/*
* 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);
}
/*
* 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;
* Compute c = a * b.
* Input is in the first len words of a and b.
* Result is returned in the first 2*len words of c.
+ *
+ * 'scratch' must point to an array of BignumInt of size at least
+ * mul_compute_scratch(len). (This covers the needs of internal_mul
+ * and all its recursive calls to itself.)
*/
#define KARATSUBA_THRESHOLD 50
-static void internal_mul(BignumInt *a, BignumInt *b,
- BignumInt *c, int len)
+static int mul_compute_scratch(int len)
+{
+ int ret = 0;
+ while (len > KARATSUBA_THRESHOLD) {
+ int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
+ int midlen = botlen + 1;
+ ret += 4*midlen;
+ len = midlen;
+ }
+ return ret;
+}
+static void internal_mul(const BignumInt *a, const BignumInt *b,
+ BignumInt *c, int len, BignumInt *scratch)
{
- int i, j;
- BignumDblInt t;
-
if (len > KARATSUBA_THRESHOLD) {
+ int i;
/*
* Karatsuba divide-and-conquer algorithm. Cut each input in
int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
int midlen = botlen + 1;
- BignumInt *scratch;
BignumDblInt carry;
/*
* place.
*/
+#ifdef KARA_DEBUG
+ printf("a1,a0 = 0x");
+ for (i = 0; i < len; i++) {
+ if (i == toplen) printf(", 0x");
+ printf("%0*x", BIGNUM_INT_BITS/4, a[len - 1 - i]);
+ }
+ printf("\n");
+ printf("b1,b0 = 0x");
+ for (i = 0; i < len; i++) {
+ if (i == toplen) printf(", 0x");
+ printf("%0*x", BIGNUM_INT_BITS/4, b[len - 1 - i]);
+ }
+ printf("\n");
+#endif
+
/* a_1 b_1 */
- internal_mul(a, b, c, toplen);
+ internal_mul(a + botlen, b + botlen, c + 2*botlen, toplen, scratch);
+#ifdef KARA_DEBUG
+ printf("a1b1 = 0x");
+ for (i = 0; i < 2*toplen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, c[2*len - 1 - i]);
+ }
+ printf("\n");
+#endif
/* a_0 b_0 */
- internal_mul(a + toplen, b + toplen, c + 2*toplen, botlen);
+ internal_mul(a, b, c, botlen, scratch);
+#ifdef KARA_DEBUG
+ printf("a0b0 = 0x");
+ for (i = 0; i < 2*botlen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, c[2*botlen - 1 - i]);
+ }
+ printf("\n");
+#endif
- /*
- * We must allocate scratch space for the central coefficient,
- * and also for the two input values that we multiply when
- * computing it. Since either or both may carry into the
- * (botlen+1)th word, we must use a slightly longer length
- * 'midlen'.
+ /* Zero padding. botlen exceeds toplen by at most 1, and we'll set
+ * the extra carry explicitly below, so we only need to zero at most
+ * one of the top words here.
*/
- scratch = snewn(4 * midlen, BignumInt);
+ scratch[midlen - 2] = scratch[2*midlen - 2] = 0;
- /* Zero padding. midlen exceeds toplen by at most 2, so just
- * zero the first two words of each input and the rest will be
- * copied over. */
- scratch[0] = scratch[1] = scratch[midlen] = scratch[midlen+1] = 0;
-
- for (j = 0; j < toplen; j++) {
- scratch[midlen - toplen + j] = a[j]; /* a_1 */
- scratch[2*midlen - toplen + j] = b[j]; /* b_1 */
+ for (i = 0; i < toplen; i++) {
+ scratch[i] = a[i + botlen]; /* a_1 */
+ scratch[midlen + i] = b[i + botlen]; /* b_1 */
}
/* compute a_1 + a_0 */
- scratch[0] = internal_add(scratch+1, a+toplen, scratch+1, botlen);
+ scratch[midlen - 1] = internal_add(scratch, a, scratch, botlen);
+#ifdef KARA_DEBUG
+ printf("a1plusa0 = 0x");
+ for (i = 0; i < midlen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, scratch[midlen - 1 - i]);
+ }
+ printf("\n");
+#endif
/* compute b_1 + b_0 */
- scratch[midlen] = internal_add(scratch+midlen+1, b+toplen,
- scratch+midlen+1, botlen);
+ scratch[2*midlen - 1] = internal_add(scratch+midlen, b,
+ scratch+midlen, botlen);
+#ifdef KARA_DEBUG
+ printf("b1plusb0 = 0x");
+ for (i = 0; i < midlen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]);
+ }
+ printf("\n");
+#endif
/*
* Now we can do the third multiplication.
