X-Git-Url: https://git.distorted.org.uk/u/mdw/putty/blobdiff_plain/4800a5e5eac9615f0685f0b7717173dcd3eaf18b..refs/heads/u/mdw/ec.wip:/sshbn.c diff --git a/sshbn.c b/sshbn.c index 24f3ca6b..580dd6f6 100644 --- a/sshbn.c +++ b/sshbn.c @@ -8,95 +8,7 @@ #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 BignumInt *Bignum; - +#include "bn-internal.h" #include "ssh.h" BignumInt bnZero[1] = { 0 }; @@ -161,7 +73,7 @@ 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 +82,7 @@ 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 +93,7 @@ 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 +102,7 @@ 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; @@ -259,9 +171,6 @@ static void internal_mul(const BignumInt *a, const BignumInt *b, 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 @@ -273,63 +182,64 @@ static void internal_mul(const BignumInt *a, const BignumInt *b, 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("%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[i]); + printf("%0*x", BIGNUM_INT_BITS/4, b[len - 1 - i]); } printf("\n"); #endif /* a_1 b_1 */ - internal_mul(a, b, c, toplen, scratch); + 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[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, scratch); + 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*toplen+i]); + printf("%0*x", BIGNUM_INT_BITS/4, c[2*botlen - 1 - 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; + /* 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[midlen - 2] = scratch[2*midlen - 2] = 0; for (i = 0; i < toplen; i++) { - scratch[midlen - toplen + i] = a[i]; /* a_1 */ - scratch[2*midlen - toplen + i] = b[i]; /* b_1 */ + 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[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[midlen+i]); + printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]); } printf("\n"); #endif @@ -342,7 +252,7 @@ static void internal_mul(const BignumInt *a, const BignumInt *b, #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("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]); } printf("\n"); #endif @@ -352,25 +262,24 @@ static void internal_mul(const BignumInt *a, const BignumInt *b, * 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; + scratch[2*botlen - 2] = scratch[2*botlen - 1] = 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); + 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[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[2*midlen+i]); + printf("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]); } printf("\n"); #endif @@ -381,21 +290,19 @@ static void internal_mul(const BignumInt *a, const 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); - i = 2*len - botlen - 2*midlen - 1; + carry = internal_add(c + botlen, scratch, c + botlen, 2*midlen); + i = botlen + 2*midlen; while (carry) { - assert(i >= 0); + assert(i <= 2*len); carry += c[i]; c[i] = (BignumInt)carry; carry >>= BIGNUM_INT_BITS; - i--; + i++; } #ifdef KARA_DEBUG printf("ab = 0x"); for (i = 0; i < 2*len; i++) { - printf("%0*x", BIGNUM_INT_BITS/4, c[i]); + printf("%0*x", BIGNUM_INT_BITS/4, c[2*len - i]); } printf("\n"); #endif @@ -404,7 +311,7 @@ static void internal_mul(const BignumInt *a, const BignumInt *b, int i; BignumInt carry; BignumDblInt t; - const BignumInt *ap, *bp; + const BignumInt *ap, *alim = a + len, *bp, *blim = b + len; BignumInt *cp, *cps; /* @@ -414,9 +321,9 @@ static void internal_mul(const BignumInt *a, const BignumInt *b, for (i = 0; i < 2 * len; i++) c[i] = 0; - for (cps = c + 2*len, ap = a + len; ap-- > a; cps--) { + for (cps = c, ap = a; ap < alim; ap++, cps++) { carry = 0; - for (cp = cps, bp = b + len; cp--, bp-- > b ;) { + for (cp = cps, bp = b, i = blim - bp; i--; bp++, cp++) { t = (MUL_WORD(*ap, *bp) + carry) + *cp; *cp = (BignumInt) t; carry = (BignumInt)(t >> BIGNUM_INT_BITS); @@ -480,34 +387,32 @@ static void internal_mul_low(const BignumInt *a, const BignumInt *b, */ /* a_0 b_0 */ - internal_mul(a + toplen, b + toplen, scratch + 2*toplen, botlen, - scratch + 2*len); + internal_mul(a, b, scratch + 2*toplen, botlen, scratch + 2*len); /* a_1 b_0 */ - internal_mul_low(a, b + len - toplen, scratch + toplen, toplen, + internal_mul_low(a + botlen, b, scratch + toplen, toplen, scratch + 2*len); /* a_0 b_1 */ - internal_mul_low(a + len - toplen, b, scratch, toplen, - scratch + 2*len); + internal_mul_low(a, b + botlen, 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]; + 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 - toplen, - c, toplen); + internal_add(scratch, scratch + 2*toplen + botlen, + c + botlen, toplen); } else { int i; BignumInt carry; BignumDblInt t; - const BignumInt *ap, *bp; - BignumInt *cp, *cps; + const BignumInt *ap, *alim = a + len, *bp; + BignumInt *cp, *cps, *clim = c + len; /* * Multiply in the ordinary O(N^2) way. @@ -516,9 +421,9 @@ static void internal_mul_low(const BignumInt *a, const BignumInt *b, for (i = 0; i < len; i++) c[i] = 0; - for (cps = c + len, ap = a + len; ap-- > a; cps--) { + for (cps = c, ap = a; ap < alim; ap++, cps++) { carry = 0; - for (cp = cps, bp = b + len; bp--, cp-- > c ;) { + for (cp = cps, bp = b, i = clim - cp; i--; bp++, cp++) { t = (MUL_WORD(*ap, *bp) + carry) + *cp; *cp = (BignumInt) t; carry = (BignumInt)(t >> BIGNUM_INT_BITS); @@ -528,13 +433,13 @@ static void internal_mul_low(const BignumInt *a, const BignumInt *b, } /* - * Montgomery reduction. Expects x to be a big-endian array of 2*len + * 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 big-endian arrays of 'len' BignumInts + * 'n' and 'mninv' should be little-endian arrays of 'len' BignumInts * each, containing respectively n and the multiplicative inverse of * -n mod r. * @@ -552,7 +457,7 @@ static void monty_reduce(BignumInt *x, const BignumInt *n, * 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); + internal_mul_low(x, mninv, tmp, len, tmp + 3*len); /* * Compute t = (mn+x)/r in ordinary, non-modular, integer @@ -566,7 +471,7 @@ static void monty_reduce(BignumInt *x, const BignumInt *n, 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; + x[i] = x[len + i], x[len + i] = 0; /* * Reduce t mod n. This doesn't require a full-on division by n, @@ -580,12 +485,12 @@ static void monty_reduce(BignumInt *x, const BignumInt *n, * + yielding 0 <= (mn+x)/r < 2n as required. */ if (!carry) { - for (i = 0; i < len; i++) - if (x[len + i] != n[i]) + for (i = len; i-- > 0; ) + if (x[i] != n[i]) break; } - if (carry || i >= len || x[len + i] > n[i]) - internal_sub(x+len, n, x+len, len); + if (carry || i < 0 || x[i] > n[i]) + internal_sub(x, n, x, len); } static void internal_add_shifted(BignumInt *number, @@ -609,11 +514,10 @@ 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, @@ -621,29 +525,22 @@ 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) { @@ -670,7 +567,7 @@ 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--; @@ -681,33 +578,65 @@ 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, the pedestrian way. */ @@ -731,36 +660,31 @@ Bignum modpow_simple(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); @@ -780,10 +704,10 @@ Bignum modpow_simple(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, scratch); + 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, scratch); + internal_mul(b, n, a, mlen, scratch); internal_mod(a, mlen * 2, m, mlen, NULL, 0); } else { BignumInt *t; @@ -799,18 +723,15 @@ Bignum modpow_simple(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]--; @@ -887,17 +808,16 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) 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. + * 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[len - 1 - j] = mod[j + 1]; + n[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); + 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); @@ -907,13 +827,13 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) /* 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); + 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[2*len - 1 - j] = (j < (int)rn[0] ? rn[j + 1] : 0); + a[j] = (j < (int)rn[0] ? rn[j + 1] : 0); freebn(rn); /* Scratch space for multiplies */ @@ -934,10 +854,10 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) /* Main computation */ while (i < (int)exp[0]) { while (j >= 0) { - internal_mul(a + len, a + len, b, len, scratch); + 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 + len, x, a, len, scratch); + internal_mul(b, x, a, len, scratch); monty_reduce(a, n, mninv, scratch, len); } else { BignumInt *t; @@ -960,7 +880,7 @@ Bignum modpow(Bignum base_in, Bignum exp, Bignum mod) /* Copy result to buffer */ result = newbn(mod[0]); for (i = 0; i < len; i++) - result[result[0] - i] = a[i + len]; + result[i + 1] = a[i]; while (result[0] > 1 && result[result[0]] == 0) result[0]--; @@ -1000,39 +920,39 @@ Bignum modmul(Bignum p, Bignum q, Bignum mod) 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); @@ -1047,19 +967,16 @@ Bignum modmul(Bignum p, Bignum q, Bignum mod) /* 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]--; @@ -1097,21 +1014,17 @@ 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 */ @@ -1120,30 +1033,26 @@ 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 */ @@ -1364,8 +1273,8 @@ Bignum bigmuladd(Bignum a, Bignum b, Bignum addend) 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, @@ -1377,10 +1286,10 @@ 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; @@ -1476,6 +1385,78 @@ Bignum bigsub(Bignum a, Bignum b) } /* + * Return a bignum which is the result of shifting another left by N bits. + * If N is negative then you get a right shift instead. + */ +Bignum biglsl(Bignum x, int n) +{ + Bignum d; + unsigned o, i; + + if (!n || !x[0]) + return copybn(x); + else if (n < 0) + return biglsr(x, -n); + + o = n/BIGNUM_INT_BITS; + n %= BIGNUM_INT_BITS; + d = newbn(x[0] + o + !!n); + + for (i = 1; i <= o; i++) + d[i] = 0; + + if (!n) { + for (i = 1; i <= x[0]; i++) + d[o + i] = x[i]; + } else { + d[o + 1] = x[1] << n; + for (i = 2; i <= x[0]; i--) + d[o + i] = (x[i] << n) | (x[i - 1] >> (BIGNUM_INT_BITS - n)); + d[o + x[0] + 1] = x[x[0]] >> (BIGNUM_INT_BITS - n); + } + + bn_restore_invariant(d); + return d; +} + +/* + * Return a bignum which is the result of shifting another right by N bits + * (discarding the least significant N bits, and shifting zeroes in at the + * most significant end). If N is negative then you get a left shift + * instead. + */ +Bignum biglsr(Bignum x, int n) +{ + Bignum d; + unsigned o, i; + + if (!n || !x[0]) + return copybn(x); + else if (n < 0) + return biglsl(x, -n); + + o = n/BIGNUM_INT_BITS; + n %= BIGNUM_INT_BITS; + d = newbn(x[0]); + + if (!n) { + for (i = o + 1; i <= x[0]; i++) + d[i - o] = x[i]; + } else { + d[1] = x[o + 1] >> n; + for (i = o + 2; i < x[0]; i++) + d[i - o] = x[ + d[o + x[0] + 1] = x[x[0]] >> (BIGNUM_INT_BITS - n); + for (i = x[0]; i > 1; i--) + d[o + i] = (x[i] << n) | (x[i - 1] >> (BIGNUM_INT_BITS - n)); + d[o + 1] = x[1] << n; + } + + bn_restore_invariant(d); + return d; +} + +/* * 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. @@ -1677,6 +1658,203 @@ Bignum modinv(Bignum number, Bignum modulus) } /* + * Extract the largest power of 2 dividing x, storing it in p2, and returning + * the product of the remaining factors. + */ +static Bignum extract_p2(Bignum x, unsigned *p2) +{ + unsigned i, j, k, n; + Bignum y; + + /* If x is zero then the following won't work. And if x is odd then + * there's nothing very useful to do. + */ + if (!x[0] || (x[1] & 1)) { + *p2 = 0; + return copybn(x); + } + + /* Find the power of two. */ + for (i = 0; !x[i + 1]; i++); + for (j = 0; !