--- /dev/null
+/*
+ * Copyright (c) 2018 Thomas Pornin <pornin@bolet.org>
+ *
+ * Permission is hereby granted, free of charge, to any person obtaining
+ * a copy of this software and associated documentation files (the
+ * "Software"), to deal in the Software without restriction, including
+ * without limitation the rights to use, copy, modify, merge, publish,
+ * distribute, sublicense, and/or sell copies of the Software, and to
+ * permit persons to whom the Software is furnished to do so, subject to
+ * the following conditions:
+ *
+ * The above copyright notice and this permission notice shall be
+ * included in all copies or substantial portions of the Software.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+ * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+ * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+ * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
+ * BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
+ * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+ * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+ * SOFTWARE.
+ */
+
+#include "inner.h"
+
+/*
+ * In this file, we handle big integers with a custom format, i.e.
+ * without the usual one-word header. Value is split into 31-bit words,
+ * each stored in a 32-bit slot (top bit is zero) in little-endian
+ * order. The length (in words) is provided explicitly. In some cases,
+ * the value can be negative (using two's complement representation). In
+ * some cases, the top word is allowed to have a 32th bit.
+ */
+
+/*
+ * Negate big integer conditionally. The value consists of 'len' words,
+ * with 31 bits in each word (the top bit of each word should be 0,
+ * except possibly for the last word). If 'ctl' is 1, the negation is
+ * computed; otherwise, if 'ctl' is 0, then the value is unchanged.
+ */
+static void
+cond_negate(uint32_t *a, size_t len, uint32_t ctl)
+{
+ size_t k;
+ uint32_t cc, xm;
+
+ cc = ctl;
+ xm = -ctl >> 1;
+ for (k = 0; k < len; k ++) {
+ uint32_t aw;
+
+ aw = a[k];
+ aw = (aw ^ xm) + cc;
+ a[k] = aw & 0x7FFFFFFF;
+ cc = aw >> 31;
+ }
+}
+
+/*
+ * Finish modular reduction. Rules on input parameters:
+ *
+ * if neg = 1, then -m <= a < 0
+ * if neg = 0, then 0 <= a < 2*m
+ *
+ * If neg = 0, then the top word of a[] may use 32 bits.
+ *
+ * Also, modulus m must be odd.
+ */
+static void
+finish_mod(uint32_t *a, size_t len, const uint32_t *m, uint32_t neg)
+{
+ size_t k;
+ uint32_t cc, xm, ym;
+
+ /*
+ * First pass: compare a (assumed nonnegative) with m.
+ * Note that if the final word uses the top extra bit, then
+ * subtracting m must yield a value less than 2^31, since we
+ * assumed that a < 2*m.
+ */
+ cc = 0;
+ for (k = 0; k < len; k ++) {
+ uint32_t aw, mw;
+
+ aw = a[k];
+ mw = m[k];
+ cc = (aw - mw - cc) >> 31;
+ }
+
+ /*
+ * At this point:
+ * if neg = 1, then we must add m (regardless of cc)
+ * if neg = 0 and cc = 0, then we must subtract m
+ * if neg = 0 and cc = 1, then we must do nothing
+ */
+ xm = -neg >> 1;
+ ym = -(neg | (1 - cc));
+ cc = neg;
+ for (k = 0; k < len; k ++) {
+ uint32_t aw, mw;
+
+ aw = a[k];
+ mw = (m[k] ^ xm) & ym;
+ aw = aw - mw - cc;
+ a[k] = aw & 0x7FFFFFFF;
+ cc = aw >> 31;
+ }
+}
+
+/*
+ * Compute:
+ * a <- (a*pa+b*pb)/(2^31)
+ * b <- (a*qa+b*qb)/(2^31)
+ * The division is assumed to be exact (i.e. the low word is dropped).
+ * If the final a is negative, then it is negated. Similarly for b.
+ * Returned value is the combination of two bits:
+ * bit 0: 1 if a had to be negated, 0 otherwise
+ * bit 1: 1 if b had to be negated, 0 otherwise
+ *
+ * Factors pa, pb, qa and qb must be at most 2^31 in absolute value.
+ * Source integers a and b must be nonnegative; top word is not allowed
+ * to contain an extra 32th bit.
