1*0957b409SSimon J. Gerraty /*
2*0957b409SSimon J. Gerraty * Copyright (c) 2017 Thomas Pornin <pornin@bolet.org>
3*0957b409SSimon J. Gerraty *
4*0957b409SSimon J. Gerraty * Permission is hereby granted, free of charge, to any person obtaining
5*0957b409SSimon J. Gerraty * a copy of this software and associated documentation files (the
6*0957b409SSimon J. Gerraty * "Software"), to deal in the Software without restriction, including
7*0957b409SSimon J. Gerraty * without limitation the rights to use, copy, modify, merge, publish,
8*0957b409SSimon J. Gerraty * distribute, sublicense, and/or sell copies of the Software, and to
9*0957b409SSimon J. Gerraty * permit persons to whom the Software is furnished to do so, subject to
10*0957b409SSimon J. Gerraty * the following conditions:
11*0957b409SSimon J. Gerraty *
12*0957b409SSimon J. Gerraty * The above copyright notice and this permission notice shall be
13*0957b409SSimon J. Gerraty * included in all copies or substantial portions of the Software.
14*0957b409SSimon J. Gerraty *
15*0957b409SSimon J. Gerraty * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
16*0957b409SSimon J. Gerraty * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
17*0957b409SSimon J. Gerraty * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
18*0957b409SSimon J. Gerraty * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
19*0957b409SSimon J. Gerraty * BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
20*0957b409SSimon J. Gerraty * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
21*0957b409SSimon J. Gerraty * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
22*0957b409SSimon J. Gerraty * SOFTWARE.
23*0957b409SSimon J. Gerraty */
24*0957b409SSimon J. Gerraty
25*0957b409SSimon J. Gerraty #include "inner.h"
26*0957b409SSimon J. Gerraty
27*0957b409SSimon J. Gerraty /*
28*0957b409SSimon J. Gerraty * Constant-time division. The divisor must not be larger than 16 bits,
29*0957b409SSimon J. Gerraty * and the quotient must fit on 17 bits.
30*0957b409SSimon J. Gerraty */
31*0957b409SSimon J. Gerraty static uint32_t
divrem16(uint32_t x,uint32_t d,uint32_t * r)32*0957b409SSimon J. Gerraty divrem16(uint32_t x, uint32_t d, uint32_t *r)
33*0957b409SSimon J. Gerraty {
34*0957b409SSimon J. Gerraty int i;
35*0957b409SSimon J. Gerraty uint32_t q;
36*0957b409SSimon J. Gerraty
37*0957b409SSimon J. Gerraty q = 0;
38*0957b409SSimon J. Gerraty d <<= 16;
39*0957b409SSimon J. Gerraty for (i = 16; i >= 0; i --) {
40*0957b409SSimon J. Gerraty uint32_t ctl;
41*0957b409SSimon J. Gerraty
42*0957b409SSimon J. Gerraty ctl = LE(d, x);
43*0957b409SSimon J. Gerraty q |= ctl << i;
44*0957b409SSimon J. Gerraty x -= (-ctl) & d;
45*0957b409SSimon J. Gerraty d >>= 1;
46*0957b409SSimon J. Gerraty }
47*0957b409SSimon J. Gerraty if (r != NULL) {
48*0957b409SSimon J. Gerraty *r = x;
49*0957b409SSimon J. Gerraty }
50*0957b409SSimon J. Gerraty return q;
51*0957b409SSimon J. Gerraty }
52*0957b409SSimon J. Gerraty
53*0957b409SSimon J. Gerraty /* see inner.h */
54*0957b409SSimon J. Gerraty void
br_i15_muladd_small(uint16_t * x,uint16_t z,const uint16_t * m)55*0957b409SSimon J. Gerraty br_i15_muladd_small(uint16_t *x, uint16_t z, const uint16_t *m)
56*0957b409SSimon J. Gerraty {
57*0957b409SSimon J. Gerraty /*
58*0957b409SSimon J. Gerraty * Constant-time: we accept to leak the exact bit length of the
59*0957b409SSimon J. Gerraty * modulus m.