*/
- internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen);
+ internal_mul(scratch, scratch + midlen, scratch + 2*midlen, midlen,
+ scratch + 4*midlen);
+#ifdef KARA_DEBUG
+ printf("a1plusa0timesb1plusb0 = 0x");
+ for (i = 0; i < 2*midlen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]);
+ }
+ printf("\n");
+#endif
/*
* Now we can reuse the first half of 'scratch' to compute the
* sum of the outer two coefficients, to subtract from that
* product to obtain the middle one.
*/
- scratch[0] = scratch[1] = scratch[2] = scratch[3] = 0;
- for (j = 0; j < 2*toplen; j++)
- scratch[2*midlen - 2*toplen + j] = c[j];
- scratch[1] = internal_add(scratch+2, c + 2*toplen,
- scratch+2, 2*botlen);
+ scratch[2*botlen - 2] = scratch[2*botlen - 1] = 0;
+ for (i = 0; i < 2*toplen; i++)
+ scratch[i] = c[2*botlen + i];
+ scratch[2*botlen] = internal_add(scratch, c, scratch, 2*botlen);
+ scratch[2*botlen + 1] = 0;
+#ifdef KARA_DEBUG
+ printf("a1b1plusa0b0 = 0x");
+ for (i = 0; i < 2*midlen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, scratch[2*midlen - 1 - i]);
+ }
+ printf("\n");
+#endif
- internal_sub(scratch + 2*midlen, scratch,
- scratch + 2*midlen, 2*midlen);
+ internal_sub(scratch + 2*midlen, scratch, scratch, 2*midlen);
+#ifdef KARA_DEBUG
+ printf("a1b0plusa0b1 = 0x");
+ for (i = 0; i < 2*midlen; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, scratch[4*midlen - 1 - i]);
+ }
+ printf("\n");
+#endif
/*
* And now all we need to do is to add that middle coefficient
* further up the output, but we can be sure it won't
* propagate right the way off the top.
*/
- carry = internal_add(c + 2*len - botlen - 2*midlen,
- scratch + 2*midlen,
- c + 2*len - botlen - 2*midlen, 2*midlen);
- j = 2*len - botlen - 2*midlen - 1;
+ carry = internal_add(c + botlen, scratch, c + botlen, 2*midlen);
+ i = botlen + 2*midlen;
while (carry) {
- assert(j >= 0);
- carry += c[j];
- c[j] = (BignumInt)carry;
+ assert(i <= 2*len);
+ carry += c[i];
+ c[i] = (BignumInt)carry;
carry >>= BIGNUM_INT_BITS;
+ i++;
+ }
+#ifdef KARA_DEBUG
+ printf("ab = 0x");
+ for (i = 0; i < 2*len; i++) {
+ printf("%0*x", BIGNUM_INT_BITS/4, c[2*len - i]);
+ }
+ printf("\n");
+#endif
+
+ } else {
+ int i;
+ BignumInt carry;
+ BignumDblInt t;
+ const BignumInt *ap, *alim = a + len, *bp, *blim = b + len;
+ BignumInt *cp, *cps;
+
+ /*
+ * Multiply in the ordinary O(N^2) way.
+ */
+
+ for (i = 0; i < 2 * len; i++)
+ c[i] = 0;
+
+ for (cps = c, ap = a; ap < alim; ap++, cps++) {
+ carry = 0;
+ for (cp = cps, bp = b, i = blim - bp; i--; bp++, cp++) {
+ t = (MUL_WORD(*ap, *bp) + carry) + *cp;
+ *cp = (BignumInt) t;
+ carry = (BignumInt)(t >> BIGNUM_INT_BITS);
+ }
+ *cp = carry;
}
+ }
+}
+
+/*
+ * Variant form of internal_mul used for the initial step of
+ * Montgomery reduction. Only bothers outputting 'len' words
+ * (everything above that is thrown away).
