Merge bitcoin-core#513: Increase sparsity of pippenger fixed window naf representation

ec0a7b3 Don't touch leading zeros in wnaf_fixed. (Jonas Nick)
9e36d1b Fix bug in wnaf_fixed where the wnaf array is not completely zeroed when given a 0 scalar. (Jonas Nick)
96f68a0 Don't invert scalar in wnaf_fixed when it is even because a caller might intentionally give a scalar with many leading zeros. (Jonas Nick)
6dbb007 Increase sparsity of pippenger fixed window naf representation (Jonas Nick)

Pull request description:

  Fixes bitcoin-core#506

Tree-SHA512: 49a237a7d09c0c376ba4e6b1f522b9aff2517e420dfef9df810fd5ba920e0b98be8fe3f730b32e41b4aef475bc4cf3b13220024bd8d6f40c2744e6f392ff97a8

Author: sipa

src/ecmult_impl.h

@@ -563,53 +563,66 @@ static size_t secp256k1_strauss_max_points(secp256k1_scratch *scratch) {
  *  It has the following guarantees:
  *  - each wnaf[i] is either 0 or an odd integer between -(1 << w) and (1 << w)
  *  - the number of words set is always WNAF_SIZE(w)
- *  - the returned skew is 0 without endomorphism, or 0 or 1 with endomorphism
+ *  - the returned skew is 0 or 1
  */
 static int secp256k1_wnaf_fixed(int *wnaf, const secp256k1_scalar *s, int w) {
-    int sign = 0;
     int skew = 0;
-    int pos = 1;
-#ifndef USE_ENDOMORPHISM
-    secp256k1_scalar neg_s;
-#endif
+    int pos;
+    int max_pos;
+    int last_w;
     const secp256k1_scalar *work = s;
 
     if (secp256k1_scalar_is_zero(s)) {
-        while (pos * w < WNAF_BITS) {
+        for (pos = 0; pos < WNAF_SIZE(w); pos++) {
             wnaf[pos] = 0;
-            ++pos;
         }
         return 0;
     }
 
     if (secp256k1_scalar_is_even(s)) {
-#ifdef USE_ENDOMORPHISM
         skew = 1;
-#else
-        secp256k1_scalar_negate(&neg_s, s);
-        work = &neg_s;
-        sign = -1;
-#endif
     }
 
-    wnaf[0] = (secp256k1_scalar_get_bits_var(work, 0, w) + skew + sign) ^ sign;
+    wnaf[0] = secp256k1_scalar_get_bits_var(work, 0, w) + skew;
+    /* Compute last window size. Relevant when window size doesn't divide the
+     * number of bits in the scalar */
+    last_w = WNAF_BITS - (WNAF_SIZE(w) - 1) * w;
 
-    while (pos * w < WNAF_BITS) {
-        int now = w;
-        int val;
-        if (now + pos * w > WNAF_BITS) {
-            now = WNAF_BITS - pos * w;
+    /* Store the position of the first nonzero word in max_pos to allow
+     * skipping leading zeros when calculating the wnaf. */
+    for (pos = WNAF_SIZE(w) - 1; pos > 0; pos--) {
+        int val = secp256k1_scalar_get_bits_var(work, pos * w, pos == WNAF_SIZE(w)-1 ? last_w : w);
+        if(val != 0) {
+            break;
         }
-        val = secp256k1_scalar_get_bits_var(work, pos * w, now);
+        wnaf[pos] = 0;
+    }
+    max_pos = pos;
+    pos = 1;
+
+    while (pos <= max_pos) {
+        int val = secp256k1_scalar_get_bits_var(work, pos * w, pos == WNAF_SIZE(w)-1 ? last_w : w);
         if ((val & 1) == 0) {
-            wnaf[pos - 1] -= ((1 << w) + sign) ^ sign;
-            wnaf[pos] = (val + 1 + sign) ^ sign;
+            wnaf[pos - 1] -= (1 << w);
+            wnaf[pos] = (val + 1);
         } else {
-            wnaf[pos] = (val + sign) ^ sign;
+            wnaf[pos] = val;
+        }
+        /* Set a coefficient to zero if it is 1 or -1 and the proceeding digit
+         * is strictly negative or strictly positive respectively. Only change
+         * coefficients at previous positions because above code assumes that
+         * wnaf[pos - 1] is odd.
+         */
+        if (pos >= 2 && ((wnaf[pos - 1] == 1 && wnaf[pos - 2] < 0) || (wnaf[pos - 1] == -1 && wnaf[pos - 2] > 0))) {
+            if (wnaf[pos - 1] == 1) {
+                wnaf[pos - 2] += 1 << w;
+            } else {
+                wnaf[pos - 2] -= 1 << w;
+            }
+            wnaf[pos - 1] = 0;
         }
         ++pos;
     }
-    VERIFY_CHECK(pos == WNAF_SIZE(w));
 
     return skew;
 }
@@ -665,7 +678,6 @@ static int secp256k1_ecmult_pippenger_wnaf(secp256k1_gej *buckets, int bucket_wi
             secp256k1_ge tmp;
             int idx;
 
