1/*
2 * Copyright (c) 2016 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/* see inner.h */
28void
29br_i31_muladd_small(uint32_t *x, uint32_t z, const uint32_t *m)
30{
31	uint32_t m_bitlen;
32	unsigned mblr;
33	size_t u, mlen;
34	uint32_t a0, a1, b0, hi, g, q, tb;
35	uint32_t under, over;
36	uint32_t cc;
37
38	/*
39	 * We can test on the modulus bit length since we accept to
40	 * leak that length.
41	 */
42	m_bitlen = m[0];
43	if (m_bitlen == 0) {
44		return;
45	}
46	if (m_bitlen <= 31) {
47		uint32_t lo;
48
49		hi = x[1] >> 1;
50		lo = (x[1] << 31) | z;
51		x[1] = br_rem(hi, lo, m[1]);
52		return;
53	}
54	mlen = (m_bitlen + 31) >> 5;
55	mblr = (unsigned)m_bitlen & 31;
56
57	/*
58	 * Principle: we estimate the quotient (x*2^31+z)/m by
59	 * doing a 64/32 division with the high words.
60	 *
61	 * Let:
62	 *   w = 2^31
63	 *   a = (w*a0 + a1) * w^N + a2
64	 *   b = b0 * w^N + b2
65	 * such that:
66	 *   0 <= a0 < w
67	 *   0 <= a1 < w
68	 *   0 <= a2 < w^N
69	 *   w/2 <= b0 < w
70	 *   0 <= b2 < w^N
71	 *   a < w*b
72	 * I.e. the two top words of a are a0:a1, the top word of b is
73	 * b0, we ensured that b0 is "full" (high bit set), and a is
74	 * such that the quotient q = a/b fits on one word (0 <= q < w).
75	 *
76	 * If a = b*q + r (with 0 <= r < q), we can estimate q by
77	 * doing an Euclidean division on the top words:
78	 *   a0*w+a1 = b0*u + v  (with 0 <= v < b0)
79	 * Then the following holds:
80	 *   0 <= u <= w
81	 *   u-2 <= q <= u
82	 */
83	hi = x[mlen];
84	if (mblr == 0) {
85		a0 = x[mlen];
86		memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
87		x[1] = z;
88		a1 = x[mlen];
89		b0 = m[mlen];
90	} else {
91		a0 = ((x[mlen] << (31 - mblr)) | (x[mlen - 1] >> mblr))
92			& 0x7FFFFFFF;
93		memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
94		x[1] = z;
95		a1 = ((x[mlen] << (31 - mblr)) | (x[mlen - 1] >> mblr))
96			& 0x7FFFFFFF;
97		b0 = ((m[mlen] << (31 - mblr)) | (m[mlen - 1] >> mblr))
98			& 0x7FFFFFFF;
99	}
100
101	/*
102	 * We estimate a divisor q. If the quotient returned by br_div()
103	 * is g:
104	 * -- If a0 == b0 then g == 0; we want q = 0x7FFFFFFF.
105	 * -- Otherwise:
106	 *    -- if g == 0 then we set q = 0;
107	 *    -- otherwise, we set q = g - 1.
108	 * The properties described above then ensure that the true
109	 * quotient is q-1, q or q+1.
110	 *
111	 * Take care that a0, a1 and b0 are 31-bit words, not 32-bit. We
112	 * must adjust the parameters to br_div() accordingly.
113	 */
114	g = br_div(a0 >> 1, a1 | (a0 << 31), b0);
115	q = MUX(EQ(a0, b0), 0x7FFFFFFF, MUX(EQ(g, 0), 0, g - 1));
116
117	/*
118	 * We subtract q*m from x (with the extra high word of value 'hi').
119	 * Since q may be off by 1 (in either direction), we may have to
120	 * add or subtract m afterwards.
121	 *
122	 * The 'tb' flag will be true (1) at the end of the loop if the
123	 * result is greater than or equal to the modulus (not counting
124	 * 'hi' or the carry).
125	 */
126	cc = 0;
127	tb = 1;
128	for (u = 1; u <= mlen; u ++) {
129		uint32_t mw, zw, xw, nxw;
130		uint64_t zl;
131
132		mw = m[u];
133		zl = MUL31(mw, q) + cc;
134		cc = (uint32_t)(zl >> 31);
135		zw = (uint32_t)zl & (uint32_t)0x7FFFFFFF;
136		xw = x[u];
137		nxw = xw - zw;
138		cc += nxw >> 31;
139		nxw &= 0x7FFFFFFF;
140		x[u] = nxw;
141		tb = MUX(EQ(nxw, mw), tb, GT(nxw, mw));
142	}
143
144	/*
145	 * If we underestimated q, then either cc < hi (one extra bit
146	 * beyond the top array word), or cc == hi and tb is true (no
147	 * extra bit, but the result is not lower than the modulus). In
148	 * these cases we must subtract m once.
149	 *
150	 * Otherwise, we may have overestimated, which will show as
151	 * cc > hi (thus a negative result). Correction is adding m once.
152	 */
153	over = GT(cc, hi);
154	under = ~over & (tb | LT(cc, hi));
155	br_i31_add(x, m, over);
156	br_i31_sub(x, m, under);
157}
158