#include <string.h>
#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.
- * <http://msdn2.microsoft.com/en-us/library/bf1dw62z.aspx> */
-#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 };
/*
* 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,
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;
/*
* 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,
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;
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
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
#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
* 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
* 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
int i;
BignumInt carry;
BignumDblInt t;
- const BignumInt *ap, *bp;
+ const BignumInt *ap, *alim = a + len, *bp, *blim = b + len;
BignumInt *cp, *cps;
/*
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);
*/
/* 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.
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);
}
/*
- * 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.
*
* 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
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,
* + 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,
* 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,
{
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) {
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--;
}
}
+ 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.
*/
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);
/* 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;
/* 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]--;
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);
/* 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 */
/* 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;
/* 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]--;
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);
/* 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]--;
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 */
/* 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 */
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,
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;
}
/*
+ * 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.
}
/*
+ * 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.
*/
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);
projects / u / mdw / putty / blobdiff