Add secp256k1_scalar_half for halving scalars (+ tests/benchmarks).

Co-authored-by: Jonas Nick <[email protected]>
Co-authored-by: Tim Ruffing <[email protected]>

Author: 3 people

src/bench_internal.c

@@ -98,6 +98,18 @@ static void bench_scalar_negate(void* arg, int iters) {
     }
 }
 
+static void bench_scalar_half(void* arg, int iters) {
+    int i;
+    bench_inv *data = (bench_inv*)arg;
+    secp256k1_scalar s = data->scalar[0];
+
+    for (i = 0; i < iters; i++) {
+        secp256k1_scalar_half(&s, &s);
+    }
+
+    data->scalar[0] = s;
+}
+
 static void bench_scalar_mul(void* arg, int iters) {
     int i;
     bench_inv *data = (bench_inv*)arg;
@@ -370,6 +382,7 @@ int main(int argc, char **argv) {
     int d = argc == 1; /* default */
     print_output_table_header_row();
 
+    if (d || have_flag(argc, argv, "scalar") || have_flag(argc, argv, "half")) run_benchmark("scalar_half", bench_scalar_half, bench_setup, NULL, &data, 10, iters*100);
     if (d || have_flag(argc, argv, "scalar") || have_flag(argc, argv, "add")) run_benchmark("scalar_add", bench_scalar_add, bench_setup, NULL, &data, 10, iters*100);
     if (d || have_flag(argc, argv, "scalar") || have_flag(argc, argv, "negate")) run_benchmark("scalar_negate", bench_scalar_negate, bench_setup, NULL, &data, 10, iters*100);
     if (d || have_flag(argc, argv, "scalar") || have_flag(argc, argv, "mul")) run_benchmark("scalar_mul", bench_scalar_mul, bench_setup, NULL, &data, 10, iters*10);

src/scalar.h

@@ -67,6 +67,9 @@ static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_sc
 /** Compute the complement of a scalar (modulo the group order). */
 static void secp256k1_scalar_negate(secp256k1_scalar *r, const secp256k1_scalar *a);
 
+/** Multiply a scalar with the multiplicative inverse of 2. */
+static void secp256k1_scalar_half(secp256k1_scalar *r, const secp256k1_scalar *a);
+
 /** Check whether a scalar equals zero. */
 static int secp256k1_scalar_is_zero(const secp256k1_scalar *a);
 

src/scalar_4x64_impl.h

@@ -199,6 +199,47 @@ static void secp256k1_scalar_negate(secp256k1_scalar *r, const secp256k1_scalar
     secp256k1_scalar_verify(r);
 }
 
+static void secp256k1_scalar_half(secp256k1_scalar *r, const secp256k1_scalar *a) {
+    /* Writing `/` for field division and `//` for integer division, we compute
+     *
+     *   a/2 = (a - (a&1))/2 + (a&1)/2
+     *       = (a >> 1) + (a&1 ?    1/2 : 0)
+     *       = (a >> 1) + (a&1 ? n//2+1 : 0),
+     *
+     * where n is the group order and in the last equality we have used 1/2 = n//2+1 (mod n).
+     * For n//2, we have the constants SECP256K1_N_H_0, ...
+     *
+     * This sum does not overflow. The most extreme case is a = -2, the largest odd scalar. Here:
+     * - the left summand is:  a >> 1 = (a - a&1)/2 = (n-2-1)//2           = (n-3)//2
+     * - the right summand is: a&1 ? n//2+1 : 0 = n//2+1 = (n-1)//2 + 2//2 = (n+1)//2
+     * Together they sum to (n-3)//2 + (n+1)//2 = (2n-2)//2 = n - 1, which is less than n.
+     */
+    uint64_t mask = -(uint64_t)(a->d[0] & 1U);
+    secp256k1_uint128 t;
+    secp256k1_scalar_verify(a);
+
+    secp256k1_u128_from_u64(&t, (a->d[0] >> 1) | (a->d[1] << 63));
+    secp256k1_u128_accum_u64(&t, (SECP256K1_N_H_0 + 1U) & mask);
+    r->d[0] = secp256k1_u128_to_u64(&t); secp256k1_u128_rshift(&t, 64);
+    secp256k1_u128_accum_u64(&t, (a->d[1] >> 1) | (a->d[2] << 63));
+    secp256k1_u128_accum_u64(&t, SECP256K1_N_H_1 & mask);
+    r->d[1] = secp256k1_u128_to_u64(&t); secp256k1_u128_rshift(&t, 64);
+    secp256k1_u128_accum_u64(&t, (a->d[2] >> 1) | (a->d[3] << 63));
+    secp256k1_u128_accum_u64(&t, SECP256K1_N_H_2 & mask);
+    r->d[2] = secp256k1_u128_to_u64(&t); secp256k1_u128_rshift(&t, 64);
+    r->d[3] = secp256k1_u128_to_u64(&t) + (a->d[3] >> 1) + (SECP256K1_N_H_3 & mask);
+#ifdef VERIFY
+    /* The line above only computed the bottom 64 bits of r->d[3]; redo the computation
+     * in full 128 bits to make sure the top 64 bits are indeed zero. */
+    secp256k1_u128_accum_u64(&t, a->d[3] >> 1);
+    secp256k1_u128_accum_u64(&t, SECP256K1_N_H_3 & mask);
+    secp256k1_u128_rshift(&t, 64);
+    VERIFY_CHECK(secp256k1_u128_to_u64(&t) == 0);
+
+    secp256k1_scalar_verify(r);
+#endif
+}
+
 SECP256K1_INLINE static int secp256k1_scalar_is_one(const secp256k1_scalar *a) {
     secp256k1_scalar_verify(a);
 

