_BITS); + } + *cp = carry; + } + } +} + +/* + * 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, BignumInt *scratch) +{ + if (len > KARATSUBA_THRESHOLD) { + int i; + + /* + * 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 */ + + /* + * 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. + */ + + /* a_0 b_0 */ + internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen, + scratch + 2*len); + + /* a_1 b_0 */ + internal_mul_low(a, b + len - toplen, scratch + toplen, toplen, + scratch + 2*len); + + /* a_0 b_1 */ + internal_mul_low(a + len - toplen, b, scratch, toplen, + scratch + 2*len); + + /* Copy the bottom half of the big coefficient into place */ + for (i = 0; i < botlen; i++) + c[toplen + i] = scratch[2*toplen + botlen + i]; + + /* 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); + + } else { + int i; + BignumInt carry; + BignumDblInt t; + const BignumInt *ap, *bp; + BignumInt *cp, *cps; + + /* + * Multiply in the ordinary O(N^2) way. + */ + + for (i = 0; i < len; i++) + c[i] = 0; + + for (cps = c + len, ap = a + len; ap-- > a; cps--) { + carry = 0; + for (cp = cps, bp = b + len; bp--, cp-- > c ;) { + t = (MUL_WORD(*ap, *bp) + carry) + *cp; + *cp = (BignumInt) t; + carry = (BignumInt)(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 BignumInt used as scratch space, of length at + * least 3*len + mul_compute_scratch(len). + */ +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, tmp + 3*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, tmp + 3*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, @@ -195,6 +607,7 @@ static void internal_add_shifted(BignumInt *number, addend = (BignumDblInt)n << bshift; while (addend) { + assert(word <= number[0]); addend += number[word]; number[word] = (BignumInt) addend & BIGNUM_INT_MASK; addend >>= BIGNUM_INT_BITS; @@ -221,6 +634,7 @@ static void internal_mod(BignumInt *a, int alen, int i, k; m0 = m[0]; + assert(m0 >> (BIGNUM_INT_BITS-1) == 1); if (mlen > 1) m1 = m[1]; else @@ -306,13 +720,13 @@ static void internal_mod(BignumInt *a, int alen, } /* - * Compute (base ^ exp) % mod. + * Compute (base ^ exp) % mod, the pedestrian way. */ -Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) +Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) { - BignumInt *a, *b, *n, *m; + BignumInt *a, *b, *n, *m, *scratch; int mshift; - int mlen, i, j; + int mlen, scratchlen, i, j; Bignum base, result; /* @@ -359,6 +773,10 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) a[i] = 0; a[2 * mlen - 1] = 1; + /* Scratch space for multiplies */ + scratchlen = mul_compute_scratch(mlen); + scratch = snewn(scratchlen, BignumInt); + /* Skip leading zero bits of exp. */ i = 0; j = BIGNUM_INT_BITS-1; @@ -373,10 +791,10 @@ 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_mul(a + mlen, a + mlen, b, mlen, scratch); internal_mod(b, mlen * 2, m, mlen, NULL, 0); if ((exp[exp[0] - i] & (1 << j)) != 0) { - internal_mul(b + mlen, n, a, mlen); + internal_mul(b + mlen, n, a, mlen, scratch); internal_mod(a, mlen * 2, m, mlen, NULL, 0); } else { BignumInt *t; @@ -408,17 +826,15 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) result[0]--; /* Free temporary arrays */ - for (i = 0; i < 2 * mlen; i++) - a[i] = 0; + smemclr(a, 2 * mlen * sizeof(*a)); sfree(a); - for (i = 0; i < 2 * mlen; i++) - b[i] = 0; + smemclr(scratch, scratchlen * sizeof(*scratch)); + sfree(scratch); + smemclr(b, 2 * mlen * sizeof(*b)); sfree(b); - for (i = 0; i < mlen; i++) - m[i] = 0; + smemclr(m, mlen * sizeof(*m)); sfree(m); - for (i = 0; i < mlen; i++) - n[i] = 0; + smemclr(n, mlen * sizeof(*n)); sfree(n); freebn(base); @@ -427,17 +843,169 