+ */
+static void internal_mul_low(const BignumInt *a, const BignumInt *b,
+ BignumInt *c, int len, BignumInt *scratch)
+{
+ if (len > KARATSUBA_THRESHOLD) {
+ int i;
+
+ /*
+ * Karatsuba-aware version of internal_mul_low. As before, we
+ * express each input value as a shifted combination of two
+ * halves:
+ *
+ * a = a_1 D + a_0
+ * b = b_1 D + b_0
+ *
+ * Then the full product is, as before,
+ *
+ * ab = a_1 b_1 D^2 + (a_1 b_0 + a_0 b_1) D + a_0 b_0
+ *
+ * Provided we choose D on the large side (so that a_0 and b_0
+ * are _at least_ as long as a_1 and b_1), we don't need the
+ * topmost term at all, and we only need half of the middle
+ * term. So there's no point in doing the proper Karatsuba
+ * optimisation which computes the middle term using the top
+ * one, because we'd take as long computing the top one as
+ * just computing the middle one directly.
+ *
+ * So instead, we do a much more obvious thing: we call the
+ * fully optimised internal_mul to compute a_0 b_0, and we
+ * recursively call ourself to compute the _bottom halves_ of
+ * a_1 b_0 and a_0 b_1, each of which we add into the result
+ * in the obvious way.
+ *
+ * In other words, there's no actual Karatsuba _optimisation_
+ * in this function; the only benefit in doing it this way is
+ * that we call internal_mul proper for a large part of the
+ * work, and _that_ can optimise its operation.
+ */
+
+ int toplen = len/2, botlen = len - toplen; /* botlen is the bigger */
+
+ /*
+ * Scratch space for the various bits and pieces we're going
+ * to be adding together: we need botlen*2 words for a_0 b_0
+ * (though we may end up throwing away its topmost word), and
+ * toplen words for each of a_1 b_0 and a_0 b_1. That adds up
+ * to exactly 2*len.
+ */
+
+ /* a_0 b_0 */
+ internal_mul(a, b, scratch + 2*toplen, botlen, scratch + 2*len);
+
+ /* a_1 b_0 */
+ internal_mul_low(a + botlen, b, scratch + toplen, toplen,
+ scratch + 2*len);
+
+ /* a_0 b_1 */
+ internal_mul_low(a, b + botlen, scratch, toplen, scratch + 2*len);
- /* Free scratch. */
- for (j = 0; j < 4 * midlen; j++)
- scratch[j] = 0;
- sfree(scratch);
+ /* Copy the bottom half of the big coefficient into place */
+ for (i = 0; i < botlen; i++)
+ c[i] = scratch[2*toplen + i];
+
+ /* Add the two small coefficients, throwing away the returned carry */
+ internal_add(scratch, scratch + toplen, scratch, toplen);
+
+ /* And add that to the large coefficient, leaving the result in c. */
+ internal_add(scratch, scratch + 2*toplen + botlen,
+ c + botlen, toplen);
} else {
+ int i;
+ BignumInt carry;
+ BignumDblInt t;
+ const BignumInt *ap, *alim = a + len, *bp;
+ BignumInt *cp, *cps, *clim = c + len;
/*
* Multiply in the ordinary O(N^2) way.
*/
- for (j = 0; j < 2 * len; j++)
- c[j] = 0;
+ for (i = 0; i < len; i++)
+ c[i] = 0;
- for (i = len - 1; i >= 0; i--) {
- t = 0;
- for (j = len - 1; j >= 0; j--) {
- t += MUL_WORD(a[i], (BignumDblInt) b[j]);
- t += (BignumDblInt) c[i + j + 1];
- c[i + j + 1] = (BignumInt) t;
- t = t >> BIGNUM_INT_BITS;
+ for (cps = c, ap = a; ap < alim; ap++, cps++) {
+ carry = 0;
+ 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);
}
- c[i] = (BignumInt) t;
}
}
}
+/*
+ * Montgomery reduction. Expects x to be a little-endian array of 2*len
+ * BignumInts whose value satisfies 0 <= x < rn (where r = 2^(len *
+ * BIGNUM_INT_BITS) is the Montgomery base). Returns in the same array
+ * a value x' which is congruent to xr^{-1} mod n, and satisfies 0 <=
+ * x' < n.
