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 */
 28  void
 29  br_i32_muladd_small(uint32_t *x, uint32_t z, const uint32_t *m)
 30  {
 31  	uint32_t m_bitlen;
 32  	size_t u, mlen;
 33  	uint32_t a0, a1, b0, hi, g, q, tb;
 34  	uint32_t chf, clow, under, over;
 35  	uint64_t cc;
 36  
 37  	/*
 38  	 * We can test on the modulus bit length since we accept to
 39  	 * leak that length.
 40  	 */
 41  	m_bitlen = m[0];
 42  	if (m_bitlen == 0) {
 43  		return;
 44  	}
 45  	if (m_bitlen <= 32) {
 46  		x[1] = br_rem(x[1], z, m[1]);
 47  		return;
 48  	}
 49  	mlen = (m_bitlen + 31) >> 5;
 50  
 51  	/*
 52  	 * Principle: we estimate the quotient (x*2^32+z)/m by
 53  	 * doing a 64/32 division with the high words.
 54  	 *
 55  	 * Let:
 56  	 *   w = 2^32
 57  	 *   a = (w*a0 + a1) * w^N + a2
 58  	 *   b = b0 * w^N + b2
 59  	 * such that:
 60  	 *   0 <= a0 < w
 61  	 *   0 <= a1 < w
 62  	 *   0 <= a2 < w^N
 63  	 *   w/2 <= b0 < w
 64  	 *   0 <= b2 < w^N
 65  	 *   a < w*b
 66  	 * I.e. the two top words of a are a0:a1, the top word of b is
 67  	 * b0, we ensured that b0 is "full" (high bit set), and a is
 68  	 * such that the quotient q = a/b fits on one word (0 <= q < w).
 69  	 *
 70  	 * If a = b*q + r (with 0 <= r < q), we can estimate q by
 71  	 * doing an Euclidean division on the top words:
 72  	 *   a0*w+a1 = b0*u + v  (with 0 <= v < w)
 73  	 * Then the following holds:
 74  	 *   0 <= u <= w
 75  	 *   u-2 <= q <= u
 76  	 */
 77  	a0 = br_i32_word(x, m_bitlen - 32);
 78  	hi = x[mlen];
 79  	memmove(x + 2, x + 1, (mlen - 1) * sizeof *x);
 80  	x[1] = z;
 81  	a1 = br_i32_word(x, m_bitlen - 32);
 82  	b0 = br_i32_word(m, m_bitlen - 32);
 83  
 84  	/*
 85  	 * We estimate a divisor q. If the quotient returned by br_div()
 86  	 * is g:
 87  	 * -- If a0 == b0 then g == 0; we want q = 0xFFFFFFFF.
 88  	 * -- Otherwise:
 89  	 *    -- if g == 0 then we set q = 0;
 90  	 *    -- otherwise, we set q = g - 1.
 91  	 * The properties described above then ensure that the true
 92  	 * quotient is q-1, q or q+1.
 93  	 */
 94  	g = br_div(a0, a1, b0);
 95  	q = MUX(EQ(a0, b0), 0xFFFFFFFF, MUX(EQ(g, 0), 0, g - 1));
 96  
 97  	/*
 98  	 * We subtract q*m from x (with the extra high word of value 'hi').
 99  	 * Since q may be off by 1 (in either direction), we may have to
100  	 * add or subtract m afterwards.
101  	 *
102  	 * The 'tb' flag will be true (1) at the end of the loop if the
103  	 * result is greater than or equal to the modulus (not counting
104  	 * 'hi' or the carry).
105  	 */
106  	cc = 0;
107  	tb = 1;
108  	for (u = 1; u <= mlen; u ++) {
109  		uint32_t mw, zw, xw, nxw;
110  		uint64_t zl;
111  
112  		mw = m[u];
113  		zl = MUL(mw, q) + cc;
114  		cc = (uint32_t)(zl >> 32);
115  		zw = (uint32_t)zl;
116  		xw = x[u];
117  		nxw = xw - zw;
118  		cc += (uint64_t)GT(nxw, xw);
119  		x[u] = nxw;
120  		tb = MUX(EQ(nxw, mw), tb, GT(nxw, mw));
121  	}
122  
123  	/*
124  	 * If we underestimated q, then either cc < hi (one extra bit
125  	 * beyond the top array word), or cc == hi and tb is true (no
126  	 * extra bit, but the result is not lower than the modulus). In
127  	 * these cases we must subtract m once.
128  	 *
129  	 * Otherwise, we may have overestimated, which will show as
130  	 * cc > hi (thus a negative result). Correction is adding m once.
131  	 */
132  	chf = (uint32_t)(cc >> 32);
133  	clow = (uint32_t)cc;
134  	over = chf | GT(clow, hi);
135  	under = ~over & (tb | (~chf & LT(clow, hi)));
136  	br_i32_add(x, m, over);
137  	br_i32_sub(x, m, under);
138  }