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) } /* + * Compute (base ^ exp) % mod. Uses the Montgomery multiplication + * technique where possible, falling back to modpow_simple otherwise. + */ +Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) +{ + BignumInt *a, *b, *x, *n, *mninv, *scratch; + int len, scratchlen, i, j; + Bignum base, base2, r, rn, inv, result; + + /* + * The most significant word of mod needs to be non-zero. It + * should already be, but let's make sure. + */ + assert(mod[mod[0]] != 0); + + /* + * mod had better be odd, or we can't do Montgomery multiplication + * using a power of two at all. + */ + if (!(mod[1] & 1)) + return modpow_simple(base_in, exp, mod); + + /* + * Make sure the base is smaller than the modulus, by reducing + * it modulo the modulus if not. + */ + base = bigmod(base_in, mod); + + /* + * 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); + assert(inv); /* cannot fail, since mod is odd and r is a power of 2 */ + + /* + * Multiply the base by r mod n, to get it into Montgomery + * representation. + */ + base2 = modmul(base, r, mod); + freebn(base); + base = base2; + + 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 < (int)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 < (int)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 < (int)rn[0] ? rn[j + 1] : 0); + freebn(rn); + + /* Scratch space for multiplies */ + scratchlen = 3*len + mul_compute_scratch(len); + scratch = snewn(scratchlen, BignumInt); + + /* Skip leading zero bits of exp. */ + i = 0; + j = BIGNUM_INT_BITS-1; + while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { + j--; + if (j < 0) { + i++; + j = BIGNUM_INT_BITS-1; + } + } + + /* Main computation */ + while (i < (int)exp[0]) { + while (j >= 0) { + internal_mul(a + len, a + len, b, len, scratch); + monty_reduce(b, n, mninv, scratch, len); + if ((exp[exp[0] - i] & (1 << j)) != 0) { + internal_mul(b + len, x, a, len, scratch); + monty_reduce(a, n, mninv, scratch, len); + } else { + BignumInt *t; + t = a; + a = b; + b = t; + } + j--; + } + i++; + j = BIGNUM_INT_BITS-1; + } + + /* + * Final monty_reduce to get back from the adjusted Montgomery + * representation. + */ + monty_reduce(a, n, mninv, scratch, len); + + /* Copy result to buffer */ + result = newbn(mod[0]); + 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 */ + smemclr(scratch, scratchlen * sizeof(*scratch)); + sfree(scratch); + smemclr(a, 2 * len * sizeof(*a)); + sfree(a); + smemclr(b, 2 * len * sizeof(*b)); + sfree(b); + smemclr(mninv, len * sizeof(*mninv)); + sfree(mninv); + smemclr(n, len * sizeof(*n)); + sfree(n); + smemclr(x, len * sizeof(*x)); + sfree(x); + + return result; +} + +/* * Compute (p * q) % mod. * The most significant word of mod MUST be non-zero. * We assume that the result array is the same size as the mod array. */ Bignum modmul(Bignum p, Bignum q, Bignum mod) { - BignumInt *a, *n, *m, *o; - int mshift; + BignumInt *a, *n, *m, *o, *scratch; + int mshift, scratchlen; int pqlen, mlen, rlen, i, j; Bignum result; + /* + * The most significant word of mod needs to be non-zero. It + * should already be, but let's make sure. + */ + assert(mod[mod[0]] != 0); + /* Allocate m of size mlen, copy mod to m */ /* We use big endian internally */ mlen = mod[0]; @@ -457,6 +1025,13 @@ Bignum modmul(Bignum p, Bignum q, Bignum mod) pqlen = (p[0] > q[0] ? p[0] : q[0]); + /* + * Make sure that we're allowing enough space. The shifting below + * will underflow the vectors we allocate if pqlen is too small. + */ + if (2*pqlen <= mlen) + pqlen = mlen/2 + 1; + /* Allocate n of size pqlen, copy p to n */ n = snewn(pqlen, BignumInt); i = pqlen - p[0]; @@ -476,8 +1051,12 @@ Bignum modmul(Bignum