152 lines
4.2 KiB
C
152 lines
4.2 KiB
C
#include "tommath_private.h"
|
|
#ifdef S_MP_MUL_KARATSUBA_C
|
|
/* LibTomMath, multiple-precision integer library -- Tom St Denis */
|
|
/* SPDX-License-Identifier: Unlicense */
|
|
|
|
/* c = |a| * |b| using Karatsuba Multiplication using
|
|
* three half size multiplications
|
|
*
|
|
* Let B represent the radix [e.g. 2**MP_DIGIT_BIT] and
|
|
* let n represent half of the number of digits in
|
|
* the min(a,b)
|
|
*
|
|
* a = a1 * B**n + a0
|
|
* b = b1 * B**n + b0
|
|
*
|
|
* Then, a * b =>
|
|
a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0
|
|
*
|
|
* Note that a1b1 and a0b0 are used twice and only need to be
|
|
* computed once. So in total three half size (half # of
|
|
* digit) multiplications are performed, a0b0, a1b1 and
|
|
* (a1+b1)(a0+b0)
|
|
*
|
|
* Note that a multiplication of half the digits requires
|
|
* 1/4th the number of single precision multiplications so in
|
|
* total after one call 25% of the single precision multiplications
|
|
* are saved. Note also that the call to mp_mul can end up back
|
|
* in this function if the a0, a1, b0, or b1 are above the threshold.
|
|
* This is known as divide-and-conquer and leads to the famous
|
|
* O(N**lg(3)) or O(N**1.584) work which is asymptotically lower than
|
|
* the standard O(N**2) that the baseline/comba methods use.
|
|
* Generally though the overhead of this method doesn't pay off
|
|
* until a certain size (N ~ 80) is reached.
|
|
*/
|
|
mp_err s_mp_mul_karatsuba(const mp_int *a, const mp_int *b, mp_int *c)
|
|
{
|
|
mp_int x0, x1, y0, y1, t1, x0y0, x1y1;
|
|
int B;
|
|
mp_err err;
|
|
|
|
/* min # of digits */
|
|
B = MP_MIN(a->used, b->used);
|
|
|
|
/* now divide in two */
|
|
B = B >> 1;
|
|
|
|
/* init copy all the temps */
|
|
if ((err = mp_init_size(&x0, B)) != MP_OKAY) {
|
|
goto LBL_ERR;
|
|
}
|
|
if ((err = mp_init_size(&x1, a->used - B)) != MP_OKAY) {
|
|
goto X0;
|
|
}
|
|
if ((err = mp_init_size(&y0, B)) != MP_OKAY) {
|
|
goto X1;
|
|
}
|
|
if ((err = mp_init_size(&y1, b->used - B)) != MP_OKAY) {
|
|
goto Y0;
|
|
}
|
|
|
|
/* init temps */
|
|
if ((err = mp_init_size(&t1, B * 2)) != MP_OKAY) {
|
|
goto Y1;
|
|
}
|
|
if ((err = mp_init_size(&x0y0, B * 2)) != MP_OKAY) {
|
|
goto T1;
|
|
}
|
|
if ((err = mp_init_size(&x1y1, B * 2)) != MP_OKAY) {
|
|
goto X0Y0;
|
|
}
|
|
|
|
/* now shift the digits */
|
|
x0.used = y0.used = B;
|
|
x1.used = a->used - B;
|
|
y1.used = b->used - B;
|
|
|
|
/* we copy the digits directly instead of using higher level functions
|
|
* since we also need to shift the digits
|
|
*/
|
|
s_mp_copy_digs(x0.dp, a->dp, x0.used);
|
|
s_mp_copy_digs(y0.dp, b->dp, y0.used);
|
|
s_mp_copy_digs(x1.dp, a->dp + B, x1.used);
|
|
s_mp_copy_digs(y1.dp, b->dp + B, y1.used);
|
|
|
|
/* only need to clamp the lower words since by definition the
|
|
* upper words x1/y1 must have a known number of digits
|
|
*/
|
|
mp_clamp(&x0);
|
|
mp_clamp(&y0);
|
|
|
|
/* now calc the products x0y0 and x1y1 */
|
|
/* after this x0 is no longer required, free temp [x0==t2]! */
|
|
if ((err = mp_mul(&x0, &y0, &x0y0)) != MP_OKAY) {
|
|
goto X1Y1; /* x0y0 = x0*y0 */
|
|
}
|
|
if ((err = mp_mul(&x1, &y1, &x1y1)) != MP_OKAY) {
|
|
goto X1Y1; /* x1y1 = x1*y1 */
|
|
}
|
|
|
|
/* now calc x1+x0 and y1+y0 */
|
|
if ((err = s_mp_add(&x1, &x0, &t1)) != MP_OKAY) {
|
|
goto X1Y1; /* t1 = x1 - x0 */
|
|
}
|
|
if ((err = s_mp_add(&y1, &y0, &x0)) != MP_OKAY) {
|
|
goto X1Y1; /* t2 = y1 - y0 */
|
|
}
|
|
if ((err = mp_mul(&t1, &x0, &t1)) != MP_OKAY) {
|
|
goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */
|
|
}
|
|
|
|
/* add x0y0 */
|
|
if ((err = mp_add(&x0y0, &x1y1, &x0)) != MP_OKAY) {
|
|
goto X1Y1; /* t2 = x0y0 + x1y1 */
|
|
}
|
|
if ((err = s_mp_sub(&t1, &x0, &t1)) != MP_OKAY) {
|
|
goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */
|
|
}
|
|
|
|
/* shift by B */
|
|
if ((err = mp_lshd(&t1, B)) != MP_OKAY) {
|
|
goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
|
|
}
|
|
if ((err = mp_lshd(&x1y1, B * 2)) != MP_OKAY) {
|
|
goto X1Y1; /* x1y1 = x1y1 << 2*B */
|
|
}
|
|
|
|
if ((err = mp_add(&x0y0, &t1, &t1)) != MP_OKAY) {
|
|
goto X1Y1; /* t1 = x0y0 + t1 */
|
|
}
|
|
if ((err = mp_add(&t1, &x1y1, c)) != MP_OKAY) {
|
|
goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */
|
|
}
|
|
|
|
X1Y1:
|
|
mp_clear(&x1y1);
|
|
X0Y0:
|
|
mp_clear(&x0y0);
|
|
T1:
|
|
mp_clear(&t1);
|
|
Y1:
|
|
mp_clear(&y1);
|
|
Y0:
|
|
mp_clear(&y0);
|
|
X1:
|
|
mp_clear(&x1);
|
|
X0:
|
|
mp_clear(&x0);
|
|
LBL_ERR:
|
|
return err;
|
|
}
|
|
#endif
|