+ *
+ * 'n' and 'mninv' should be little-endian arrays of 'len' BignumInts
+ * each, containing respectively n and the multiplicative inverse of
+ * -n mod r.
+ *
+ * 'tmp' is an array of BignumInt used as scratch space, of length at
+ * least 3*len + mul_compute_scratch(len).
+ */
+static void monty_reduce(BignumInt *x, const BignumInt *n,
+ const BignumInt *mninv, BignumInt *tmp, int len)
+{
+ int i;
+ BignumInt carry;
+
+ /*
+ * Multiply x by (-n)^{-1} mod r. This gives us a value m such
+ * that mn is congruent to -x mod r. Hence, mn+x is an exact
+ * multiple of r, and is also (obviously) congruent to x mod n.
+ */
+ internal_mul_low(x, mninv, tmp, len, tmp + 3*len);
+
+ /*
+ * Compute t = (mn+x)/r in ordinary, non-modular, integer
+ * arithmetic. By construction this is exact, and is congruent mod
+ * n to x * r^{-1}, i.e. the answer we want.
+ *
+ * The following multiply leaves that answer in the _most_
+ * significant half of the 'x' array, so then we must shift it
+ * down.
+ */
+ internal_mul(tmp, n, tmp+len, len, tmp + 3*len);
+ carry = internal_add(x, tmp+len, x, 2*len);
+ for (i = 0; i < len; i++)
+ x[i] = x[len + i], x[len + i] = 0;
+
+ /*
+ * Reduce t mod n. This doesn't require a full-on division by n,
+ * but merely a test and single optional subtraction, since we can
+ * show that 0 <= t < 2n.
+ *
+ * Proof:
+ * + we computed m mod r, so 0 <= m < r.
+ * + so 0 <= mn < rn, obviously
+ * + hence we only need 0 <= x < rn to guarantee that 0 <= mn+x < 2rn
+ * + yielding 0 <= (mn+x)/r < 2n as required.
+ */
+ if (!carry) {
+ for (i = len; i-- > 0; )
+ if (x[i] != n[i])
+ break;
+ }
+ if (carry || i < 0 || x[i] > n[i])
+ internal_sub(x, n, x, len);
+}
+
static void internal_add_shifted(BignumInt *number,
unsigned n, int shift)
{
* 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.
+ * Compute (base ^ exp) % mod, the pedestrian way.
*/
-Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)
+Bignum modpow_simple(Bignum base_in, Bignum exp, Bignum mod)
{
- BignumInt *a, *b, *n, *m;
+ BignumInt *a, *b, *n, *m, *scratch;
int mshift;
- int mlen, i, j;
+ int mlen, scratchlen, i, j;
Bignum base, result;
/*
base = bigmod(base_in, mod);
/* Allocate m of size mlen, copy mod to m */
- /* We use big endian internally */
mlen = mod[0];
m = snewn(mlen, BignumInt);
for (j = 0; j < mlen; j++)
- m[j] = mod[mod[0] - j];
+ m[j] = mod[j + 1];
/* Shift m left to make msb bit set */
for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
- if ((m[0] << mshift) & BIGNUM_TOP_BIT)
+ if ((m[mlen - 1] << mshift) & BIGNUM_TOP_BIT)
break;
- if (mshift) {
- for (i = 0; i < mlen - 1; i++)
- m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
- m[mlen - 1] = m[mlen - 1] << mshift;
- }
+ if (mshift)
+ shift_left(m, mlen, mshift);
/* Allocate n of size mlen, copy base to n */
n = snewn(mlen, BignumInt);
- i = mlen - base[0];
- for (j = 0; j < i; j++)
- n[j] = 0;
- for (j = 0; j < (int)base[0]; j++)
- n[i + j] = base[base[0] - j];
+ for (i = 0; i < (int)base[0]; i++)
+ n[i] = base[i + 1];
+ for (; i < mlen; i++)
+ n[i] = 0;
/* Allocate a and b of size 2*mlen. Set a = 1 */
a = snewn(2 * mlen, BignumInt);
b = snewn(2 * mlen, BignumInt);
- for (i = 0; i < 2 * mlen; i++)
+ a[0] = 1;
+ for (i = 1; i < 2 * mlen; i++)
a[i] = 0;
- a[2 * mlen - 1] = 1;
+
+ /* Scratch space for multiplies */
+ scratchlen = mul_compute_scratch(mlen);
+ scratch = snewn(scratchlen, BignumInt);
/* Skip leading zero bits of exp. */
i = 0;
/* Main computation */
while (i < (int)exp[0]) {
while (j >= 0) {
- internal_mul(a + mlen, a + mlen, b, mlen);
+ internal_mul(a, a, b, mlen, scratch);
internal_mod(b, mlen * 2, m, mlen, NULL, 0);
if ((exp[exp[0] - i] & (1 << j)) != 0) {
- internal_mul(b + mlen, n, a, mlen);
+ internal_mul(b, n, a, mlen, scratch);
internal_mod(a, mlen * 2, m, mlen, NULL, 0);
} else {
BignumInt *t;
/* 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]--;
for (i = 0; i < 2 * mlen; i++)
a[i] = 0;
sfree(a);
+ for (i = 0; i < scratchlen; i++)
+ scratch[i] = 0;
+ sfree(scratch);
for (i = 0; i < 2 * mlen; i++)
b[i] = 0;
sfree(b);
}
/*
+ * Compute (base ^ exp) % mod. Uses the Montgomery multiplication
+ * technique where possible, falling back to modpow_simple otherwise.
