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 */ 31 static uint32_t 32 divrem16(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 */ 54 void 55 br_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