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