X-Git-Url: https://git.distorted.org.uk/~mdw/sgt/putty/blobdiff_plain/0c431b2f818d6696270774c88fc48e4b68b9c3fa..refs/heads/mdw/ec:/sshbn.c diff --git a/sshbn.c b/sshbn.c index 65c0629e..ccb710d9 100644 --- a/sshbn.c +++ b/sshbn.c @@ -148,7 +148,7 @@ 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); } @@ -161,7 +161,7 @@ Bignum bn_power_2(int n) /* * Internal addition. Sets c = a - b, where 'a', 'b' and 'c' are all - * big-endian arrays of 'len' BignumInts. Returns a BignumInt carried + * little-endian arrays of 'len' BignumInts. Returns a BignumInt carried * off the top. */ static BignumInt internal_add(const BignumInt *a, const BignumInt *b, @@ -170,7 +170,7 @@ static BignumInt internal_add(const BignumInt *a, const BignumInt *b, int i; BignumDblInt carry = 0; - for (i = len-1; i >= 0; i--) { + for (i = 0; i < len; i++) { carry += (BignumDblInt)a[i] + b[i]; c[i] = (BignumInt)carry; carry >>= BIGNUM_INT_BITS; @@ -181,7 +181,7 @@ static BignumInt internal_add(const BignumInt *a, const BignumInt *b, /* * Internal subtraction. Sets c = a - b, where 'a', 'b' and 'c' are - * all big-endian arrays of 'len' BignumInts. Any borrow from the top + * all little-endian arrays of 'len' BignumInts. Any borrow from the top * is ignored. */ static void internal_sub(const BignumInt *a, const BignumInt *b, @@ -190,7 +190,7 @@ static void internal_sub(const BignumInt *a, const BignumInt *b, int i; BignumDblInt carry = 1; - for (i = len-1; i >= 0; i--) { + for (i = 0; i < len; i++) { carry += (BignumDblInt)a[i] + (b[i] ^ BIGNUM_INT_MASK); c[i] = (BignumInt)carry; carry >>= BIGNUM_INT_BITS; @@ -201,15 +201,28 @@ static void internal_sub(const BignumInt *a, const BignumInt *b, * 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.) */ #define KARATSUBA_THRESHOLD 50 -static void internal_mul(BignumInt *a, BignumInt *b, - BignumInt *c, int len) +static int mul_compute_scratch(int len) +{ + 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) { - int i, j; - BignumDblInt t; - if (len > KARATSUBA_THRESHOLD) { + int i; /* * Karatsuba divide-and-conquer algorithm. Cut each input in @@ -245,7 +258,6 @@ static void internal_mul(BignumInt *a, BignumInt *b, int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */ int midlen = botlen + 1; - BignumInt *scratch; BignumDblInt carry; /* @@ -254,55 +266,111 @@ static void internal_mul(BignumInt *a, BignumInt *b, * 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[len - 1 - 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[len - 1 - i]); + } + printf("\n"); +#endif + /* a_1 b_1 */ - internal_mul(a, b, c, toplen); + internal_mul(a + botlen, b + botlen, c + 2*botlen, toplen, scratch); +#ifdef KARA_DEBUG + printf("a1b1 = 0x"); + for (i = 0; i < 2*toplen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, c[2*len - 1 - i]); + } + printf("\n"); +#endif /* a_0 b_0 */ - internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen); + internal_mul(a, b, c, botlen, scratch); +#ifdef KARA_DEBUG + printf("a0b0 = 0x"); + for (i = 0; i < 2*botlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, c[2*botlen - 1 - i]); + } + printf("\n"); +#endif - /* - * We must allocate scratch space for the central coefficient, - * and also for the two input values that we multiply when - * computing it. Since either or both may carry into the - * (botlen+1)th word, we must use a slightly longer length - * 'midlen'. + /* Zero padding. botlen exceeds toplen by at most 1, and we'll set + * the extra carry explicitly below, so we only need to zero at most + * one of the top words here. */ - scratch = snewn(4 * midlen, BignumInt); + scratch[midlen - 2] = scratch[2*midlen - 2] = 0; - /* 