+ */
+Bignum modpow(Bignum base_in, Bignum exp, Bignum mod)
+{
+ BignumInt *a, *b, *x, *n, *mninv, *scratch;
+ int len, scratchlen, i, j;
+ Bignum base, base2, r, rn, inv, result;
+
+ /*
+ * The most significant word of mod needs to be non-zero. It
+ * should already be, but let's make sure.
+ */
+ assert(mod[mod[0]] != 0);
+
+ /*
+ * mod had better be odd, or we can't do Montgomery multiplication
+ * using a power of two at all.
+ */
+ if (!(mod[1] & 1))
+ return modpow_simple(base_in, exp, mod);
+
+ /*
+ * Make sure the base is smaller than the modulus, by reducing
+ * it modulo the modulus if not.
+ */
+ base = bigmod(base_in, mod);
+
+ /*
+ * Compute the inverse of n mod r, for monty_reduce. (In fact we
+ * want the inverse of _minus_ n mod r, but we'll sort that out
+ * below.)
+ */
+ len = mod[0];
+ r = bn_power_2(BIGNUM_INT_BITS * len);
+ inv = modinv(mod, r);
+
+ /*
+ * Multiply the base by r mod n, to get it into Montgomery
+ * representation.
+ */
+ base2 = modmul(base, r, mod);
+ freebn(base);
+ base = base2;
+
+ rn = bigmod(r, mod); /* r mod n, i.e. Montgomerified 1 */
+
+ freebn(r); /* won't need this any more */
+
+ /*
+ * Set up internal arrays of the right lengths containing the base,
+ * the modulus, and the modulus's inverse.
+ */
+ n = snewn(len, BignumInt);
+ for (j = 0; j < len; j++)
+ n[j] = mod[j + 1];
+
+ mninv = snewn(len, BignumInt);
+ for (j = 0; j < len; j++)
+ mninv[j] = (j < (int)inv[0] ? inv[j + 1] : 0);
+ freebn(inv); /* we don't need this copy of it any more */
+ /* Now negate mninv mod r, so it's the inverse of -n rather than +n. */
+ x = snewn(len, BignumInt);
+ for (j = 0; j < len; j++)
+ x[j] = 0;
+ internal_sub(x, mninv, mninv, len);
+
+ /* x = snewn(len, BignumInt); */ /* already done above */
+ for (j = 0; j < len; j++)
+ x[j] = (j < (int)base[0] ? base[j + 1] : 0);
+ freebn(base); /* we don't need this copy of it any more */
+
+ a = snewn(2*len, BignumInt);
+ b = snewn(2*len, BignumInt);
+ for (j = 0; j < len; j++)
+ a[j] = (j < (int)rn[0] ? rn[j + 1] : 0);
+ freebn(rn);
+
+ /* Scratch space for multiplies */
+ scratchlen = 3*len + mul_compute_scratch(len);
+ scratch = snewn(scratchlen, BignumInt);
+
+ /* Skip leading zero bits of exp. */
+ i = 0;
+ j = BIGNUM_INT_BITS-1;
+ while (i < (int)exp[0] && (exp[exp[0] - i] & (1 << j)) == 0) {
+ j--;
+ if (j < 0) {
+ i++;
+ j = BIGNUM_INT_BITS-1;
+ }
+ }
+
+ /* Main computation */
+ while (i < (int)exp[0]) {
+ while (j >= 0) {
+ internal_mul(a, a, b, len, scratch);
+ monty_reduce(b, n, mninv, scratch, len);
+ if ((exp[exp[0] - i] & (1 << j)) != 0) {
+ internal_mul(b, x, a, len, scratch);
+ monty_reduce(a, n, mninv, scratch, len);
+ } else {
+ BignumInt *t;
+ t = a;
+ a = b;
+ b = t;
+ }
+ j--;
+ }
+ i++;
+ j = BIGNUM_INT_BITS-1;
+ }
+
+ /*
+ * Final monty_reduce to get back from the adjusted Montgomery
+ * representation.
