Minor optimizations to _scalar_inverse to save 4M

- Precalculate x^5 and use for "01010" patterns during accumulation. (net -2M)
- Further use of x^5 to allow shorter addition chain (net -2M)

Author: peterdettman

src/scalar_impl.h

@@ -66,76 +66,70 @@ static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar
 #else
     secp256k1_scalar *t;
     int i;
-    /* First compute x ^ (2^N - 1) for some values of N. */
-    secp256k1_scalar x2, x3, x4, x6, x7, x8, x15, x30, x60, x120, x127;
+    /* First compute xN as x ^ (2^N - 1) for some values of N,
+     * and uM as x ^ M for some values of M. */
+    secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126;
+    secp256k1_scalar u2, u5;
 
-    secp256k1_scalar_sqr(&x2,  x);
-    secp256k1_scalar_mul(&x2, &x2,  x);
+    secp256k1_scalar_sqr(&u2, x);
+    secp256k1_scalar_mul(&x2, &u2,  x);
+    secp256k1_scalar_mul(&u5, &u2, &x2);
+    secp256k1_scalar_mul(&x3, &u5,  &u2);
 
-    secp256k1_scalar_sqr(&x3, &x2);
-    secp256k1_scalar_mul(&x3, &x3,  x);
-
-    secp256k1_scalar_sqr(&x4, &x3);
-    secp256k1_scalar_mul(&x4, &x4,  x);
-
-    secp256k1_scalar_sqr(&x6, &x4);
-    secp256k1_scalar_sqr(&x6, &x6);
-    secp256k1_scalar_mul(&x6, &x6, &x2);
-
-    secp256k1_scalar_sqr(&x7, &x6);
-    secp256k1_scalar_mul(&x7, &x7,  x);
+    secp256k1_scalar_sqr(&x6, &x3);
+    for (i = 0; i < 2; i++) {
+        secp256k1_scalar_sqr(&x6, &x6);
+    }
+    secp256k1_scalar_mul(&x6, &x6, &x3);
 
-    secp256k1_scalar_sqr(&x8, &x7);
-    secp256k1_scalar_mul(&x8, &x8,  x);
+    secp256k1_scalar_sqr(&x8, &x6);
+    secp256k1_scalar_sqr(&x8, &x8);
+    secp256k1_scalar_mul(&x8, &x8,  &x2);
 
-    secp256k1_scalar_sqr(&x15, &x8);
-    for (i = 0; i < 6; i++) {
-        secp256k1_scalar_sqr(&x15, &x15);
+    secp256k1_scalar_sqr(&x14, &x8);
+    for (i = 0; i < 5; i++) {
+        secp256k1_scalar_sqr(&x14, &x14);
     }
-    secp256k1_scalar_mul(&x15, &x15, &x7);
+    secp256k1_scalar_mul(&x14, &x14, &x6);
 
-    secp256k1_scalar_sqr(&x30, &x15);
-    for (i = 0; i < 14; i++) {
-        secp256k1_scalar_sqr(&x30, &x30);
+    secp256k1_scalar_sqr(&x28, &x14);
+    for (i = 0; i < 13; i++) {
+        secp256k1_scalar_sqr(&x28, &x28);
     }
-    secp256k1_scalar_mul(&x30, &x30, &x15);
+    secp256k1_scalar_mul(&x28, &x28, &x14);
 
-    secp256k1_scalar_sqr(&x60, &x30);
-    for (i = 0; i < 29; i++) {
-        secp256k1_scalar_sqr(&x60, &x60);
+    secp256k1_scalar_sqr(&x56, &x28);
+    for (i = 0; i < 27; i++) {
+        secp256k1_scalar_sqr(&x56, &x56);
     }
-    secp256k1_scalar_mul(&x60, &x60, &x30);
+    secp256k1_scalar_mul(&x56, &x56, &x28);
 
-    secp256k1_scalar_sqr(&x120, &x60);
-    for (i = 0; i < 59; i++) {
-        secp256k1_scalar_sqr(&x120, &x120);
+    secp256k1_scalar_sqr(&x112, &x56);
+    for (i = 0; i < 55; i++) {
+        secp256k1_scalar_sqr(&x112, &x112);
     }
-    secp256k1_scalar_mul(&x120, &x120, &x60);
+    secp256k1_scalar_mul(&x112, &x112, &x56);
 
-    secp256k1_scalar_sqr(&x127, &x120);
-    for (i = 0; i < 6; i++) {
-        secp256k1_scalar_sqr(&x127, &x127);
+    secp256k1_scalar_sqr(&x126, &x112);
+    for (i = 0; i < 13; i++) {
+        secp256k1_scalar_sqr(&x126, &x126);
     }
-    secp256k1_scalar_mul(&x127, &x127, &x7);
+    secp256k1_scalar_mul(&x126, &x126, &x14);
 
-    /* Then accumulate the final result (t starts at x127). */
-    t = &x127;
-    for (i = 0; i < 2; i++) { /* 0 */
+    /* Then accumulate the final result (t starts at x126). */
+    t = &x126;
+    for (i = 0; i < 3; i++) {
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
     for (i = 0; i < 4; i++) { /* 0 */
         secp256k1_scalar_sqr(t, t);
     }
     secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 2; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 2; i++) { /* 0 */
+    for (i = 0; i < 4; i++) { /* 0 */
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
     for (i = 0; i < 2; i++) { /* 0 */
         secp256k1_scalar_sqr(t, t);
     }
@@ -160,34 +154,26 @@ static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar
         secp256k1_scalar_sqr(t, t);
     }
     secp256k1_scalar_mul(t, t, &x2); /* 11 */
-    for (i = 0; i < 2; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 2; i++) { /* 0 */
+    for (i = 0; i < 4; i++) { /* 0 */
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 5; i++) { /* 0 */
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 4; i++) { /* 0 */
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, &x4); /* 1111 */
-    for (i = 0; i < 2; i++) { /* 0 */
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 3; i++) {
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
     for (i = 0; i < 3; i++) { /* 00 */
         secp256k1_scalar_sqr(t, t);
     }
     secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 4; i++) { /* 000 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 2; i++) { /* 0 */
+    for (i = 0; i < 6; i++) { /* 000 */
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
     for (i = 0; i < 10; i++) { /* 0000000 */
         secp256k1_scalar_sqr(t, t);
     }
@@ -212,14 +198,14 @@ static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar
         secp256k1_scalar_sqr(t, t);
     }
     secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 5; i++) { /* 0 */
+    for (i = 0; i < 4; i++) { /* 0 */
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, &x4); /* 1111 */
-    for (i = 0; i < 2; i++) { /* 0 */
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 3; i++) {
         secp256k1_scalar_sqr(t, t);
     }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
     for (i = 0; i < 5; i++) { /* 000 */
         secp256k1_scalar_sqr(t, t);
     }