60*0957b409SSimon J. Gerraty */
61*0957b409SSimon J. Gerraty unsigned m_bitlen, mblr;
62*0957b409SSimon J. Gerraty size_t u, mlen;
63*0957b409SSimon J. Gerraty uint32_t hi, a0, a, b, q;
64*0957b409SSimon J. Gerraty uint32_t cc, tb, over, under;
65*0957b409SSimon J. Gerraty
66*0957b409SSimon J. Gerraty /*
67*0957b409SSimon J. Gerraty * Simple case: the modulus fits on one word.
68*0957b409SSimon J. Gerraty */
69*0957b409SSimon J. Gerraty m_bitlen = m[0];
70*0957b409SSimon J. Gerraty if (m_bitlen == 0) {
71*0957b409SSimon J. Gerraty return;
72*0957b409SSimon J. Gerraty }
73*0957b409SSimon J. Gerraty if (m_bitlen <= 15) {
74*0957b409SSimon J. Gerraty uint32_t rem;
75*0957b409SSimon J. Gerraty
76*0957b409SSimon J. Gerraty divrem16(((uint32_t)x[1] << 15) | z, m[1], &rem);
77*0957b409SSimon J. Gerraty x[1] = rem;
78*0957b409SSimon J. Gerraty return;
79*0957b409SSimon J. Gerraty }
80*0957b409SSimon J. Gerraty mlen = (m_bitlen + 15) >> 4;
81*0957b409SSimon J. Gerraty mblr = m_bitlen & 15;
82*0957b409SSimon J. Gerraty
83*0957b409SSimon J. Gerraty /*
84*0957b409SSimon J. Gerraty * Principle: we estimate the quotient (x*2^15+z)/m by
85*0957b409SSimon J. Gerraty * doing a 30/15 division with the high words.
86*0957b409SSimon J. Gerraty *
87*0957b409SSimon J. Gerraty * Let:
88*0957b409SSimon J. Gerraty * w = 2^15
89*0957b409SSimon J. Gerraty * a = (w*a0 + a1) * w^N + a2
90*0957b409SSimon J. Gerraty * b = b0 * w^N + b2
91*0957b409SSimon J. Gerraty * such that:
92*0957b409SSimon J. Gerraty * 0 <= a0 < w
93*0957b409SSimon J. Gerraty * 0 <= a1 < w
94*0957b409SSimon J. Gerraty * 0 <= a2 < w^N
95*0957b409SSimon J. Gerraty * w/2 <= b0 < w
96*0957b409SSimon J. Gerraty * 0 <= b2 < w^N
97*0957b409SSimon J. Gerraty * a < w*b
98*0957b409SSimon J. Gerraty * I.e. the two top words of a are a0:a1, the top word of b is
99*0957b409SSimon J. Gerraty * b0, we ensured that b0 is "full" (high bit set), and a is
100*0957b409SSimon J. Gerraty * such that the quotient q = a/b fits on one word (0 <= q < w).
101*0957b409SSimon J. Gerraty *
102*0957b409SSimon J. Gerraty * If a = b*q + r (with 0 <= r < q), then we can estimate q by
103*0957b409SSimon J. Gerraty * using a division on the top words:
104*0957b409SSimon J. Gerraty * a0*w + a1 = b0*u + v (with 0 <= v < b0)
105*0957b409SSimon J. Gerraty * Then the following holds:
106*0957b409SSimon J. Gerraty * 0 <= u <= w
107*0957b409SSimon J. Gerraty * u-2 <= q <= u
108*0957b409SSimon J. Gerraty */
109*0957b409SSimon J. Gerraty hi = x[mlen];
110*0957b409SSimon J. Gerraty if (mblr == 0) {
111*0957b409SSimon J. Gerraty a0 = x[mlen];
112*0957b409SSimon J. Gerraty memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
113*0957b409SSimon J. Gerraty x[1] = z;
114*0957b409SSimon J. Gerraty a = (a0 << 15) + x[mlen];
115*0957b409SSimon J. Gerraty b = m[mlen];
116*0957b409SSimon J. Gerraty } else {
117*0957b409SSimon J. Gerraty a0 = (x[mlen] << (15 - mblr)) | (x[mlen - 1] >> mblr);
118*0957b409SSimon J. Gerraty memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
119*0957b409SSimon J. Gerraty x[1] = z;
120*0957b409SSimon J. Gerraty a = (a0 << 15) | (((x[mlen] << (15 - mblr))
121*0957b409SSimon J. Gerraty | (x[mlen - 1] >> mblr)) & 0x7FFF);
122*0957b409SSimon J. Gerraty b = (m[mlen] << (15 - mblr)) | (m[mlen - 1] >> mblr);
123*0957b409SSimon J. Gerraty }
124*0957b409SSimon J. Gerraty q = divrem16(a, b, NULL);
125*0957b409SSimon J. Gerraty
126*0957b409SSimon J. Gerraty /*
127*0957b409SSimon J. Gerraty * We computed an estimate for q, but the real one may be q,
128*0957b409SSimon J. Gerraty * q-1 or q-2; moreover, the division may have returned a value
129*0957b409SSimon J. Gerraty * 8000 or even 8001 if the two high words were identical, and
130*0957b409SSimon J. Gerraty * we want to avoid values beyond 7FFF. We thus adjust q so
131*0957b409SSimon J. Gerraty * that the "true" multiplier will be q+1, q or q-1, and q is
132*0957b409SSimon J. Gerraty * in the 0000..7FFF range.