+ */
+ monty_reduce(a, n, mninv, scratch, len);
+
+ /* Copy result to buffer */
+ result = newbn(mod[0]);
+ for (i = 0; i < len; i++)
+ result[i + 1] = a[i];
+ while (result[0] > 1 && result[result[0]] == 0)
+ result[0]--;
+
+ /* Free temporary arrays */
+ for (i = 0; i < scratchlen; i++)
+ scratch[i] = 0;
+ sfree(scratch);
+ for (i = 0; i < 2 * len; i++)
+ a[i] = 0;
+ sfree(a);
+ for (i = 0; i < 2 * len; i++)
+ b[i] = 0;
+ sfree(b);
+ for (i = 0; i < len; i++)
+ mninv[i] = 0;
+ sfree(mninv);
+ for (i = 0; i < len; i++)
+ n[i] = 0;
+ sfree(n);
+ for (i = 0; i < len; i++)
+ x[i] = 0;
+ sfree(x);
+
+ return result;
+}
+
+/*
* Compute (p * q) % mod.
* The most significant word of mod MUST be non-zero.
* We assume that the result array is the same size as the mod array.
*/
Bignum modmul(Bignum p, Bignum q, Bignum mod)
{
- BignumInt *a, *n, *m, *o;
- int mshift;
+ BignumInt *a, *n, *m, *o, *scratch;
+ int mshift, scratchlen;
int pqlen, mlen, rlen, i, j;
Bignum result;
/* Allocate m of size mlen, copy mod to m */
- /* We use big endian internally */
mlen = mod[0];
m = snewn(mlen, BignumInt);
for (j = 0; j < mlen; j++)
- m[j] = mod[mod[0] - j];
+ m[j] = mod[j + 1];
/* Shift m left to make msb bit set */
for (mshift = 0; mshift < BIGNUM_INT_BITS-1; mshift++)
- if ((m[0] << mshift) & BIGNUM_TOP_BIT)
+ if ((m[mlen - 1] << mshift) & BIGNUM_TOP_BIT)
break;
- if (mshift) {
- for (i = 0; i < mlen - 1; i++)
- m[i] = (m[i] << mshift) | (m[i + 1] >> (BIGNUM_INT_BITS - mshift));
- m[mlen - 1] = m[mlen - 1] << mshift;
- }
+ if (mshift)
+ shift_left(m, mlen, mshift);
pqlen = (p[0] > q[0] ? p[0] : q[0]);
+ /* Make sure that we're allowing enough space. The shifting below will
+ * underflow the vectors we allocate if `pqlen' is too small.