((x[i + 1] >> j) & 1); j++); + *p2 = i*BIGNUM_INT_BITS + j; + + /* Work out how big the copy should be. */ + n = x[0] - i - 1; + if (x[x[0]] >> j) n++; + + /* Copy and shift down. */ + y = newbn(n); + for (k = 1; k <= n; k++) { + y[k] = x[k + i] >> j; + if (j && k < x[0]) y[k] |= x[k + i + 1] << (BIGNUM_INT_BITS - j); + } + + /* Done. */ + return y; +} + +/* + * Kronecker symbol (a|n). The result is always in { -1, 0, +1 }, and is + * zero if and only if a and n have a nontrivial common factor. Most + * usefully, if n is prime, this is the Legendre symbol, taking the value +1 + * if a is a quadratic residue mod n, and -1 otherwise; i.e., (a|p) == + * a^{(p-1)/2} (mod p). + */ +int kronecker(Bignum a, Bignum n) +{ + unsigned s, nn; + int r = +1; + Bignum t; + + /* Special case for n = 0. This is the same convention PARI uses, + * except that we can't represent negative numbers. + */ + if (bignum_cmp(n, Zero) == 0) { + if (bignum_cmp(a, One) == 0) return +1; + else return 0; + } + + /* Write n = 2^s t, with t odd. If s > 0 and a is even, then the answer + * is zero; otherwise throw in a factor of (-1)^s if a == 3 or 5 (mod 8). + * + * At this point, we have a copy of n, and must remember to free it when + * we're done. It's convenient to take a copy of a at the same time. + */ + a = copybn(a); + n = extract_p2(n, &s); + + if (s && (!a[0] || !(a[1] & 1))) { r = 0; goto done; } + else if ((s & 1) && ((a[1] & 7) == 3 || (a[1] & 7) == 5)) r = -r; + + /* If n is (now) a unit then we're done. */ + if (bignum_cmp(n, One) == 0) goto done; + + /* Reduce a modulo n before we go any further. */ + if (bignum_cmp(a, n) >= 0) { t = bigmod(a, n); freebn(a); a = t; } + + /* Main loop. */ + for (;;) { + if (bignum_cmp(a, Zero) == 0) { r = 0; goto done; } + + /* Strip out and handle powers of two from a. */ + t = extract_p2(a, &s); freebn(a); a = t; + nn = n[1] & 7; + if ((s & 1) && (nn == 3 || nn == 5)) r = -r; + if (bignum_cmp(a, One) == 0) break; + + /* Swap, applying quadratic reciprocity. */ + if ((nn & 3) == 3 && (a[1] & 3) == 3) r = -r; + t = bigmod(n, a); freebn(n); n = a; a = t; + } + + /* Tidy up: we're done. */ +done: + freebn(a); freebn(n); + return r; +} + +/* + * Modular square root. We must have p prime: extracting square roots modulo + * composites is equivalent to factoring (but we don't check: you'll just get + * the wrong answer). Returns NULL if x is not a quadratic residue mod p. + */ +Bignum modsqrt(Bignum x, Bignum p) +{ + Bignum xinv, b, c, r, t, z, X, mone; + unsigned i, j, s; + + /* If x is not a quadratic residue then we will not go to space today. */ + if (kronecker(x, p) != +1) return NULL; + + /* We need a quadratic nonresidue from somewhere. Exactly half of all + * units mod p are quadratic residues, but no efficient deterministic + * algorithm for finding one is known. So pick at random: we don't + * expect this to take long. + */ + z = newbn(p[0]); + do { + for (i = 1; i <= p[0]; i++) z[i] = rand(); + z[0] = p[0]; bn_restore_invariant(z); + } while (kronecker(z, p) != -1); + b = bigmod(z, p); freebn(z); + + /* We need to compute a few things before we really get started. */ + xinv = modinv(x, p); /* x^{-1} mod p */ + mone = bigsub(p, One); /* p - 1 == -1 (mod p) */ + t = extract_p2(mone, &s); /* 2^s t = p - 1 */ + c = modpow(b, t, p); /* b^t (mod p) */ + z = bigadd(t, One); freebn(t); t = z; /* (t + 1) */ + shift_right(t + 1, t[0], 1); if (!t[t[0]]) t[0]--; + r = modpow(x, t, p); /* x^{(t+1)/2} (mod p) */ + freebn(b); freebn(mone); freebn(t); + + /* OK, so how does this work anyway? + * + * We know that x^t is somewhere in the order-2^s subgroup of GF(p)^*; + * and g = c^{-1} is a generator for this subgroup (since we know that + * g^{2^{s-1}} = b^{(p-1)/2} = (b|p) = -1); so x^t = g^m for some m. In + * fact, we know that m is even because x is a square. Suppose we can + * determine m; then we know that x^t/g^m = 1, so x^{t+1}/c^m = x -- but + * both t + 1 and m are even, so x^{(t+1)/2}/g^{m/2} is a square root of + * x. + * + * Conveniently, finding the discrete log of an element X in a group of + * order 2^s is easy. Write X = g^m = g^{m_0+2k'}; then X^{2^{s-1}} = + * g^{m_0 2^{s-1}} c^{m' 2^s} = g^{m_0 2^{s-1}} is either -1 or +1, + * telling us that m_0 is 1 or 0 respectively. Then X/g^{m_0} = + * (g^2)^{m'} has order 2^{s-1} so we can continue inductively. What we + * end up with at the end is X/g^m. + * + * There are a few wrinkles. As we proceed through the induction, the + * generator for the subgroup will be c^{-2}, since we know that m is + * even. While we want the discrete log of X = x^t, we're actually going + * to keep track of r, which will eventually be x^{(t+1)/2}/g^{m/2} = + * x^{(t+1)/2} c^m, recovering X/g^m = r^2/x as we go. We don't actually + * form the discrete log explicitly, because the final result will + * actually be the square root we want. + */ + for (i = 1; i < s; i++) { + + /* Determine X. We could optimize this, only recomputing it when + * it's been invalidated, but that's fiddlier and this isn't + * performance critical. + */ + z = modmul(r, r, p); + X = modmul(z, xinv, p); + freebn(z); + + /* Determine X^{2^{s-1-i}}. */ + for (j = i + 1; j < s; j++) + z = modmul(X, X, p), freebn(X), X = z; + + /* Maybe accumulate a factor of c. */ + if (bignum_cmp(X, One) != 0) + z = modmul(r, c, p), freebn(r), r = z; + + /* Move on to the next smaller subgroup. */ + z = modmul(c, c, p), freebn(c), c = z; + freebn(X); + } + + /* Of course, there are two square roots of x. For predictability's sake + * we'll always return the one in [1..(p - 1)/2]. The other is, of + * course, p - r. + */ + z = bigsub(p, r); + if (bignum_cmp(r, z) < 0) + freebn(z); + else { + freebn(r); + r = z; + } + + /* We're done. */ + freebn(xinv); freebn(c); + return r; +} + +/* * Render a bignum into decimal. Return a malloced string holding * the decimal representation. */ @@ -1902,6 +2080,42 @@ int main(int argc, char **argv) freebn(modulus); freebn(expected); freebn(answer); + } else if (!strcmp(buf, "modsqrt")) { + Bignum x, p, expected, answer; + + if (ptrnum != 3) { + printf("%d: modsqrt with %d parameters, expected 3\n", line, ptrnum); + exit(1); + } + + x = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]); + p = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]); + expected = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]); + answer = modsqrt(x, p); + if (!answer) + answer = copybn(Zero); + + if (bignum_cmp(expected, answer) == 0) { + passes++; + } else { + char *xs = bignum_decimal(x); + char *ps = bignum_decimal(p); + char *qs = bignum_decimal(answer); + char *ws = bignum_decimal(expected); + + printf("%d: fail: sqrt(%s) mod %s gave %s expected %s\n", + line, xs, ps, qs, ws); + fails++; + + sfree(xs); + sfree(ps); + sfree(qs); + sfree(ws); + } + freebn(p); + freebn(x); + freebn(expected); + freebn(answer); } else { printf("%d: unrecognised test keyword: '%s'\n", line, buf); exit(1);