-#ifdef USE_ENDOMORPHISM
             if (i == 0) {
                 /* correct for wnaf skew */
                 int skew = point_state.skew_na;
@@ -674,7 +686,6 @@ static int secp256k1_ecmult_pippenger_wnaf(secp256k1_gej *buckets, int bucket_wi
                     secp256k1_gej_add_ge_var(&buckets[0], &buckets[0], &tmp, NULL);
                 }
             }
-#endif
             if (n > 0) {
                 idx = (n - 1)/2;
                 secp256k1_gej_add_ge_var(&buckets[idx], &buckets[idx], &pt[point_state.input_pos], NULL);

src/tests.c

@@ -3022,8 +3022,7 @@ void test_fixed_wnaf(const secp256k1_scalar *number, int w) {
     for (i = WNAF_SIZE(w)-1; i >= 0; --i) {
         secp256k1_scalar t;
         int v = wnaf[i];
-        CHECK(v != 0); /* check nonzero */
-        CHECK(v & 1);  /* check parity */
+        CHECK(v == 0 || v & 1);  /* check parity */
         CHECK(v > -(1 << w)); /* check range above */
         CHECK(v < (1 << w));  /* check range below */
 
@@ -3041,20 +3040,70 @@ void test_fixed_wnaf(const secp256k1_scalar *number, int w) {
     CHECK(secp256k1_scalar_eq(&x, &num));
 }
 
-void test_fixed_wnaf_zero(int w) {
+/* Checks that the first 8 elements of wnaf are equal to wnaf_expected and the
+ * rest is 0.*/
+void test_fixed_wnaf_small_helper(int *wnaf, int *wnaf_expected, int w) {
+    int i;
+    for (i = WNAF_SIZE(w)-1; i >= 8; --i) {
+        CHECK(wnaf[i] == 0);
+    }
+    for (i = 7; i >= 0; --i) {
+        CHECK(wnaf[i] == wnaf_expected[i]);
+    }
+}
+
+void test_fixed_wnaf_small(void) {
+    int w = 4;
     int wnaf[256] = {0};
     int i;
     int skew;
     secp256k1_scalar num;
 
     secp256k1_scalar_set_int(&num, 0);
     skew = secp256k1_wnaf_fixed(wnaf, &num, w);
-
     for (i = WNAF_SIZE(w)-1; i >= 0; --i) {
         int v = wnaf[i];
         CHECK(v == 0);
     }
     CHECK(skew == 0);
+
+    secp256k1_scalar_set_int(&num, 1);
+    skew = secp256k1_wnaf_fixed(wnaf, &num, w);
+    for (i = WNAF_SIZE(w)-1; i >= 1; --i) {
+        int v = wnaf[i];
+        CHECK(v == 0);
+    }
+    CHECK(wnaf[0] == 1);
+    CHECK(skew == 0);
+
+    {
+        int wnaf_expected[8] = { 0xf, 0xf, 0xf, 0xf, 0xf, 0xf, 0xf, 0xf };
+        secp256k1_scalar_set_int(&num, 0xffffffff);
+        skew = secp256k1_wnaf_fixed(wnaf, &num, w);
+        test_fixed_wnaf_small_helper(wnaf, wnaf_expected, w);
+        CHECK(skew == 0);
+    }
+    {
+        int wnaf_expected[8] = { -1, -1, -1, -1, -1, -1, -1, 0xf };
+        secp256k1_scalar_set_int(&num, 0xeeeeeeee);
+        skew = secp256k1_wnaf_fixed(wnaf, &num, w);
+        test_fixed_wnaf_small_helper(wnaf, wnaf_expected, w);
+        CHECK(skew == 1);
+    }
+    {
+        int wnaf_expected[8] = { 1, 0, 1, 0, 1, 0, 1, 0 };
+        secp256k1_scalar_set_int(&num, 0x01010101);
+        skew = secp256k1_wnaf_fixed(wnaf, &num, w);
+        test_fixed_wnaf_small_helper(wnaf, wnaf_expected, w);
+        CHECK(skew == 0);
+    }
+    {
+        int wnaf_expected[8] = { -0xf, 0, 0xf, -0xf, 0, 0xf, 1, 0 };
+        secp256k1_scalar_set_int(&num, 0x01ef1ef1);
+        skew = secp256k1_wnaf_fixed(wnaf, &num, w);
+        test_fixed_wnaf_small_helper(wnaf, wnaf_expected, w);
+        CHECK(skew == 0);
+    }
 }
 
 void run_wnaf(void) {
@@ -3068,7 +3117,7 @@ void run_wnaf(void) {
     n.d[0] = 2;
     test_constant_wnaf(&n, 4);
     /* Test 0 */
-    test_fixed_wnaf_zero(4);
+    test_fixed_wnaf_small();
     /* Random tests */
     for (i = 0; i < count; i++) {
         random_scalar_order(&n);