+ */
+ if (2*pqlen <= mlen)
+ pqlen = mlen/2 + 1;
+
/* Allocate n of size pqlen, copy p to n */
n = snewn(pqlen, BignumInt);
- i = pqlen - p[0];
- for (j = 0; j < i; j++)
- n[j] = 0;
- for (j = 0; j < (int)p[0]; j++)
- n[i + j] = p[p[0] - j];
+ for (i = 0; i < (int)p[0]; i++)
+ n[i] = p[i + 1];
+ for (; i < pqlen; i++)
+ n[i] = 0;
/* Allocate o of size pqlen, copy q to o */
o = snewn(pqlen, BignumInt);
- i = pqlen - q[0];
- for (j = 0; j < i; j++)
- o[j] = 0;
- for (j = 0; j < (int)q[0]; j++)
- o[i + j] = q[q[0] - j];
+ for (i = 0; i < (int)q[0]; i++)
+ o[i] = q[i + 1];
+ for (; i < pqlen; i++)
+ o[i] = 0;
/* Allocate a of size 2*pqlen for result */
a = snewn(2 * pqlen, BignumInt);
+ /* Scratch space for multiplies */
+ scratchlen = mul_compute_scratch(pqlen);
+ scratch = snewn(scratchlen, BignumInt);
+
/* Main computation */
- internal_mul(n, o, a, pqlen);
+ internal_mul(n, o, a, pqlen, scratch);
internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
/* Fixup result in case the modulus was shifted */
if (mshift) {
- for (i = 2 * pqlen - mlen - 1; i < 2 * pqlen - 1; i++)
- a[i] = (a[i] << mshift) | (a[i + 1] >> (BIGNUM_INT_BITS - mshift));
- a[2 * pqlen - 1] = a[2 * pqlen - 1] << mshift;
- internal_mod(a, pqlen * 2, m, mlen, NULL, 0);
- for (i = 2 * pqlen - 1; i >= 2 * pqlen - mlen; i--)
- a[i] = (a[i] >> mshift) | (a[i - 1] << (BIGNUM_INT_BITS - mshift));
+ shift_left(a, mlen + 1, mshift);
+ internal_mod(a, mlen + 1, m, mlen, NULL, 0);
+ shift_right(a, mlen, mshift);
}
/* Copy result to buffer */
rlen = (mlen < pqlen * 2 ? mlen : pqlen * 2);
result = newbn(rlen);
for (i = 0; i < rlen; i++)
- result[result[0] - i] = a[i + 2 * pqlen - rlen];
+ result[i + 1] = a[i];
while (result[0] > 1 && result[result[0]] == 0)
result[0]--;
/* Free temporary arrays */
+ for (i = 0; i < scratchlen; i++)
+ scratch[i] = 0;
+ sfree(scratch);
for (i = 0; i < 2 * pqlen; i++)
a[i] = 0;
sfree(a);
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 */
int alen = a[0], blen = b[0];
int mlen = (alen > blen ? alen : blen);
int rlen, i, maxspot;
+ int wslen;
BignumInt *workspace;
Bignum ret;
- /* mlen space for a, mlen space for b, 2*mlen for result */
- workspace = snewn(mlen * 4, BignumInt);
+ /* mlen space for a, mlen space for b, 2*mlen for result,
+ * plus scratch space for multiplication */
+ wslen = mlen * 4 + mul_compute_scratch(mlen);
+ workspace = snewn(wslen, BignumInt);
for (i = 0; i < mlen; i++) {
- workspace[0 * mlen + i] = (mlen - i <= (int)a[0] ? a[mlen - i] : 0);
- workspace[1 * mlen + i] = (mlen - i <= (int)b[0] ? b[mlen - i] : 0);
+ workspace[0 * mlen + i] = i < (int)a[0] ? a[i + 1] : 0;
+ workspace[1 * mlen + i] = i < (int)b[0] ? b[i + 1] : 0;
}
internal_mul(workspace + 0 * mlen, workspace + 1 * mlen,
- workspace + 2 * mlen, mlen);
+ workspace + 2 * mlen, mlen, workspace + 4 * mlen);
/* now just copy the result back */
rlen = alen + blen + 1;
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;
}
ret[0] = maxspot;
+ for (i = 0; i < wslen; i++)
+ workspace[i] = 0;
sfree(workspace);
return ret;
}
}
/*
+ * 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.
sfree(workspace);
return ret;
}
+
+#ifdef TESTBN
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <ctype.h>
+
+/*
+ * gcc -Wall -g -O0 -DTESTBN -o testbn sshbn.c misc.c conf.c tree234.c unix/uxmisc.c -I. -I unix -I charset
+ *
+ * Then feed to this program's standard input the output of
+ * testdata/bignum.py .
+ */
+
+void modalfatalbox(char *p, ...)