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 (j = 0; j < toplen; j++) { - scratch[midlen - toplen + j] = a[j]; /* a_1 */ - scratch[2*midlen - toplen + j] = b[j]; /* b_1 */ + for (i = 0; i < toplen; i++) { + scratch[i] = a[i + botlen]; /* a_1 */ + scratch[midlen + i] = b[i + botlen]; /* b_1 */ } /* compute a_1 + a_0 */ - scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen); + scratch[midlen - 1] = internal_add(scratch, a, scratch, botlen); +#ifdef KARA_DEBUG + printf("a1plusa0 = 0x"); + for (i = 0; i < midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen - 1 - i]); + } + printf("\n"); +#endif /* compute b_1 + b_0 */ - scratch[midlen] = internal_add(scratch+midlen+1, b+toplen, - scratch+midlen+1, botlen); + scratch[2*midlen - 1] = internal_add(scratch+midlen, b, + scratch+midlen, botlen); +#ifdef KARA_DEBUG + printf("b1plusb0 = 0x"); + for (i = 0; i < midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]); + } + printf("\n"); +#endif /* * Now we can do the third multiplication. */ - internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen); + 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[4*midlen - 1 - 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 (j = 0; j < 2*toplen; j++) - scratch[2*midlen - 2*toplen + j] = c[j]; - scratch[1] = internal_add(scratch+2, c + 2*toplen, - scratch+2, 2*botlen); + scratch[2*botlen - 2] = scratch[2*botlen - 1] = 0; + for (i = 0; i < 2*toplen; i++) + scratch[i] = c[2*botlen + i]; + scratch[2*botlen] = internal_add(scratch, c, scratch, 2*botlen); + scratch[2*botlen + 1] = 0; +#ifdef KARA_DEBUG + printf("a1b1plusa0b0 = 0x"); + for (i = 0; i < 2*midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]); + } + printf("\n"); +#endif - internal_sub(scratch + 2*midlen, scratch, - scratch + 2*midlen, 2*midlen); + internal_sub(scratch + 2*midlen, scratch, scratch, 2*midlen); +#ifdef KARA_DEBUG + printf("a1b0plusa0b1 = 0x"); + for (i = 0; i < 2*midlen; i++) { + printf("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]); + } + printf("\n"); +#endif /* * And now all we need to do is to add that middle coefficient @@ -310,44 +378,209 @@ static void internal_mul(BignumInt *a, BignumInt *b, * 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); - j = 2*len - botlen - 2*midlen - 1; + carry = internal_add(c + botlen, scratch, c + botlen, 2*midlen); + i = botlen + 2*midlen; while (carry) { - assert(j >= 0); - carry += c[j]; - c[j] = (BignumInt)carry; + assert(i <= 2*len); + 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[2*len - i]); + } + printf("\n"); +#endif + + } else { + int i; + BignumInt carry; + BignumDblInt t; + const BignumInt *ap, *alim = a + len, *bp, *blim = b + len; + BignumInt *cp, *cps; + + /* + * Multiply in the ordinary O(N^2) way. + */ + + for (i = 0; i < 2 * len; i++) + c[i] = 0; + + for (cps = c, ap = a; ap < alim; ap++, cps++) { + carry = 0; + for (cp = cps, bp = b; bp < blim; bp++, cp++) { + 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, b, scratch + 2*toplen, botlen, scratch + 2*len); + + /* a_1 b_0 */ + internal_mul_low(a + botlen, b, scratch + toplen, toplen, + scratch + 2*len); + + /* a_0 b_1 */ + internal_mul_low(a, b + botlen, scratch, toplen, scratch + 2*len); - /* Free scratch. */ - for (j = 0; j < 4 * midlen; j++) - scratch[j] = 0; - sfree(scratch); + /* Copy the bottom half of the big coefficient into place */ + for (i = 0; i < botlen; i++) + c[i] = scratch[2*toplen + 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, + c + botlen, toplen); } else { + int i; + BignumInt carry; + BignumDblInt t; + const BignumInt *ap, *alen = a + len, *bp; + BignumInt *cp, *cps, *clen = c + len; /* * Multiply