+ */
+static uint32_t
+co_reduce(uint32_t *a, uint32_t *b, size_t len,
+ int64_t pa, int64_t pb, int64_t qa, int64_t qb)
+{
+ size_t k;
+ int64_t cca, ccb;
+ uint32_t nega, negb;
+
+ cca = 0;
+ ccb = 0;
+ for (k = 0; k < len; k ++) {
+ uint32_t wa, wb;
+ uint64_t za, zb;
+ uint64_t tta, ttb;
+
+ /*
+ * Since:
+ * |pa| <= 2^31
+ * |pb| <= 2^31
+ * 0 <= wa <= 2^31 - 1
+ * 0 <= wb <= 2^31 - 1
+ * |cca| <= 2^32 - 1
+ * Then:
+ * |za| <= (2^31-1)*(2^32) + (2^32-1) = 2^63 - 1
+ *
+ * Thus, the new value of cca is such that |cca| <= 2^32 - 1.
+ * The same applies to ccb.
+ */
+ wa = a[k];
+ wb = b[k];
+ za = wa * (uint64_t)pa + wb * (uint64_t)pb + (uint64_t)cca;
+ zb = wa * (uint64_t)qa + wb * (uint64_t)qb + (uint64_t)ccb;
+ if (k > 0) {
+ a[k - 1] = za & 0x7FFFFFFF;
+ b[k - 1] = zb & 0x7FFFFFFF;
+ }
+
+ /*
+ * For the new values of cca and ccb, we need a signed
+ * right-shift; since, in C, right-shifting a signed
+ * negative value is implementation-defined, we use a
+ * custom portable sign extension expression.
+ */
+#define M ((uint64_t)1 << 32)
+ tta = za >> 31;
+ ttb = zb >> 31;
+ tta = (tta ^ M) - M;
+ ttb = (ttb ^ M) - M;
+ cca = *(int64_t *)&tta;
+ ccb = *(int64_t *)&ttb;
+#undef M
+ }
+ a[len - 1] = (uint32_t)cca;
+ b[len - 1] = (uint32_t)ccb;
+
+ nega = (uint32_t)((uint64_t)cca >> 63);
+ negb = (uint32_t)((uint64_t)ccb >> 63);
+ cond_negate(a, len, nega);
+ cond_negate(b, len, negb);
+ return nega | (negb << 1);
+}
+
+/*
+ * Compute:
+ * a <- (a*pa+b*pb)/(2^31) mod m
+ * b <- (a*qa+b*qb)/(2^31) mod m
+ *
+ * m0i is equal to -1/m[0] mod 2^31.
+ *
+ * Factors pa, pb, qa and qb must be at most 2^31 in absolute value.
+ * Source integers a and b must be nonnegative; top word is not allowed
+ * to contain an extra 32th bit.
+ */
+static void
+co_reduce_mod(uint32_t *a, uint32_t *b, size_t len,
+ int64_t pa, int64_t pb, int64_t qa, int64_t qb,
+ const uint32_t *m, uint32_t m0i)
+{
+ size_t k;
+ int64_t cca, ccb;
+ uint32_t fa, fb;
+
+ cca = 0;
+ ccb = 0;
+ fa = ((a[0] * (uint32_t)pa + b[0] * (uint32_t)pb) * m0i) & 0x7FFFFFFF;
+ fb = ((a[0] * (uint32_t)qa + b[0] * (uint32_t)qb) * m0i) & 0x7FFFFFFF;
+ for (k = 0; k < len; k ++) {
+ uint32_t wa, wb;
+ uint64_t za, zb;
+ uint64_t tta, ttb;
+
+ /*
+ * In this loop, carries 'cca' and 'ccb' always fit on
+ * 33 bits (in absolute value).
+ */
+ wa = a[k];
+ wb = b[k];
+ za = wa * (uint64_t)pa + wb * (uint64_t)pb
+ + m[k] * (uint64_t)fa + (uint64_t)cca;
+ zb = wa * (uint64_t)qa + wb * (uint64_t)qb
+ + m[k] * (uint64_t)fb + (uint64_t)ccb;
+ if (k > 0) {
+ a[k - 1] = (uint32_t)za & 0x7FFFFFFF;
+ b[k - 1] = (uint32_t)zb & 0x7FFFFFFF;
+ }
+
+#define M ((uint64_t)1 << 32)
+ tta = za >> 31;
+ ttb = zb >> 31;
+ tta = (tta ^ M) - M;
+ ttb = (ttb ^ M) - M;
+ cca = *(int64_t *)&tta;
+ ccb = *(int64_t *)&ttb;
+#undef M
+ }
+ a[len - 1] = (uint32_t)cca;
+ b[len - 1] = (uint32_t)ccb;
+
+ /*
+ * At this point:
+ * -m <= a < 2*m
+ * -m <= b < 2*m
+ * (this is a case of Montgomery reduction)
+ * The top word of 'a' and 'b' may have a 32-th bit set.
+ * We may have to add or subtract the modulus.