+{
+ va_list ap;
+ fprintf(stderr, "FATAL ERROR: ");
+ va_start(ap, p);
+ vfprintf(stderr, p, ap);
+ va_end(ap);
+ fputc('\n', stderr);
+ exit(1);
+}
+
+#define fromxdigit(c) ( (c)>'9' ? ((c)&0xDF) - 'A' + 10 : (c) - '0' )
+
+int main(int argc, char **argv)
+{
+ char *buf;
+ int line = 0;
+ int passes = 0, fails = 0;
+
+ while ((buf = fgetline(stdin)) != NULL) {
+ int maxlen = strlen(buf);
+ unsigned char *data = snewn(maxlen, unsigned char);
+ unsigned char *ptrs[5], *q;
+ int ptrnum;
+ char *bufp = buf;
+
+ line++;
+
+ q = data;
+ ptrnum = 0;
+
+ while (*bufp && !isspace((unsigned char)*bufp))
+ bufp++;
+ if (bufp)
+ *bufp++ = '\0';
+
+ while (*bufp) {
+ char *start, *end;
+ int i;
+
+ while (*bufp && !isxdigit((unsigned char)*bufp))
+ bufp++;
+ start = bufp;
+
+ if (!*bufp)
+ break;
+
+ while (*bufp && isxdigit((unsigned char)*bufp))
+ bufp++;
+ end = bufp;
+
+ if (ptrnum >= lenof(ptrs))
+ break;
+ ptrs[ptrnum++] = q;
+
+ for (i = -((end - start) & 1); i < end-start; i += 2) {
+ unsigned char val = (i < 0 ? 0 : fromxdigit(start[i]));
+ val = val * 16 + fromxdigit(start[i+1]);
+ *q++ = val;
+ }
+
+ ptrs[ptrnum] = q;
+ }
+
+ if (!strcmp(buf, "mul")) {
+ Bignum a, b, c, p;
+
+ if (ptrnum != 3) {
+ printf("%d: mul with %d parameters, expected 3\n", line, ptrnum);
+ exit(1);
+ }
+ a = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
+ b = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
+ c = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
+ p = bigmul(a, b);
+
+ if (bignum_cmp(c, p) == 0) {
+ passes++;
+ } else {
+ char *as = bignum_decimal(a);
+ char *bs = bignum_decimal(b);
+ char *cs = bignum_decimal(c);
+ char *ps = bignum_decimal(p);
+
+ printf("%d: fail: %s * %s gave %s expected %s\n",
+ line, as, bs, ps, cs);
+ fails++;
+
+ sfree(as);
+ sfree(bs);
+ sfree(cs);
+ sfree(ps);
+ }
+ freebn(a);
+ freebn(b);
+ freebn(c);
+ freebn(p);
+ } else if (!strcmp(buf, "pow")) {
+ Bignum base, expt, modulus, expected, answer;
+
+ if (ptrnum != 4) {
+ printf("%d: mul with %d parameters, expected 4\n", line, ptrnum);
+ exit(1);
+ }
+
+ base = bignum_from_bytes(ptrs[0], ptrs[1]-ptrs[0]);
+ expt = bignum_from_bytes(ptrs[1], ptrs[2]-ptrs[1]);
+ modulus = bignum_from_bytes(ptrs[2], ptrs[3]-ptrs[2]);
+ expected = bignum_from_bytes(ptrs[3], ptrs[4]-ptrs[3]);
+ answer = modpow(base, expt, modulus);
+
+ if (bignum_cmp(expected, answer) == 0) {
+ passes++;
+ } else {
+ char *as = bignum_decimal(base);
+ char *bs = bignum_decimal(expt);
+ char *cs = bignum_decimal(modulus);
+ char *ds = bignum_decimal(answer);
+ char *ps = bignum_decimal(expected);
+
+ printf("%d: fail: %s ^ %s mod %s gave %s expected %s\n",
+ line, as, bs, cs, ds, ps);
+ fails++;
+
+ sfree(as);
+ sfree(bs);
+ sfree(cs);
+ sfree(ds);
+ sfree(ps);
+ }
+ freebn(base);
+ freebn(expt);
+ freebn(modulus);
+ freebn(expected);
+ freebn(answer);
+ } else {
+ printf("%d: unrecognised test keyword: '%s'\n", line, buf);
+ exit(1);
+ }
+
+ sfree(buf);
+ sfree(data);
+ }
+
+ printf("passed %d failed %d total %d\n", passes, fails, passes+fails);
+ return fails != 0;
+}
+
+#endif
projects / u / mdw / putty / blobdiff