br_i31_montymul(uint32_t *d, const uint32_t *x, const uint32_t *y,
const uint32_t *m, uint32_t m0i)
{
+ /*
+ * Each outer loop iteration computes:
+ * d <- (d + xu*y + f*m) / 2^31
+ * We have xu <= 2^31-1 and f <= 2^31-1.
+ * Thus, if d <= 2*m-1 on input, then:
+ * 2*m-1 + 2*(2^31-1)*m <= (2^32)*m-1
+ * and the new d value is less than 2*m.
+ *
+ * We represent d over 31-bit words, with an extra word 'dh'
+ * which can thus be only 0 or 1.
+ */
size_t len, len4, u, v;
- uint64_t dh;
+ uint32_t dh;
len = (m[0] + 31) >> 5;
len4 = len & ~(size_t)3;
br_i31_zero(d, m[0]);
dh = 0;
for (u = 0; u < len; u ++) {
+ /*
+ * The carry for each operation fits on 32 bits:
+ * d[v+1] <= 2^31-1
+ * xu*y[v+1] <= (2^31-1)*(2^31-1)
+ * f*m[v+1] <= (2^31-1)*(2^31-1)
+ * r <= 2^32-1
+ * (2^31-1) + 2*(2^31-1)*(2^31-1) + (2^32-1) = 2^63 - 2^31
+ * After division by 2^31, the new r is then at most 2^32-1
+ *
+ * Using a 32-bit carry has performance benefits on 32-bit
+ * systems; however, on 64-bit architectures, we prefer to
+ * keep the carry (r) in a 64-bit register, thus avoiding some
+ * "clear high bits" operations.
+ */
uint32_t f, xu;
- uint64_t r, zh;
+#if BR_64
+ uint64_t r;
+#else
+ uint32_t r;
+#endif
xu = x[u + 1];
f = MUL31_lo((d[1] + MUL31_lo(x[u + 1], y[1])), m0i);
d[v] = (uint32_t)z & 0x7FFFFFFF;
}
- zh = dh + r;
- d[len] = (uint32_t)zh & 0x7FFFFFFF;
- dh = zh >> 31;
+ /*
+ * Since the new dh can only be 0 or 1, the addition of
+ * the old dh with the carry MUST fit on 32 bits, and
+ * thus can be done into dh itself.
+ */
+ dh += r;
+ d[len] = dh & 0x7FFFFFFF;
+ dh >>= 31;
}
/*