X-Git-Url: https://git.distorted.org.uk/u/mdw/putty/blobdiff_plain/3d88e64dfcf5dc0fd361ce0c504c67a9196ce44c..HEAD:/sshbn.c diff --git a/sshbn.c b/sshbn.c index d404ed06..a5e0552f 100644 --- a/sshbn.c +++ b/sshbn.c @@ -3,23 +3,110 @@ */ #include +#include #include #include +#include #include "misc.h" +/* + * Usage notes: + * * Do not call the DIVMOD_WORD macro with expressions such as array + * subscripts, as some implementations object to this (see below). + * * Note that none of the division methods below will cope if the + * quotient won't fit into BIGNUM_INT_BITS. Callers should be careful + * to avoid this case. + * If this condition occurs, in the case of the x86 DIV instruction, + * an overflow exception will occur, which (according to a correspondent) + * will manifest on Windows as something like + * 0xC0000095: Integer overflow + * The C variant won't give the right answer, either. + */ + +#if defined __GNUC__ && defined __i386__ +typedef unsigned long BignumInt; +typedef unsigned long long BignumDblInt; +#define BIGNUM_INT_MASK 0xFFFFFFFFUL +#define BIGNUM_TOP_BIT 0x80000000UL +#define BIGNUM_INT_BITS 32 +#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) +#define DIVMOD_WORD(q, r, hi, lo, w) \ + __asm__("div %2" : \ + "=d" (r), "=a" (q) : \ + "r" (w), "d" (hi), "a" (lo)) +#elif defined _MSC_VER && defined _M_IX86 +typedef unsigned __int32 BignumInt; +typedef unsigned __int64 BignumDblInt; +#define BIGNUM_INT_MASK 0xFFFFFFFFUL +#define BIGNUM_TOP_BIT 0x80000000UL +#define BIGNUM_INT_BITS 32 +#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) +/* Note: MASM interprets array subscripts in the macro arguments as + * assembler syntax, which gives the wrong answer. Don't supply them. + * */ +#define DIVMOD_WORD(q, r, hi, lo, w) do { \ + __asm mov edx, hi \ + __asm mov eax, lo \ + __asm div w \ + __asm mov r, edx \ + __asm mov q, eax \ +} while(0) +#elif defined _LP64 +/* 64-bit architectures can do 32x32->64 chunks at a time */ +typedef unsigned int BignumInt; +typedef unsigned long BignumDblInt; +#define BIGNUM_INT_MASK 0xFFFFFFFFU +#define BIGNUM_TOP_BIT 0x80000000U +#define BIGNUM_INT_BITS 32 +#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) +#define DIVMOD_WORD(q, r, hi, lo, w) do { \ + BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ + q = n / w; \ + r = n % w; \ +} while (0) +#elif defined _LLP64 +/* 64-bit architectures in which unsigned long is 32 bits, not 64 */ +typedef unsigned long BignumInt; +typedef unsigned long long BignumDblInt; +#define BIGNUM_INT_MASK 0xFFFFFFFFUL +#define BIGNUM_TOP_BIT 0x80000000UL +#define BIGNUM_INT_BITS 32 +#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) +#define DIVMOD_WORD(q, r, hi, lo, w) do { \ + BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ + q = n / w; \ + r = n % w; \ +} while (0) +#else +/* Fallback for all other cases */ +typedef unsigned short BignumInt; +typedef unsigned long BignumDblInt; +#define BIGNUM_INT_MASK 0xFFFFU +#define BIGNUM_TOP_BIT 0x8000U +#define BIGNUM_INT_BITS 16 +#define MUL_WORD(w1, w2) ((BignumDblInt)w1 * w2) +#define DIVMOD_WORD(q, r, hi, lo, w) do { \ + BignumDblInt n = (((BignumDblInt)hi) << BIGNUM_INT_BITS) | lo; \ + q = n / w; \ + r = n % w; \ +} while (0) +#endif + +#define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8) + #define BIGNUM_INTERNAL -typedef unsigned short *Bignum; +typedef BignumInt *Bignum; #include "ssh.h" -unsigned short bnZero[1] = { 0 }; -unsigned short bnOne[2] = { 1, 1 }; +BignumInt bnZero[1] = { 0 }; +BignumInt bnOne[2] = { 1, 1 }; /* - * The Bignum format is an array of `unsigned short'. The first + * The Bignum format is an array of `BignumInt'. The first * element of the array counts the remaining elements. The - * remaining elements express the actual number, base 2^16, _least_ + * remaining elements express the actual number, base 2^BIGNUM_INT_BITS, _least_ * significant digit first. (So it's trivial to extract the bit * with value 2^n for any n.) * @@ -34,7 +121,11 @@ Bignum Zero = bnZero, One = bnOne; static Bignum newbn(int length) { - Bignum b = snewn(length + 1, unsigned short); + 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)); @@ -50,7 +141,7 @@ void bn_restore_invariant(Bignum b) Bignum copybn(Bignum orig) { - Bignum b = snewn(orig[0] + 1, unsigned short); + Bignum b = snewn(orig[0] + 1, BignumInt); if (!b) abort(); /* FIXME */ memcpy(b, orig, (orig[0] + 1) * sizeof(*b)); @@ -62,57 +153,464 @@ 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 / 16 + 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(unsigned short *a, unsigned short *b, - unsigned short *c, int len) +#define KARATSUBA_THRESHOLD 50 +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; - unsigned long ai, t; + 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 - for (j = 0; j < 2 * len; j++) - c[j] = 0; + /* + * 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 - for (i = len - 1; i >= 0; i--) { - ai = a[i]; - t = 0; - for (j = len - 1; j >= 0; j--) { - t += ai * (unsigned long) b[j]; - t += (unsigned long) c[i + j + 1]; - c[i + j + 1] = (unsigned short) t; - t = t >> 16; - } - c[i] = (unsigned short) t; + /* 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; + } } } -static void internal_add_shifted(unsigned short *number, +/* + * 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, unsigned n, int shift) { - int word = 1 + (shift / 16); - int bshift = shift % 16; - unsigned long addend; + int word = 1 + (shift / BIGNUM_INT_BITS); + int bshift = shift % BIGNUM_INT_BITS; + BignumDblInt addend; - addend = n << bshift; + addend = (BignumDblInt)n << bshift; while (addend) { + assert(word <= number[0]); addend += number[word]; - number[word] = (unsigned short) addend & 0xFFFF; - addend >>= 16; + number[word] = (BignumInt) addend & BIGNUM_INT_MASK; + addend >>= BIGNUM_INT_BITS; word++; } } @@ -127,22 +625,23 @@ static void internal_add_shifted(unsigned short *number, * rather than the internal bigendian format. Quotient parts are shifted * left by `qshift' before adding into quot. */ -static void internal_mod(unsigned short *a, int alen, - unsigned short *m, int mlen, - unsigned short *quot, int qshift) +static void internal_mod(BignumInt *a, int alen, + BignumInt *m, int mlen, + BignumInt *quot, int qshift) { - unsigned short m0, m1; + BignumInt m0, m1; unsigned int h; int i, k; m0 = m[0]; + assert(m0 >> (BIGNUM_INT_BITS-1) == 1); if (mlen > 1) m1 = m[1]; else m1 = 0; for (i = 0; i <= alen - mlen; i++) { - unsigned long t; + BignumDblInt t; unsigned int q, r, c, ai1; if (i == 0) { @@ -158,30 +657,50 @@ static void internal_mod(unsigned short *a, int alen, ai1 = a[i + 1]; /* Find q = h:a[i] / m0 */ - t = ((unsigned long) h << 16) + a[i]; - q = t / m0; - r = t % m0; - - /* Refine our estimate of q by looking at - h:a[i]:a[i+1] / m0:m1 */ - t = (long) m1 *(long) q; - if (t > ((unsigned long) r << 16) + ai1) { - q--; - t -= m1; - r = (r + m0) & 0xffff; /* overflow? */ - if (r >= (unsigned long) m0 && - t > ((unsigned long) r << 16) + ai1) q--; + if (h >= m0) { + /* + * Special case. + * + * To illustrate it, suppose a BignumInt is 8 bits, and + * we are dividing (say) A1:23:45:67 by A1:B2:C3. Then + * our initial division will be 0xA123 / 0xA1, which + * will give a quotient of 0x100 and a divide overflow. + * However, the invariants in this division algorithm + * are not violated, since the full number A1:23:... is + * _less_ than the quotient prefix A1:B2:... and so the + * following correction loop would have sorted it out. + * + * In this situation we set q to be the largest + * quotient we _can_ stomach (0xFF, of course). + */ + q = BIGNUM_INT_MASK; + } else { + /* Macro doesn't want an array subscript expression passed + * into it (see definition), so use a temporary. */ + BignumInt tmplo = a[i]; + DIVMOD_WORD(q, r, h, tmplo, m0); + + /* Refine our estimate of q by looking at + h:a[i]:a[i+1] / m0:m1 */ + t = MUL_WORD(m1, q); + if (t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) { + q--; + t -= m1; + r = (r + m0) & BIGNUM_INT_MASK; /* overflow? */ + if (r >= (BignumDblInt) m0 && + t > ((BignumDblInt) r << BIGNUM_INT_BITS) + ai1) q--; + } } /* Subtract q * m from a[i...] */ c = 0; for (k = mlen - 1; k >= 0; k--) { - t = (long) q *(long) m[k]; + t = MUL_WORD(q, m[k]); t += c; - c = t >> 16; - if ((unsigned short) t > a[i + k]) + c = (unsigned)(t >> BIGNUM_INT_BITS); + if ((BignumInt) t > a[i + k]) c++; - a[i + k] -= (unsigned short) t; + a[i + k] -= (BignumInt) t; } /* Add back m in case of borrow */ @@ -190,82 +709,95 @@ static void internal_mod(unsigned short *a, int alen, for (k = mlen - 1; k >= 0; k--) { t += m[k]; t += a[i + k]; - a[i + k] = (unsigned short) t; - t = t >> 16; + a[i + k] = (BignumInt) t; + t = t >> BIGNUM_INT_BITS; } q--; } if (quot) - internal_add_shifted(quot, q, qshift + 16 * (alen - mlen - i)); + internal_add_shifted(quot, q, qshift + BIGNUM_INT_BITS * (alen - mlen - i)); } } /* - * Compute (base ^ exp) % mod. - * The base MUST be smaller than the modulus. - * The most significant word of mod MUST be non-zero. - * We assume that the result array is the same size as the mod array. + * Compute (base ^ exp) % mod, the pedestrian way. */ -Bignum modpow(Bignum base, Bignum exp, Bignum mod) +Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod) { - unsigned short *a, *b, *n, *m; + BignumInt *a, *b, *n, *m, *scratch; int mshift; - int mlen, i, j; - Bignum result; + int mlen, scratchlen, i, j; + Bignum base, 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); + + /* + * Make sure the base is smaller than the modulus, by reducing + * it modulo the modulus if not. + */ + 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, unsigned short); + m = snewn(mlen, BignumInt); for (j = 0; j < mlen; j++) m[j] = mod[mod[0] - j]; /* Shift m left to make msb bit set */ - for (mshift = 0; mshift < 15; mshift++) - if ((m[0] << mshift) & 0x8000) + 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] >> (16 - mshift)); + m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); m[mlen - 1] = m[mlen - 1] << mshift; } /* Allocate n of size mlen, copy base to n */ - n = snewn(mlen, unsigned short); + n = snewn(mlen, BignumInt); i = mlen - base[0]; for (j = 0; j < i; j++) n[j] = 0; - for (j = 0; j < base[0]; j++) + for (j = 0; j < (int)base[0]; j++) n[i + j] = base[base[0] - j]; /* Allocate a and b of size 2*mlen. Set a = 1 */ - a = snewn(2 * mlen, unsigned short); - b = snewn(2 * mlen, unsigned short); + 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; + /* Scratch space for multiplies */ + scratchlen = mul_compute_scratch(mlen); + scratch = snewn(scratchlen, BignumInt); + /* Skip leading zero bits of exp. */ i = 0; - j = 15; - while (i < exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { + j = BIGNUM_INT_BITS-1; + while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) { j--; if (j < 0) { i++; - j = 15; + j = BIGNUM_INT_BITS-1; } } /* Main computation */ - while (i < exp[0]) { + 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 { - unsigned short *t; + BignumInt *t; t = a; a = b; b = t; @@ -273,17 +805,17 @@ Bignum modpow(Bignum base, Bignum exp, Bignum mod) j--; } i++; - j = 15; + 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] >> (16 - mshift)); + 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] << (16 - mshift)); + a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); } /* Copy result to buffer */ @@ -294,18 +826,164 @@ Bignum modpow(Bignum base, 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); + + return result; +} + +/* + * 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; } @@ -317,61 +995,78 @@ Bignum modpow(Bignum base, Bignum exp, Bignum mod) */ Bignum modmul(Bignum p, Bignum q, Bignum mod) { - unsigned short *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]; - m = snewn(mlen, unsigned short); + m = snewn(mlen, BignumInt); for (j = 0; j < mlen; j++) m[j] = mod[mod[0] - j]; /* Shift m left to make msb bit set */ - for (mshift = 0; mshift < 15; mshift++) - if ((m[0] << mshift) & 0x8000) + 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] >> (16 - mshift)); + m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); m[mlen - 1] = m[mlen - 1] << 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, unsigned short); + n = snewn(pqlen, BignumInt); i = pqlen - p[0]; for (j = 0; j < i; j++) n[j] = 0; - for (j = 0; j < p[0]; j++) + for (j = 0; j < (int)p[0]; j++) n[i + j] = p[p[0] - j]; /* Allocate o of size pqlen, copy q to o */ - o = snewn(pqlen, unsigned short); + o = snewn(pqlen, BignumInt); i = pqlen - q[0]; for (j = 0; j < i; j++) o[j] = 0; - for (j = 0; j < q[0]; j++) + for (j = 0; j < (int)q[0]; j++) o[i + j] = q[q[0] - j]; /* Allocate a of size 2*pqlen for result */ - a = snewn(2 * pqlen, unsigned short); + 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] >> (16 - mshift)); + 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] << (16 - mshift)); + a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift)); } /* Copy result to buffer */ @@ -383,17 +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; @@ -408,24 +1101,30 @@ Bignum modmul(Bignum p, Bignum q, Bignum mod) */ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) { - unsigned short *n, *m; + BignumInt *n, *m; 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]; - m = snewn(mlen, unsigned short); + m = snewn(mlen, BignumInt); for (j = 0; j < mlen; j++) m[j] = mod[mod[0] - j]; /* Shift m left to make msb bit set */ - for (mshift = 0; mshift < 15; mshift++) - if ((m[0] << mshift) & 0x8000) + 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] >> (16 - mshift)); + m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift)); m[mlen - 1] = m[mlen - 1] << mshift; } @@ -435,10 +1134,10 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) plen = mlen + 1; /* Allocate n of size plen, copy p to n */ - n = snewn(plen, unsigned short); + n = snewn(plen, BignumInt); for (j = 0; j < plen; j++) n[j] = 0; - for (j = 1; j <= p[0]; j++) + for (j = 1; j <= (int)p[0]; j++) n[plen - j] = p[j]; /* Main computation */ @@ -447,27 +1146,25 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) /* 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] >> (16 - mshift)); + n[i] = (n[i] << mshift) | (n[i + 1] >> (BIGNUM_INT_BITS - mshift)); n[plen - 1] = n[plen - 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] << (16 - mshift)); + n[i] = (n[i] >> mshift) | (n[i - 1] << (BIGNUM_INT_BITS - mshift)); } /* Copy result to buffer */ if (result) { - for (i = 1; i <= result[0]; i++) { + for (i = 1; i <= (int)result[0]; i++) { int j = plen - i; result[i] = j >= 0 ? n[j] : 0; } } /* 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); } @@ -477,8 +1174,8 @@ static void bigdivmod(Bignum p, Bignum mod, Bignum result, Bignum quotient) void decbn(Bignum bn) { int i = 1; - while (i < bn[0] && bn[i] == 0) - bn[i++] = 0xFFFF; + while (i < (int)bn[0] && bn[i] == 0) + bn[i++] = BIGNUM_INT_MASK; bn[i]--; } @@ -487,17 +1184,16 @@ Bignum bignum_from_bytes(const unsigned char *data, int nbytes) Bignum result; int w, i; - w = (nbytes + 1) / 2; /* bytes -> words */ + assert(nbytes >= 0 && nbytes < INT_MAX/8); + + w = (nbytes + BIGNUM_INT_BYTES - 1) / BIGNUM_INT_BYTES; /* bytes->words */ result = newbn(w); for (i = 1; i <= w; i++) result[i] = 0; for (i = nbytes; i--;) { unsigned char byte = *data++; - if (i & 1) - result[1 + i / 2] |= byte << 8; - else - result[1 + i / 2] |= byte; + result[1 + i / BIGNUM_INT_BYTES] |= byte << (8*i % BIGNUM_INT_BITS); } while (result[0] > 1 && result[result[0]] == 0) @@ -506,20 +1202,26 @@ Bignum bignum_from_bytes(const unsigned char *data, int nbytes) } /* - * Read an ssh1-format bignum from a data buffer. Return the number - * of bytes consumed. + * Read an SSH-1-format bignum from a data buffer. Return the number + * of bytes consumed, or -1 if there wasn't enough data. */ -int ssh1_read_bignum(const unsigned char *data, Bignum * result) +int ssh1_read_bignum(const unsigned char *data, int len, Bignum * result) { const unsigned char *p = data; int i; int w, b; + if (len < 2) + return -1; + w = 0; for (i = 0; i < 2; i++) w = (w << 8) + *p++; b = (w + 7) / 8; /* bits -> bytes */ + if (len < b+2) + return -1; + if (!result) /* just return length */ return b + 2; @@ -529,18 +1231,18 @@ int ssh1_read_bignum(const unsigned char *data, Bignum * result) } /* - * Return the bit count of a bignum, for ssh1 encoding. + * Return the bit count of a bignum, for SSH-1 encoding. */ int bignum_bitcount(Bignum bn) { - int bitcount = bn[0] * 16 - 1; + int bitcount = bn[0] * BIGNUM_INT_BITS - 1; while (bitcount >= 0 - && (bn[bitcount / 16 + 1] >> (bitcount % 16)) == 0) bitcount--; + && (bn[bitcount / BIGNUM_INT_BITS + 1] >> (bitcount % BIGNUM_INT_BITS)) == 0) bitcount--; return bitcount + 1; } /* - * Return the byte length of a bignum when ssh1 encoded. + * Return the byte length of a bignum when SSH-1 encoded. */ int ssh1_bignum_length(Bignum bn) { @@ -548,7 +1250,7 @@ int ssh1_bignum_length(Bignum bn) } /* - * Return the byte length of a bignum when ssh2 encoded. + * Return the byte length of a bignum when SSH-2 encoded. */ int ssh2_bignum_length(Bignum bn) { @@ -560,12 +1262,11 @@ int ssh2_bignum_length(Bignum bn) */ int bignum_byte(Bignum bn, int i) { - if (i >= 2 * bn[0]) + if (i < 0 || i >= (int)(BIGNUM_INT_BYTES * bn[0])) return 0; /* beyond the end */ - else if (i & 1) - return (bn[i / 2 + 1] >> 8) & 0xFF; else - return (bn[i / 2 + 1]) & 0xFF; + return (bn[i / BIGNUM_INT_BYTES + 1] >> + ((i % BIGNUM_INT_BYTES)*8)) & 0xFF; } /* @@ -573,10 +1274,10 @@ int bignum_byte(Bignum bn, int i) */ int bignum_bit(Bignum bn, int i) { - if (i >= 16 * bn[0]) + if (i < 0 || i >= (int)(BIGNUM_INT_BITS * bn[0])) return 0; /* beyond the end */ else - return (bn[i / 16 + 1] >> (i % 16)) & 1; + return (bn[i / BIGNUM_INT_BITS + 1] >> (i % BIGNUM_INT_BITS)) & 1; } /* @@ -584,11 +1285,11 @@ int bignum_bit(Bignum bn, int i) */ void bignum_set_bit(Bignum bn, int bitnum, int value) { - if (bitnum >= 16 * bn[0]) + if (bitnum < 0 || bitnum >= (int)(BIGNUM_INT_BITS * bn[0])) abort(); /* beyond the end */ else { - int v = bitnum / 16 + 1; - int mask = 1 << (bitnum % 16); + int v = bitnum / BIGNUM_INT_BITS + 1; + int mask = 1 << (bitnum % BIGNUM_INT_BITS); if (value) bn[v] |= mask; else @@ -597,7 +1298,7 @@ void bignum_set_bit(Bignum bn, int bitnum, int value) } /* - * Write a ssh1-format bignum into a buffer. It is assumed the + * Write a SSH-1-format bignum into a buffer. It is assumed the * buffer is big enough. Returns the number of bytes used. */ int ssh1_write_bignum(void *data, Bignum bn) @@ -620,10 +1321,21 @@ 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) { - unsigned short aval = (i > amax ? 0 : a[i]); - unsigned short bval = (i > bmax ? 0 : b[i]); + BignumInt aval = (i > amax ? 0 : a[i]); + BignumInt bval = (i > bmax ? 0 : b[i]); if (aval < bval) return -1; if (aval > bval) @@ -640,21 +1352,23 @@ Bignum bignum_rshift(Bignum a, int shift) { Bignum ret; int i, shiftw, shiftb, shiftbb, bits; - unsigned short ai, ai1; + BignumInt ai, ai1; + + assert(shift >= 0); bits = bignum_bitcount(a) - shift; - ret = newbn((bits + 15) / 16); + ret = newbn((bits + BIGNUM_INT_BITS - 1) / BIGNUM_INT_BITS); if (ret) { - shiftw = shift / 16; - shiftb = shift % 16; - shiftbb = 16 - shiftb; + shiftw = shift / BIGNUM_INT_BITS; + shiftb = shift % BIGNUM_INT_BITS; + shiftbb = BIGNUM_INT_BITS - shiftb; ai1 = a[shiftw + 1]; - for (i = 1; i <= ret[0]; i++) { + for (i = 1; i <= (int)ret[0]; i++) { ai = ai1; - ai1 = (i + shiftw + 1 <= a[0] ? a[i + shiftw + 1] : 0); - ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & 0xFFFF; + ai1 = (i + shiftw + 1 <= (int)a[0] ? a[i + shiftw + 1] : 0); + ret[i] = ((ai >> shiftb) | (ai1 << shiftbb)) & BIGNUM_INT_MASK; } } @@ -669,26 +1383,29 @@ 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; - unsigned short *workspace; + int wslen; + BignumInt *workspace; Bignum ret; - /* mlen space for a, mlen space for b, 2*mlen for result */ - workspace = snewn(mlen * 4, unsigned short); + /* 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 <= a[0] ? a[mlen - i] : 0); - workspace[1 * mlen + i] = (mlen - i <= b[0] ? b[mlen - i] : 0); + 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; - if (addend && rlen <= addend[0]) + if (addend && rlen <= (int)addend[0]) rlen = addend[0] + 1; ret = newbn(rlen); maxspot = 0; - for (i = 1; i <= ret[0]; i++) { + for (i = 1; i <= (int)ret[0]; i++) { ret[i] = (i <= 2 * mlen ? workspace[4 * mlen - i] : 0); if (ret[i] != 0) maxspot = i; @@ -697,18 +1414,20 @@ Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) /* now add in the addend, if any */ if (addend) { - unsigned long carry = 0; + BignumDblInt carry = 0; for (i = 1; i <= rlen; i++) { - carry += (i <= ret[0] ? ret[i] : 0); - carry += (i <= addend[0] ? addend[i] : 0); - ret[i] = (unsigned short) carry & 0xFFFF; - carry >>= 16; + carry += (i <= (int)ret[0] ? ret[i] : 0); + carry += (i <= (int)addend[0] ? addend[i] : 0); + ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; + carry >>= BIGNUM_INT_BITS; if (ret[i] != 0 && i > maxspot) maxspot = i; } } ret[0] = maxspot; + smemclr(workspace, wslen * sizeof(*workspace)); + sfree(workspace); return ret; } @@ -721,6 +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. @@ -729,7 +1511,7 @@ Bignum bignum_bitmask(Bignum n) { Bignum ret = copybn(n); int i; - unsigned short j; + BignumInt j; i = ret[0]; while (n[i] == 0 && i > 0) @@ -741,20 +1523,21 @@ Bignum bignum_bitmask(Bignum n) j = 2 * j + 1; ret[i] = j; while (--i > 0) - ret[i] = 0xFFFF; + ret[i] = BIGNUM_INT_MASK; return ret; } /* * Convert a (max 32-bit) long into a bignum. */ -Bignum bignum_from_long(unsigned long n) +Bignum bignum_from_long(unsigned long nn) { Bignum ret; + BignumDblInt n = nn; ret = newbn(3); - ret[1] = (unsigned short)(n & 0xFFFF); - ret[2] = (unsigned short)((n >> 16) & 0xFFFF); + ret[1] = (BignumInt)(n & BIGNUM_INT_MASK); + ret[2] = (BignumInt)((n >> BIGNUM_INT_BITS) & BIGNUM_INT_MASK); ret[3] = 0; ret[0] = (ret[2] ? 2 : 1); return ret; @@ -763,18 +1546,18 @@ Bignum bignum_from_long(unsigned long n) /* * Add a long to a bignum. */ -Bignum bignum_add_long(Bignum number, unsigned long addend) +Bignum bignum_add_long(Bignum number, unsigned long addendx) { Bignum ret = newbn(number[0] + 1); int i, maxspot = 0; - unsigned long carry = 0; - - for (i = 1; i <= ret[0]; i++) { - carry += addend & 0xFFFF; - carry += (i <= number[0] ? number[i] : 0); - addend >>= 16; - ret[i] = (unsigned short) carry & 0xFFFF; - carry >>= 16; + BignumDblInt carry = 0, addend = addendx; + + for (i = 1; i <= (int)ret[0]; i++) { + carry += addend & BIGNUM_INT_MASK; + carry += (i <= (int)number[0] ? number[i] : 0); + addend >>= BIGNUM_INT_BITS; + ret[i] = (BignumInt) carry & BIGNUM_INT_MASK; + carry >>= BIGNUM_INT_BITS; if (ret[i] != 0) maxspot = i; } @@ -787,20 +1570,19 @@ Bignum bignum_add_long(Bignum number, unsigned long addend) */ unsigned short bignum_mod_short(Bignum number, unsigned short modulus) { - unsigned long mod, r; + BignumDblInt mod, r; int i; r = 0; mod = modulus; for (i = number[0]; i > 0; i--) - r = (r * 65536 + number[i]) % mod; + r = (r * (BIGNUM_TOP_BIT % mod) * 2 + number[i] % mod) % mod; return (unsigned short) r; } -#if 0 +#ifdef DEBUG void diagbn(char *prefix, Bignum md) { -#ifdef DEBUG int i, nibbles, morenibbles; static const char hex[] = "0123456789ABCDEF"; @@ -818,7 +1600,6 @@ void diagbn(char *prefix, Bignum md) if (prefix) debug(("\n")); -#endif } #endif @@ -875,9 +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]--; @@ -889,6 +1687,7 @@ Bignum modinv(Bignum number, Bignum modulus) x = bigmuladd(q, xp, t); sign = -sign; freebn(t); + freebn(q); } freebn(b); @@ -899,13 +1698,13 @@ Bignum modinv(Bignum number, Bignum modulus) if (sign < 0) { /* set a new x to be modulus - x */ Bignum newx = newbn(modulus[0]); - unsigned short carry = 0; + BignumInt carry = 0; int maxspot = 1; int i; - for (i = 1; i <= newx[0]; i++) { - unsigned short aword = (i <= modulus[0] ? modulus[i] : 0); - unsigned short bword = (i <= x[0] ? x[i] : 0); + for (i = 1; i <= (int)newx[0]; i++) { + BignumInt aword = (i <= (int)modulus[0] ? modulus[i] : 0); + BignumInt bword = (i <= (int)x[0] ? x[i] : 0); newx[i] = aword - bword - carry; bword = ~bword; carry = carry ? (newx[i] >= bword) : (newx[i] > bword); @@ -929,9 +1728,9 @@ char *bignum_decimal(Bignum x) { int ndigits, ndigit; int i, iszero; - unsigned long carry; + BignumDblInt carry; char *ret; - unsigned short *workspace; + BignumInt *workspace; /* * First, estimate the number of digits. Since log(10)/log(2) @@ -945,9 +1744,14 @@ char *bignum_decimal(Bignum x) * round up (rounding down might make it less than x again). * Therefore if we multiply the bit count by 28/93, rounding * up, we will have enough digits. + * + * i=0 (i.e., x=0) is an irritating special case. */ i = bignum_bitcount(x); - ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ + if (!i) + ndigits = 1; /* x = 0 */ + else + ndigits = (28 * i + 92) / 93; /* multiply by 28/93 and round up */ ndigits++; /* allow for trailing \0 */ ret = snewn(ndigits, char); @@ -956,8 +1760,8 @@ char *bignum_decimal(Bignum x) * repeatedly divide it by ten. Initialise this to the * big-endian form of the number. */ - workspace = snewn(x[0], unsigned short); - for (i = 0; i < x[0]; i++) + workspace = snewn(x[0], BignumInt); + for (i = 0; i < (int)x[0]; i++) workspace[i] = x[x[0] - i]; /* @@ -970,9 +1774,9 @@ char *bignum_decimal(Bignum x) do { iszero = 1; carry = 0; - for (i = 0; i < x[0]; i++) { - carry = (carry << 16) + workspace[i]; - workspace[i] = (unsigned short) (carry / 10); + for (i = 0; i < (int)x[0]; i++) { + carry = (carry << BIGNUM_INT_BITS) + workspace[i]; + workspace[i] = (BignumInt) (carry / 10); if (workspace[i]) iszero = 0; carry %= 10; @@ -990,5 +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