2 * Copyright (c) 2016 Thomas Pornin <pornin@bolet.org>
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29 br_i31_montymul(uint32_t *d, const uint32_t *x, const uint32_t *y,
30 const uint32_t *m, uint32_t m0i)
33 * Each outer loop iteration computes:
34 * d <- (d + xu*y + f*m) / 2^31
35 * We have xu <= 2^31-1 and f <= 2^31-1.
36 * Thus, if d <= 2*m-1 on input, then:
37 * 2*m-1 + 2*(2^31-1)*m <= (2^32)*m-1
38 * and the new d value is less than 2*m.
40 * We represent d over 31-bit words, with an extra word 'dh'
41 * which can thus be only 0 or 1.
43 size_t len, len4, u, v;
46 len = (m[0] + 31) >> 5;
47 len4 = len & ~(size_t)3;
50 for (u = 0; u < len; u ++) {
52 * The carry for each operation fits on 32 bits:
54 * xu*y[v+1] <= (2^31-1)*(2^31-1)
55 * f*m[v+1] <= (2^31-1)*(2^31-1)
57 * (2^31-1) + 2*(2^31-1)*(2^31-1) + (2^32-1) = 2^63 - 2^31
58 * After division by 2^31, the new r is then at most 2^32-1
60 * Using a 32-bit carry has performance benefits on 32-bit
61 * systems; however, on 64-bit architectures, we prefer to
62 * keep the carry (r) in a 64-bit register, thus avoiding some
63 * "clear high bits" operations.
73 f = MUL31_lo((d[1] + MUL31_lo(x[u + 1], y[1])), m0i);
76 for (v = 0; v < len4; v += 4) {
79 z = (uint64_t)d[v + 1] + MUL31(xu, y[v + 1])
80 + MUL31(f, m[v + 1]) + r;
82 d[v + 0] = (uint32_t)z & 0x7FFFFFFF;
83 z = (uint64_t)d[v + 2] + MUL31(xu, y[v + 2])
84 + MUL31(f, m[v + 2]) + r;
86 d[v + 1] = (uint32_t)z & 0x7FFFFFFF;
87 z = (uint64_t)d[v + 3] + MUL31(xu, y[v + 3])
88 + MUL31(f, m[v + 3]) + r;
90 d[v + 2] = (uint32_t)z & 0x7FFFFFFF;
91 z = (uint64_t)d[v + 4] + MUL31(xu, y[v + 4])
92 + MUL31(f, m[v + 4]) + r;
94 d[v + 3] = (uint32_t)z & 0x7FFFFFFF;
96 for (; v < len; v ++) {
99 z = (uint64_t)d[v + 1] + MUL31(xu, y[v + 1])
100 + MUL31(f, m[v + 1]) + r;
102 d[v] = (uint32_t)z & 0x7FFFFFFF;
106 * Since the new dh can only be 0 or 1, the addition of
107 * the old dh with the carry MUST fit on 32 bits, and
108 * thus can be done into dh itself.
111 d[len] = dh & 0x7FFFFFFF;
116 * We must write back the bit length because it was overwritten in
117 * the loop (not overwriting it would require a test in the loop,
118 * which would yield bigger and slower code).
123 * d[] may still be greater than m[] at that point; notably, the
124 * 'dh' word may be non-zero.
126 br_i31_sub(d, m, NEQ(dh, 0) | NOT(br_i31_sub(d, m, 0)));