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 */

27 /* see inner.h */

28 void

30 {

38 /*

39 * We can test on the modulus bit length since we accept to

40 * leak that length.

41 */

45 }

53 }

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 < w)

79 * Then the following holds:

80 * 0 <= u <= w

81 * u-2 <= q <= u

82 */

99 }

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 */

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 */

142 }

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 */

157 }