1 /* $NetBSD: bn_mp_karatsuba_mul.c,v 1.1.1.2 2014/04/24 12:45:31 pettai Exp $ */
2
3 #include <tommath.h>
4 #ifdef BN_MP_KARATSUBA_MUL_C
5 /* LibTomMath, multiple-precision integer library -- Tom St Denis
6 *
7 * LibTomMath is a library that provides multiple-precision
8 * integer arithmetic as well as number theoretic functionality.
9 *
10 * The library was designed directly after the MPI library by
11 * Michael Fromberger but has been written from scratch with
12 * additional optimizations in place.
13 *
14 * The library is free for all purposes without any express
15 * guarantee it works.
16 *
17 * Tom St Denis, tomstdenis@gmail.com, http://libtom.org
18 */
19
20 /* c = |a| * |b| using Karatsuba Multiplication using
21 * three half size multiplications
22 *
23 * Let B represent the radix [e.g. 2**DIGIT_BIT] and
24 * let n represent half of the number of digits in
25 * the min(a,b)
26 *
27 * a = a1 * B**n + a0
28 * b = b1 * B**n + b0
29 *
30 * Then, a * b =>
31 a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0
32 *
33 * Note that a1b1 and a0b0 are used twice and only need to be
34 * computed once. So in total three half size (half # of
35 * digit) multiplications are performed, a0b0, a1b1 and
36 * (a1+b1)(a0+b0)
37 *
38 * Note that a multiplication of half the digits requires
39 * 1/4th the number of single precision multiplications so in
40 * total after one call 25% of the single precision multiplications
41 * are saved. Note also that the call to mp_mul can end up back
42 * in this function if the a0, a1, b0, or b1 are above the threshold.
43 * This is known as divide-and-conquer and leads to the famous
44 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than
45 * the standard O(N**2) that the baseline/comba methods use.
46 * Generally though the overhead of this method doesn't pay off
47 * until a certain size (N ~ 80) is reached.
48 */
mp_karatsuba_mul(mp_int * a,mp_int * b,mp_int * c)49 int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
50 {
51 mp_int x0, x1, y0, y1, t1, x0y0, x1y1;
52 int B, err;
53
54 /* default the return code to an error */
55 err = MP_MEM;
56
57 /* min # of digits */
58 B = MIN (a->used, b->used);
59
60 /* now divide in two */
61 B = B >> 1;
62
63 /* init copy all the temps */
64 if (mp_init_size (&x0, B) != MP_OKAY)
65 goto ERR;
66 if (mp_init_size (&x1, a->used - B) != MP_OKAY)
67 goto X0;
68 if (mp_init_size (&y0, B) != MP_OKAY)
69 goto X1;
70 if (mp_init_size (&y1, b->used - B) != MP_OKAY)
71 goto Y0;
72
73 /* init temps */
74 if (mp_init_size (&t1, B * 2) != MP_OKAY)
75 goto Y1;
76 if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
77 goto T1;
78 if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
79 goto X0Y0;
80
81 /* now shift the digits */
82 x0.used = y0.used = B;
83 x1.used = a->used - B;
84 y1.used = b->used - B;
85
86 {
87 register int x;
88 register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
89
90 /* we copy the digits directly instead of using higher level functions
91 * since we also need to shift the digits
92 */
93 tmpa = a->dp;
94 tmpb = b->dp;
95
96 tmpx = x0.dp;
97 tmpy = y0.dp;
98 for (x = 0; x < B; x++) {
99 *tmpx++ = *tmpa++;
100 *tmpy++ = *tmpb++;
101 }
102
103 tmpx = x1.dp;
104 for (x = B; x < a->used; x++) {
105 *tmpx++ = *tmpa++;
106 }
107
108 tmpy = y1.dp;
109 for (x = B; x < b->used; x++) {
110 *tmpy++ = *tmpb++;
111 }
112 }
113
114 /* only need to clamp the lower words since by definition the
115 * upper words x1/y1 must have a known number of digits
116 */
117 mp_clamp (&x0);
118 mp_clamp (&y0);
119
120 /* now calc the products x0y0 and x1y1 */
121 /* after this x0 is no longer required, free temp [x0==t2]! */
122 if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)
123 goto X1Y1; /* x0y0 = x0*y0 */
124 if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
125 goto X1Y1; /* x1y1 = x1*y1 */
126
127 /* now calc x1+x0 and y1+y0 */
128 if (s_mp_add (&x1, &x0, &t1) != MP_OKAY)
129 goto X1Y1; /* t1 = x1 - x0 */
130 if (s_mp_add (&y1, &y0, &x0) != MP_OKAY)
131 goto X1Y1; /* t2 = y1 - y0 */
132 if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
133 goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */
134
135 /* add x0y0 */
136 if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
137 goto X1Y1; /* t2 = x0y0 + x1y1 */
138 if (s_mp_sub (&t1, &x0, &t1) != MP_OKAY)
139 goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */
140
141 /* shift by B */
142 if (mp_lshd (&t1, B) != MP_OKAY)
143 goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
144 if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
145 goto X1Y1; /* x1y1 = x1y1 << 2*B */
146
147 if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
148 goto X1Y1; /* t1 = x0y0 + t1 */
149 if (mp_add (&t1, &x1y1, c) != MP_OKAY)
150 goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */
151
152 /* Algorithm succeeded set the return code to MP_OKAY */
153 err = MP_OKAY;
154
155 X1Y1:mp_clear (&x1y1);
156 X0Y0:mp_clear (&x0y0);
157 T1:mp_clear (&t1);
158 Y1:mp_clear (&y1);
159 Y0:mp_clear (&y0);
160 X1:mp_clear (&x1);
161 X0:mp_clear (&x0);
162 ERR:
163 return err;
164 }
165 #endif
166
167 /* Source: /cvs/libtom/libtommath/bn_mp_karatsuba_mul.c,v */
168 /* Revision: 1.6 */
169 /* Date: 2006/12/28 01:25:13 */
170