in the ordinary O(N^2) way. */ - for (j = 0; j < 2 * len; j++) - c[j] = 0; + for (i = 0; i < len; i++) + c[i] = 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; + for (cps = c, ap = a; ap < alen; ap++, cps++) { + carry = 0; + for (cp = cps, bp = b; cp < clen; bp++, cp++) { + t = (MUL_WORD(*ap, *bp) + carry) + *cp; + *cp = (BignumInt) t; + carry = (BignumInt)(t >> BIGNUM_INT_BITS); } - c[i] = (BignumInt) t; } } } +/* + * Montgomery reduction. Expects x to be a little-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 little-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, 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[i] = x[len + i], x[len + 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 = len; i-- > 0; ) + if (x[i] != n[i]) + break; + } + if (carry || !i || x[i] > n[i]) + internal_sub(x, n, x, len); +} + static void internal_add_shifted(BignumInt *number, unsigned n, int shift) { @@ -369,11 +602,10 @@ static void internal_add_shifted(BignumInt *number, * Compute a = a % m. * Input in first alen words of a and first mlen words of m. * Output in first alen words of a - * (of which first alen-mlen words will be zero). + * (of which last alen-mlen words will be zero). * The MSW of m MUST have its high bit set. - * Quotient is accumulated in the `quotient' array, which is a Bignum - * rather than the internal bigendian format. Quotient parts are shifted - * left by `qshift' before adding into quot. + * Quotient is accumulated in the `quotient' array. Quotient parts + * are shifted left by `qshift' before adding into quot. */ static void internal_mod(BignumInt *a, int alen, BignumInt *m, int mlen, @@ -381,29 +613,22 @@ static void internal_mod(BignumInt *a, int alen, { BignumInt m0, m1; unsigned int h; - int i, k; + int i, j, k; - m0 = m[0]; + m0 = m[mlen - 1]; if (mlen > 1) - m1 = m[1]; + m1 = m[mlen - 2]; else m1 = 0; - for (i = 0; i <= alen - mlen; i++) { + for (i = alen, h = 0; i-- >= mlen; ) { BignumDblInt t; unsigned int q, r, c, ai1; - if (i == 0) { - h = 0; - } else { - h = a[i - 1]; - a[i - 1] = 0; - } - - if (i == alen - 1) - ai1 = 0; - else - ai1 = a[i + 1]; + if (i) + ai1 = a[i - 1]; + else + ai1 = 0; /* Find q = h:a[i] / m0 */ if (h >= m0) { @@ -430,7 +655,7 @@ static void internal_mod(BignumInt *a, int alen, DIVMOD_WORD(q, r, h, tmplo, m0); /* Refine our estimate of q by looking at - h:a[i]:a[i+1] / m0:m1 */ + h:a[i]:a[i-1] / m0:m1 */ t = MUL_WORD(m1, q); if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { q--; @@ -441,41 +666,73 @@ static void internal_mod(BignumInt *a, int alen, } } + j = i + 1 - mlen; + /* Subtract q * m from a[i...] */ c = 0; - for (k = mlen - 1; k >= 0; k--) { + for (k = 0; k < mlen; k++) { t = MUL_WORD(q, m[k]); t += c; c = (unsigned)(t >> BIGNUM_INT_BITS); - if ((BignumInt) t > a[i + k]) + if ((BignumInt) t > a[j + k]) c++; - a[i + k] -= (BignumInt) t; + a[j + k] -= (BignumInt) t; } /* Add back m in case of borrow */ if (c != h) { t = 0; - for (k = mlen - 1; k >= 0; k--) { + for (k = 0; k < mlen; k++) { t += m[k]; - t += a[i + k]; - a[i + k] = (BignumInt) t; + t += a[j + k]; + a[j + k] = (BignumInt) t; t = t >> BIGNUM_INT_BITS; } q--; } + if (quot) - internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i)); + internal_add_shifted(quot, q, + qshift + BIGNUM_INT_BITS * (i + 1 - mlen)); + + if (i >= mlen) { + h = a[i]; + a[i] = 0; + } } } +static void shift_left(BignumInt *x, int xlen, int shift) +{ + int i; + + if (!shift) + return; + for (i = xlen; --i > 0; ) + x[i] = (x[i] << shift) | (x[i - 1] >> (BIGNUM_INT_BITS - shift)); + x[0] = x[0] << shift; +} + +static void shift_right(BignumInt *x, int xlen, int