p, Bignum q, Bignum mod) /* Allocate a of size 2*pqlen for result */ a = snewn(2 * pqlen, BignumInt); + /* Scratch space for multiplies */ + scratchlen = mul_compute_scratch(pqlen); + scratch = snewn(scratchlen, BignumInt); + /* Main computation */ - internal_mul(n, o, a, pqlen); + internal_mul(n, o, a, pqlen, scratch); internal_mod(a, pqlen * 2, m, mlen, NULL, 0); /* Fixup result in case the modulus was shifted */ @@ -499,17 +1078,15 @@ Bignum modmul(Bignum p, Bignum q, Bignum mod) result[0]--; /* Free temporary arrays */ - for (i = 0; i < 2 * pqlen; i++) - a[i] = 0; + smemclr(scratch, scratchlen * sizeof(*scratch)); + sfree(scratch); + smemclr(a, 2 * pqlen * sizeof(*a)); sfree(a); - for (i = 0; i < mlen; i++) - m[i] = 0; + smemclr(m, mlen * sizeof(*m)); sfree(m); - for (i = 0; i < pqlen; i++) - n[i] = 0; + smemclr(n, pqlen * sizeof(*n)); sfree(n); - for (i = 0; i < pqlen; i++) - o[i] = 0; + smemclr(o, pqlen * sizeof(*o)); sfree(o); return result; @@ -528,6 +1105,12 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) int mshift; int plen, mlen, i, j; + /* + * The most significant word of mod needs to be non-zero. It + * should already be, but let's make sure. + */ + assert(mod[mod[0]] != 0); + /* Allocate m of size mlen, copy mod to m */ /* We use big endian internally */ mlen = mod[0]; @@ -579,11 +1162,9 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) } /* Free temporary arrays */ - for (i = 0; i < mlen; i++) - m[i] = 0; + smemclr(m, mlen * sizeof(*m)); sfree(m); - for (i = 0; i < plen; i++) - n[i] = 0; + smemclr(n, plen * sizeof(*n)); sfree(n); } @@ -603,6 +1184,8 @@ Bignum bignum_from_bytes(const unsigned char *data, int nbytes) Bignum result; int w, i; + assert(nbytes >= 0 && nbytes < INT_MAX/8); + w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ result = newbn(w); @@ -679,7 +1262,7 @@ int ssh2_bignum_length(Bignum bn) */ int bignum_byte(Bignum bn, int i) { - if (i >= (int)(BIGNUM_INT_BYTES * bn[0])) + if (i < 0 || i >= (int)(BIGNUM_INT_BYTES * bn[0])) return 0; /* beyond the end */ else return (bn[i / BIGNUM_INT_BYTES + 1] >> @@ -691,7 +1274,7 @@ int bignum_byte(Bignum bn, int i) */ int bignum_bit(Bignum bn, int i) { - if (i >= (int)(BIGNUM_INT_BITS * bn[0])) + if (i < 0 || i >= (int)(BIGNUM_INT_BITS * bn[0])) return 0; /* beyond the end */ else return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; @@ -702,7 +1285,7 @@ int bignum_bit(Bignum bn, int i) */ void bignum_set_bit(Bignum bn, int bitnum, int value) { - if (bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) + if (bitnum < 0 || bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) abort(); /* beyond the end */ else { int v = bitnum / BIGNUM_INT_BITS + 1; @@ -738,7 +1321,18 @@ int ssh1_write_bignum(void *data, Bignum bn) int bignum_cmp(Bignum a, Bignum b) { int amax = a[0], bmax = b[0]; - int i = (amax > bmax ? amax : bmax); + int i; + + /* Annoyingly we have two representations of zero */ + if (amax == 1 && a[amax] == 0) + amax = 0; + if (bmax == 1 && b[bmax] == 0) + bmax = 0; + + assert(amax == 0 || a[amax] != 0); + assert(bmax == 0 || b[bmax] != 0); + + i = (amax > bmax ? amax : bmax); while (i) { BignumInt aval = (i > amax ? 0 : a[i]); BignumInt bval = (i > bmax ? 