### diff libtommath/bn_mp_karatsuba_mul.c @ 284:eed26cff980b

author Matt Johnston Wed, 08 Mar 2006 13:23:49 +0000 5ff8218bcee9
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```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/libtommath/bn_mp_karatsuba_mul.c	Wed Mar 08 13:23:49 2006 +0000
@@ -0,0 +1,163 @@
+#include <tommath.h>
+#ifdef BN_MP_KARATSUBA_MUL_C
+/* LibTomMath, multiple-precision integer library -- Tom St Denis
+ *
+ * LibTomMath is a library that provides multiple-precision
+ * integer arithmetic as well as number theoretic functionality.
+ *
+ * The library was designed directly after the MPI library by
+ * Michael Fromberger but has been written from scratch with
+ * additional optimizations in place.
+ *
+ * The library is free for all purposes without any express
+ * guarantee it works.
+ *
+ * Tom St Denis, [email protected], http://math.libtomcrypt.org
+ */
+
+/* c = |a| * |b| using Karatsuba Multiplication using
+ * three half size multiplications
+ *
+ * Let B represent the radix [e.g. 2**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 asymptopically 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.
+ */
+int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
+{
+  mp_int  x0, x1, y0, y1, t1, x0y0, x1y1;
+  int     B, err;
+
+  /* default the return code to an error */
+  err = MP_MEM;
+
+  /* min # of digits */
+  B = MIN (a->used, b->used);
+
+  /* now divide in two */
+  B = B >> 1;
+
+  /* init copy all the temps */
+  if (mp_init_size (&x0, B) != MP_OKAY)
+    goto ERR;
+  if (mp_init_size (&x1, a->used - B) != MP_OKAY)
+    goto X0;
+  if (mp_init_size (&y0, B) != MP_OKAY)
+    goto X1;
+  if (mp_init_size (&y1, b->used - B) != MP_OKAY)
+    goto Y0;
+
+  /* init temps */
+  if (mp_init_size (&t1, B * 2) != MP_OKAY)
+    goto Y1;
+  if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
+    goto T1;
+  if (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;
+
+  {
+    register int x;
+    register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
+
+    /* we copy the digits directly instead of using higher level functions
+     * since we also need to shift the digits
+     */
+    tmpa = a->dp;
+    tmpb = b->dp;
+
+    tmpx = x0.dp;
+    tmpy = y0.dp;
+    for (x = 0; x < B; x++) {
+      *tmpx++ = *tmpa++;
+      *tmpy++ = *tmpb++;
+    }
+
+    tmpx = x1.dp;
+    for (x = B; x < a->used; x++) {
+      *tmpx++ = *tmpa++;
+    }
+
+    tmpy = y1.dp;
+    for (x = B; x < b->used; x++) {
+      *tmpy++ = *tmpb++;
+    }
+  }
+
+  /* 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 (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)
+    goto X1Y1;          /* x0y0 = x0*y0 */
+  if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
+    goto X1Y1;          /* x1y1 = x1*y1 */
+
+  /* now calc x1-x0 and y1-y0 */
+  if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
+    goto X1Y1;          /* t1 = x1 - x0 */
+  if (mp_sub (&y1, &y0, &x0) != MP_OKAY)
+    goto X1Y1;          /* t2 = y1 - y0 */
+  if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
+    goto X1Y1;          /* t1 = (x1 - x0) * (y1 - y0) */
+
+  if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
+    goto X1Y1;          /* t2 = x0y0 + x1y1 */
+  if (mp_sub (&x0, &t1, &t1) != MP_OKAY)
+    goto X1Y1;          /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */
+
+  /* shift by B */
+  if (mp_lshd (&t1, B) != MP_OKAY)
+    goto X1Y1;          /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
+  if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
+    goto X1Y1;          /* x1y1 = x1y1 << 2*B */
+
+  if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
+    goto X1Y1;          /* t1 = x0y0 + t1 */
+  if (mp_add (&t1, &x1y1, c) != MP_OKAY)
+    goto X1Y1;          /* t1 = x0y0 + t1 + x1y1 */
+
+  /* Algorithm succeeded set the return code to MP_OKAY */
+  err = MP_OKAY;
+
+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);
+ERR:
+  return err;
+}
+#endif```