133*0957b409SSimon J. Gerraty */
134*0957b409SSimon J. Gerraty q = MUX(EQ(b, a0), 0x7FFF, q - 1 + ((q - 1) >> 31));
135*0957b409SSimon J. Gerraty
136*0957b409SSimon J. Gerraty /*
137*0957b409SSimon J. Gerraty * We subtract q*m from x (x has an extra high word of value 'hi').
138*0957b409SSimon J. Gerraty * Since q may be off by 1 (in either direction), we may have to
139*0957b409SSimon J. Gerraty * add or subtract m afterwards.
140*0957b409SSimon J. Gerraty *
141*0957b409SSimon J. Gerraty * The 'tb' flag will be true (1) at the end of the loop if the
142*0957b409SSimon J. Gerraty * result is greater than or equal to the modulus (not counting
143*0957b409SSimon J. Gerraty * 'hi' or the carry).
144*0957b409SSimon J. Gerraty */
145*0957b409SSimon J. Gerraty cc = 0;
146*0957b409SSimon J. Gerraty tb = 1;
147*0957b409SSimon J. Gerraty for (u = 1; u <= mlen; u ++) {
148*0957b409SSimon J. Gerraty uint32_t mw, zl, xw, nxw;
149*0957b409SSimon J. Gerraty
150*0957b409SSimon J. Gerraty mw = m[u];
151*0957b409SSimon J. Gerraty zl = MUL15(mw, q) + cc;
152*0957b409SSimon J. Gerraty cc = zl >> 15;
153*0957b409SSimon J. Gerraty zl &= 0x7FFF;
154*0957b409SSimon J. Gerraty xw = x[u];
155*0957b409SSimon J. Gerraty nxw = xw - zl;
156*0957b409SSimon J. Gerraty cc += nxw >> 31;
157*0957b409SSimon J. Gerraty nxw &= 0x7FFF;
158*0957b409SSimon J. Gerraty x[u] = nxw;
159*0957b409SSimon J. Gerraty tb = MUX(EQ(nxw, mw), tb, GT(nxw, mw));
160*0957b409SSimon J. Gerraty }
161*0957b409SSimon J. Gerraty
162*0957b409SSimon J. Gerraty /*
163*0957b409SSimon J. Gerraty * If we underestimated q, then either cc < hi (one extra bit
164*0957b409SSimon J. Gerraty * beyond the top array word), or cc == hi and tb is true (no
165*0957b409SSimon J. Gerraty * extra bit, but the result is not lower than the modulus).
166*0957b409SSimon J. Gerraty *
167*0957b409SSimon J. Gerraty * If we overestimated q, then cc > hi.
168*0957b409SSimon J. Gerraty */
169*0957b409SSimon J. Gerraty over = GT(cc, hi);
170*0957b409SSimon J. Gerraty under = ~over & (tb | LT(cc, hi));
171*0957b409SSimon J. Gerraty br_i15_add(x, m, over);
172*0957b409SSimon J. Gerraty br_i15_sub(x, m, under);
173*0957b409SSimon J. Gerraty }
174