0 : b[i]); @@ -760,6 +1354,8 @@ Bignum bignum_rshift(Bignum a, int shift) int i, shiftw, shiftb, shiftbb, bits; BignumInt ai, ai1; + assert(shift >= 0); + bits = bignum_bitcount(a) - shift; ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); @@ -787,18 +1383,21 @@ Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) int alen = a[0], blen = b[0]; int mlen = (alen > blen ? alen : blen); int rlen, i, maxspot; + int wslen; BignumInt *workspace; Bignum ret; - /* mlen space for a, mlen space for b, 2*mlen for result */ - workspace = snewn(mlen * 4, BignumInt); + /* mlen space for a, mlen space for b, 2*mlen for result, + * plus scratch space for multiplication */ + wslen = mlen * 4 + mul_compute_scratch(mlen); + workspace = snewn(wslen, BignumInt); for (i = 0; i < mlen; i++) { workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0); workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0); } internal_mul(workspace + 0 * mlen, workspace + 1 * mlen, - workspace + 2 * mlen, mlen); + workspace + 2 * mlen, mlen, workspace + 4 * mlen); /* now just copy the result back */ rlen = alen + blen + 1; @@ -827,6 +1426,7 @@ Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) } ret[0] = maxspot; + smemclr(workspace, wslen * sizeof(*workspace)); sfree(workspace); return ret; } @@ -840,6 +1440,69 @@ Bignum bigmul(Bignum a, Bignum b) } /* + * Simple addition. + */ +Bignum bigadd(Bignum a, Bignum b) +{ + int alen = a[0], blen = b[0]; + int rlen = (alen > blen ? alen : blen) + 1; + int i, maxspot; + Bignum ret; + BignumDblInt carry; + + ret = newbn(rlen); + + carry = 0; + maxspot = 0; + for (i = 1; i <= rlen; i++) { + carry += (i <= (int)a[0] ? a[i] : 0); + carry += (i <= (int)b[0] ? b[i] : 0); + ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; + carry >>= BIGNUM_INT_BITS; + if (ret[i] != 0 && i > maxspot) + maxspot = i; + } + ret[0] = maxspot; + + return ret; +} + +/* + * Subtraction. Returns a-b, or NULL if the result would come out + * negative (recall that this entire bignum module only handles + * positive numbers). + */ +Bignum bigsub(Bignum a, Bignum b) +{ + int alen = a[0], blen = b[0]; + int rlen = (alen > blen ? alen : blen); + int i, maxspot; + Bignum ret; + BignumDblInt carry; + + ret = newbn(rlen); + + carry = 1; + maxspot = 0; + for (i = 1; i <= rlen; i++) { + carry += (i <= (int)a[0] ? a[i] : 0); + carry += (i <= (int)b[0] ? b[i] ^ BIGNUM_INT_MASK : BIGNUM_INT_MASK); + ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; + carry >>= BIGNUM_INT_BITS; + if (ret[i] != 0 && i > maxspot) + maxspot = i; + } + ret[0] = maxspot; + + if (!carry) { + freebn(ret); + return NULL; + } + + return ret; +} + +/* * Create a bignum which is the bitmask covering another one. That * is, the smallest integer which is >= N and is also one less than * a power of two. @@ -993,9 +1656,26 @@ Bignum modinv(Bignum number, Bignum modulus) Bignum x = copybn(One); int sign = +1; + assert(number[number[0]] != 0); + assert(modulus[modulus[0]] != 0); + while (bignum_cmp(b, One) != 0) { - Bignum t = newbn(b[0]); - Bignum q = newbn(a[0]); + Bignum t, q; + + if (bignum_cmp(b, Zero) == 0) { + /* + * Found a common factor between the inputs, so we cannot + * return a modular inverse at all. + */ + freebn(b); + freebn(a); + freebn(xp); + freebn(x); + return NULL; + } + + t = newbn(b[0]); + q = newbn(a[0]); bigdivmod(a, b, t, q); while (t[0] > 1 && t[t[0]] == 0) t[0]--; @@ -1114,6 +1794,208 @@ char *bignum_decimal(Bignum x) /* * Done. */ + smemclr(workspace, x[0] * sizeof(*workspace)); sfree(workspace); return ret; } + +#ifdef TESTBN + +#include +#include +#include + +/* + * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset + * + * Then feed to this program's standard input the output of + * testdata/bignum.py . + */ + +void modalfatalbox(char *p, ...) +{ + va_list ap; + fprintf(stderr, "FATAL ERROR: "); + va_start(ap, p); + vfprintf(stderr, p, ap); + va_end(ap); + fputc('\n', stderr); + exit(1); +} + +#define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' ) + +int main(int argc, char **argv) +{ + char *buf; + int line = 0; + int passes = 0, fails = 0; + + while ((buf = fgetline(stdin)) != NULL) { + int maxlen = strlen(buf); + unsigned char *data = snewn(maxlen, unsigned char); + unsigned char *ptrs[5], *q; + int ptrnum; + char *bufp = buf; + + line++; + + q = data; + ptrnum = 0; + + while (*bufp && !isspace((unsigned char)*bufp)) + bufp++; + if (bufp) + *bufp++ = '\0'; + + while (*bufp) { + char *start, *end; + int i; + + while (*bufp && !isxdigit((unsigned char)*bufp)) + bufp++; + start = bufp; + + if (!