+ */
+ finish_mod(a, len, m, (uint32_t)((uint64_t)cca >> 63));
+ finish_mod(b, len, m, (uint32_t)((uint64_t)ccb >> 63));
+}
+
+/* see inner.h */
+uint32_t
+br_i31_moddiv(uint32_t *x, const uint32_t *y, const uint32_t *m, uint32_t m0i,
+ uint32_t *t)
+{
+ /*
+ * Algorithm is an extended binary GCD. We maintain four values
+ * a, b, u and v, with the following invariants:
+ *
+ * a * x = y * u mod m
+ * b * x = y * v mod m
+ *
+ * Starting values are:
+ *
+ * a = y
+ * b = m
+ * u = x
+ * v = 0
+ *
+ * The formal definition of the algorithm is a sequence of steps:
+ *
+ * - If a is even, then a <- a/2 and u <- u/2 mod m.
+ * - Otherwise, if b is even, then b <- b/2 and v <- v/2 mod m.
+ * - Otherwise, if a > b, then a <- (a-b)/2 and u <- (u-v)/2 mod m.
+ * - Otherwise, b <- (b-a)/2 and v <- (v-u)/2 mod m.
+ *
+ * Algorithm stops when a = b. At that point, they both are equal
+ * to GCD(y,m); the modular division succeeds if that value is 1.
+ * The result of the modular division is then u (or v: both are
+ * equal at that point).
+ *
+ * Each step makes either a or b shrink by at least one bit; hence,
+ * if m has bit length k bits, then 2k-2 steps are sufficient.
+ *
+ *
+ * Though complexity is quadratic in the size of m, the bit-by-bit
+ * processing is not very efficient. We can speed up processing by
+ * remarking that the decisions are taken based only on observation
+ * of the top and low bits of a and b.
+ *
+ * In the loop below, at each iteration, we use the two top words
+ * of a and b, and the low words of a and b, to compute reduction
+ * parameters pa, pb, qa and qb such that the new values for a
+ * and b are:
+ *
+ * a' = (a*pa + b*pb) / (2^31)
+ * b' = (a*qa + b*qb) / (2^31)
+ *
+ * the division being exact.
+ *
+ * Since the choices are based on the top words, they may be slightly
+ * off, requiring an optional correction: if a' < 0, then we replace
+ * pa with -pa, and pb with -pb. The total length of a and b is
+ * thus reduced by at least 30 bits at each iteration.
+ *
+ * The stopping conditions are still the same, though: when a
+ * and b become equal, they must be both odd (since m is odd,
+ * the GCD cannot be even), therefore the next operation is a
+ * subtraction, and one of the values becomes 0. At that point,
+ * nothing else happens, i.e. one value is stuck at 0, and the
+ * other one is the GCD.
+ */
+ size_t len, k;
+ uint32_t *a, *b, *u, *v;
+ uint32_t num, r;
+
+ len = (m[0] + 31) >> 5;
+ a = t;
+ b = a + len;
+ u = x + 1;
+ v = b + len;
+ memcpy(a, y + 1, len * sizeof *y);
+ memcpy(b, m + 1, len * sizeof *m);
+ memset(v, 0, len * sizeof *v);
+
+ /*
+ * Loop below ensures that a and b are reduced by some bits each,
+ * for a total of at least 30 bits.
+ */
+ for (num = ((m[0] - (m[0] >> 5)) << 1) + 30; num >= 30; num -= 30) {
+ size_t j;
+ uint32_t c0, c1;
+ uint32_t a0, a1, b0, b1;
+ uint64_t a_hi, b_hi;
+ uint32_t a_lo, b_lo;
+ int64_t pa, pb, qa, qb;
+ int i;
+
+ /*
+ * Extract top words of a and b. If j is the highest
+ * index >= 1 such that a[j] != 0 or b[j] != 0, then we want
+ * (a[j] << 31) + a[j - 1], and (b[j] << 31) + b[j - 1].
+ * If a and b are down to one word each, then we use a[0]
+ * and b[0].
+ */
+ c0 = (uint32_t)-1;
+ c1 = (uint32_t)-1;
+ a0 = 0;
+ a1 = 0;
+ b0 = 0;
+ b1 = 0;
+ j = len;
+ while (j -- > 0) {
+ uint32_t aw, bw;
+
+ aw = a[j];
+ bw = b[j];
+ a0 ^= (a0 ^ aw) & c0;
+ a1 ^= (a1 ^ aw) & c1;
+ b0 ^= (b0 ^ bw) & c0;
+ b1 ^= (b1 ^ bw) & c1;
+ c1 = c0;
+ c0 &= (((aw | bw) + 0x7FFFFFFF) >> 31) - (uint32_t)1;
+ }
+
+ /*
+ * If c1 = 0, then we grabbed two words for a and b.