shift) +{ + int i; + + if (!shift || !xlen) + return; + xlen--; + for (i = 0; i < xlen; i++) + x[i] = (x[i] >> shift) | (x[i + 1] << (BIGNUM_INT_BITS - shift)); + x[i] = x[i] >> shift; +} + /* - * 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; /* @@ -491,36 +748,35 @@ 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]; + m[j] = mod[j + 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) + if ((m[mlen - 1] << 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; - } + if (mshift) + shift_left(m, mlen, mshift); /* 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]; + for (i = 0; i < (int)base[0]; i++) + n[i] = base[i + 1]; + for (; i < mlen; i++) + n[i] = 0; /* 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[0] = 1; + for (i = 1; i < 2 * mlen; i++) 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; @@ -536,10 +792,10 @@ 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, a, 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, n, a, mlen, scratch); internal_mod(a, mlen * 2, m, mlen, NULL, 0); } else { BignumInt *t; @@ -555,18 +811,15 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) /* 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)); + shift_left(a, mlen + 1, mshift); + internal_mod(a, mlen + 1, m, mlen, NULL, 0); + shift_right(a, mlen, mshift); } /* Copy result to buffer */ result = newbn(mod[0]); for (i = 0; i < mlen; i++) - result[result[0] - i] = a[i + mlen]; + result[i + 1] = a[i]; while (result[0] > 1 && result[result[0]] == 0) result[0]--; @@ -574,6 +827,9 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) for (i = 0; i < 2 * mlen; i++) a[i] = 0; sfree(a); + for (i = 0; i < scratchlen; i++) + scratch[i] = 0; + sfree(scratch); for (i = 0; i < 2 * mlen; i++) b[i] = 0; sfree(b); @@ -590,78 +846,232 @@ 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); + + /* + * 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 containing the base, + * the modulus, and the modulus's inverse. + */ + n = snewn(len, BignumInt); + for (j = 0; j < len; j++) + n[j] = mod[j + 1]; + + mninv = snewn(len, BignumInt); + for (j = 0; j < len; j++) + mninv[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[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[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, a, b, len, scratch); + monty_reduce(b, n, mninv, scratch, len); + if ((exp[exp[0] - i] & (1 << j)) != 0) { + internal_mul(b, 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[i + 1] = a[i]; + while (result[0] > 1 && result[result[0]] == 0) + result[0]--; + + /* Free temporary arrays */ + for (i = 0; i < scratchlen; i++) + scratch[i] = 0; + sfree(scratch); + for (i = 0; i < 2 * len; i++) + a[i] = 0; + sfree(a); + for (i = 0; i < 2 * len; i++) + b[i] = 0; + sfree(b); + for (i = 0; i < len; i++) + mninv[i] = 0; + sfree(mninv); + for (i = 0; i < len; i++) + n[i] = 0; + sfree(n); + for (i = 0; i < len; i++) + x[i] = 0; + 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; /* 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]; + m[j] = mod[j + 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) + if ((m[mlen - 1] << 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; - } + if (mshift) + shift_left(m, mlen, mshift); 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]; - for (j = 0; j < i; j++) - n[j] = 0; - for (j = 0; j < (int)p[0]; j++) - n[i + j] = p[p[0] - j]; + for (i = 0; i < (int)p[0]; i++) + n[i] = p[i + 1]; + for (; i < pqlen; i++) + n[i] = 0; /* Allocate o of size pqlen, copy q to o */ o = snewn(pqlen, BignumInt); - i = pqlen - q[0]; - for (j = 0; j < i; j++) - o[j] = 0; - for (j = 0; j < (int)q[0]; j++) - o[i + j] = q[q[0] - j]; + for (i = 0; i < (int)q[0]; i++) + o[i] = q[i + 1]; + for (; i < pqlen; i++) + o[i] = 0; /* 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 */ if (mshift) { - for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++) - a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift)); - a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift; - internal_mod(a, pqlen * 2, m, mlen, NULL, 0); - for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--) - a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); + shift_left(a, mlen + 1, mshift); + internal_mod(a, mlen + 1, m, mlen, NULL, 0); + shift_right(a, mlen, mshift); } /* Copy result to buffer */ rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2); result = newbn(rlen); for (i = 0; i < rlen; i++) - result[result[0] - i] = a[i + 2 * pqlen - rlen]; + result[i + 1] = a[i]; while (result[0] > 1 && result[result[0]] == 0) result[0]--; /* Free temporary arrays */ + for (i = 0; i < scratchlen; i++) + scratch[i] = 0; + sfree(scratch); for (i = 0; i < 2 * pqlen; i++) a[i] = 0; sfree(a); @@ -692,21 +1102,17 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) int plen, mlen, i, j; /* 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]; + m[j] = mod[j + 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) + if ((m[mlen - 1] << 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; - } + if (mshift) + shift_left(m, mlen, mshift); plen = p[0]; /* Ensure plen > mlen */ @@ -715,30 +1121,26 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) /* Allocate n of size plen, copy p to n */ n = snewn(plen, BignumInt); - for (j = 0; j < plen; j++) - n[j] = 0; - for (j = 1; j <= (int)p[0]; j++) - n[plen - j] = p[j]; + for (i = 0; i < (int)p[0]; i++) + n[i] = p[i + 1]; + for (; i < plen; i++) + n[i] = 0; /* Main computation */ internal_mod(n, plen, m, mlen, quotient, mshift); /* Fixup result in case the modulus was shifted */ if (mshift) { - for (i = plen - mlen - 1; i < plen - 1; i++) - n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift)); - n[plen - 1] = n[plen - 1] << mshift; + shift_left(n, mlen + 1, mshift); internal_mod(n, plen, m, mlen, quotient, 0); - for (i = plen - 1; i >= plen - mlen; i--) - n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift)); + shift_right(n, mlen, mshift); } /* Copy result to buffer */ if (result) { - for (i = 1; i <= (int)result[0]; i++) { - int j = plen - i; - result[i] = j >= 0 ? n[j] : 0; - } + for (i = 0; i < (int)result[0]; i++) + result[i + 1] = i < plen ? n[i] : 0; + bn_restore_invariant(result); } /* Free temporary arrays */ @@ -950,18 +1352,21 @@ 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); + workspace[0 * mlen + i] = i < (int)a[0] ? a[i + 1] : 0; + workspace[1 * mlen + i] = i < (int)b[0] ? b[i + 1] : 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; @@ -969,10 +1374,10 @@ Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) rlen = addend[0] + 1; ret = newbn(rlen); maxspot = 0; - for (i = 1; i <= (int)ret[0]; i++) { - ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0); - if (ret[i] != 0) - maxspot = i; + for (i = 0; i < (int)ret[0]; i++) { + ret[i + 1] = (i < 2 * mlen ? workspace[2 * mlen + i] : 0); + if (ret[i + 1] != 0) + maxspot = i + 1; } ret[0] = maxspot; @@ -990,6 +1395,8 @@ Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) } ret[0] = maxspot; + for (i = 0; i < wslen; i++) + workspace[i] = 0; sfree(workspace); return ret; } @@ -1003,6 +1410,69 @@ 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. @@ -1280,3 +1750,166 @@ char *bignum_decimal(Bignum x) 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, "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