*bufp) + break; + + while (*bufp && isxdigit((unsigned char)*bufp)) + bufp++; + end = bufp; + + if (ptrnum >= lenof(ptrs)) + break; + ptrs[ptrnum++] = q; + + for (i = -((end - start) & 1); i < end-start; i += 2) { + unsigned char val = (i < 0 ? 0 : fromxdigit(start[i])); + val = val * 16 + fromxdigit(start[i+1]); + *q++ = val; + } + + ptrs[ptrnum] = q; + } + + if (!strcmp(buf, "mul")) { + Bignum a, b, c, p; + + if (ptrnum != 3) { + printf("%d: mul with %d parameters, expected 3\n", line, ptrnum); + exit(1); + } + a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); + b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); + c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); + p = bigmul(a, b); + + if (bignum_cmp(c, p) == 0) { + passes++; + } else { + char *as = bignum_decimal(a); + char *bs = bignum_decimal(b); + char *cs = bignum_decimal(c); + char *ps = bignum_decimal(p); + + printf("%d: fail: %s * %s gave %s expected %s\n", + line, as, bs, ps, cs); + fails++; + + sfree(as); + sfree(bs); + sfree(cs); + sfree(ps); + } + freebn(a); + freebn(b); + freebn(c); + freebn(p); + } else if (!strcmp(buf, "modmul")) { + Bignum a, b, m, c, p; + + if (ptrnum != 4) { + printf("%d: modmul with %d parameters, expected 4\n", + line, ptrnum); + exit(1); + } + a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); + b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); + m = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); + c = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); + p = modmul(a, b, m); + + if (bignum_cmp(c, p) == 0) { + passes++; + } else { + char *as = bignum_decimal(a); + char *bs = bignum_decimal(b); + char *ms = bignum_decimal(m); + char *cs = bignum_decimal(c); + char *ps = bignum_decimal(p); + + printf("%d: fail: %s * %s mod %s gave %s expected %s\n", + line, as, bs, ms, ps, cs); + fails++; + + sfree(as); + sfree(bs); + sfree(ms); + sfree(cs); + sfree(ps); + } + freebn(a); + freebn(b); + freebn(m); + freebn(c); + freebn(p); + } else if (!strcmp(buf, "pow")) { + Bignum base, expt, modulus, expected, answer; + + if (ptrnum != 4) { + printf("%d: mul with %d parameters, expected 4\n", line, ptrnum); + exit(1); + } + + base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); + expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); + modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); + expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]); + answer = modpow(base, expt, modulus); + + if (bignum_cmp(expected, answer) == 0) { + passes++; + } else { + char *as = bignum_decimal(base); + char *bs = bignum_decimal(expt); + char *cs = bignum_decimal(modulus); + char *ds = bignum_decimal(answer); + char *ps = bignum_decimal(expected); + + printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n", + line, as, bs, cs, ds, ps); + fails++; + + sfree(as); + sfree(bs); + sfree(cs); + sfree(ds); + sfree(ps); + } + freebn(base); + freebn(expt); + freebn(modulus); + freebn(expected); + freebn(answer); + } else { + printf("%d: unrecognised test keyword: '%s'\n", line, buf); + exit(1); + } + + sfree(buf); + sfree(data); + } + + printf("passed %d failed %d total %d\n", passes, fails, passes+fails); + return fails != 0; +} + +#endif