+ * If c1 != 0 but c0 = 0, then we grabbed one word. It
+ * is not possible that c1 != 0 and c0 != 0, because that
+ * would mean that both integers are zero.
+ */
+ a1 |= a0 & c1;
+ a0 &= ~c1;
+ b1 |= b0 & c1;
+ b0 &= ~c1;
+ a_hi = ((uint64_t)a0 << 31) + a1;
+ b_hi = ((uint64_t)b0 << 31) + b1;
+ a_lo = a[0];
+ b_lo = b[0];
+
+ /*
+ * Compute reduction factors:
+ *
+ * a' = a*pa + b*pb
+ * b' = a*qa + b*qb
+ *
+ * such that a' and b' are both multiple of 2^31, but are
+ * only marginally larger than a and b.
+ */
+ pa = 1;
+ pb = 0;
+ qa = 0;
+ qb = 1;
+ for (i = 0; i < 31; i ++) {
+ /*
+ * At each iteration:
+ *
+ * a <- (a-b)/2 if: a is odd, b is odd, a_hi > b_hi
+ * b <- (b-a)/2 if: a is odd, b is odd, a_hi <= b_hi
+ * a <- a/2 if: a is even
+ * b <- b/2 if: a is odd, b is even
+ *
+ * We multiply a_lo and b_lo by 2 at each
+ * iteration, thus a division by 2 really is a
+ * non-multiplication by 2.
+ */
+ uint32_t r, oa, ob, cAB, cBA, cA;
+ uint64_t rz;
+
+ /*
+ * r = GT(a_hi, b_hi)
+ * But the GT() function works on uint32_t operands,
+ * so we inline a 64-bit version here.
+ */
+ rz = b_hi - a_hi;
+ r = (uint32_t)((rz ^ ((a_hi ^ b_hi)
+ & (a_hi ^ rz))) >> 63);
+
+ /*
+ * cAB = 1 if b must be subtracted from a
+ * cBA = 1 if a must be subtracted from b
+ * cA = 1 if a is divided by 2, 0 otherwise
+ *
+ * Rules:
+ *
+ * cAB and cBA cannot be both 1.
+ * if a is not divided by 2, b is.
+ */
+ oa = (a_lo >> i) & 1;
+ ob = (b_lo >> i) & 1;
+ cAB = oa & ob & r;
+ cBA = oa & ob & NOT(r);
+ cA = cAB | NOT(oa);
+
+ /*
+ * Conditional subtractions.
+ */
+ a_lo -= b_lo & -cAB;
+ a_hi -= b_hi & -(uint64_t)cAB;
+ pa -= qa & -(int64_t)cAB;
+ pb -= qb & -(int64_t)cAB;
+ b_lo -= a_lo & -cBA;
+ b_hi -= a_hi & -(uint64_t)cBA;
+ qa -= pa & -(int64_t)cBA;
+ qb -= pb & -(int64_t)cBA;
+
+ /*
+ * Shifting.
+ */
+ a_lo += a_lo & (cA - 1);
+ pa += pa & ((int64_t)cA - 1);
+ pb += pb & ((int64_t)cA - 1);
+ a_hi ^= (a_hi ^ (a_hi >> 1)) & -(uint64_t)cA;
+ b_lo += b_lo & -cA;
+ qa += qa & -(int64_t)cA;
+ qb += qb & -(int64_t)cA;
+ b_hi ^= (b_hi ^ (b_hi >> 1)) & ((uint64_t)cA - 1);
+ }
+
+ /*
+ * Replace a and b with new values a' and b'.
+ */
+ r = co_reduce(a, b, len, pa, pb, qa, qb);
+ pa -= pa * ((r & 1) << 1);
+ pb -= pb * ((r & 1) << 1);
+ qa -= qa * (r & 2);
+ qb -= qb * (r & 2);
+ co_reduce_mod(u, v, len, pa, pb, qa, qb, m + 1, m0i);
+ }
+
+ /*
+ * Now one of the arrays should be 0, and the other contains
+ * the GCD. If a is 0, then u is 0 as well, and v contains
+ * the division result.
+ * Result is correct if and only if GCD is 1.
+ */
+ r = (a[0] | b[0]) ^ 1;
+ u[0] |= v[0];
+ for (k = 1; k < len; k ++) {
+ r |= a[k] | b[k];
+ u[k] |= v[k];
+ }
+ return EQ0(r);
+}
projects / BearSSL / blobdiff