src/scalar_8x32_impl.h

@@ -245,6 +245,55 @@ static void secp256k1_scalar_negate(secp256k1_scalar *r, const secp256k1_scalar
     secp256k1_scalar_verify(r);
 }
 
+static void secp256k1_scalar_half(secp256k1_scalar *r, const secp256k1_scalar *a) {
+    /* Writing `/` for field division and `//` for integer division, we compute
+     *
+     *   a/2 = (a - (a&1))/2 + (a&1)/2
+     *       = (a >> 1) + (a&1 ?    1/2 : 0)
+     *       = (a >> 1) + (a&1 ? n//2+1 : 0),
+     *
+     * where n is the group order and in the last equality we have used 1/2 = n//2+1 (mod n).
+     * For n//2, we have the constants SECP256K1_N_H_0, ...
+     *
+     * This sum does not overflow. The most extreme case is a = -2, the largest odd scalar. Here:
+     * - the left summand is:  a >> 1 = (a - a&1)/2 = (n-2-1)//2           = (n-3)//2
+     * - the right summand is: a&1 ? n//2+1 : 0 = n//2+1 = (n-1)//2 + 2//2 = (n+1)//2
+     * Together they sum to (n-3)//2 + (n+1)//2 = (2n-2)//2 = n - 1, which is less than n.
+     */
+    uint32_t mask = -(uint32_t)(a->d[0] & 1U);
+    uint64_t t = (uint32_t)((a->d[0] >> 1) | (a->d[1] << 31));
+    secp256k1_scalar_verify(a);
+
+    t += (SECP256K1_N_H_0 + 1U) & mask;
+    r->d[0] = t; t >>= 32;
+    t += (uint32_t)((a->d[1] >> 1) | (a->d[2] << 31));
+    t += SECP256K1_N_H_1 & mask;
+    r->d[1] = t; t >>= 32;
+    t += (uint32_t)((a->d[2] >> 1) | (a->d[3] << 31));
+    t += SECP256K1_N_H_2 & mask;
+    r->d[2] = t; t >>= 32;
+    t += (uint32_t)((a->d[3] >> 1) | (a->d[4] << 31));
+    t += SECP256K1_N_H_3 & mask;
+    r->d[3] = t; t >>= 32;
+    t += (uint32_t)((a->d[4] >> 1) | (a->d[5] << 31));
+    t += SECP256K1_N_H_4 & mask;
+    r->d[4] = t; t >>= 32;
+    t += (uint32_t)((a->d[5] >> 1) | (a->d[6] << 31));
+    t += SECP256K1_N_H_5 & mask;
+    r->d[5] = t; t >>= 32;
+    t += (uint32_t)((a->d[6] >> 1) | (a->d[7] << 31));
+    t += SECP256K1_N_H_6 & mask;
+    r->d[6] = t; t >>= 32;
+    r->d[7] = (uint32_t)t + (uint32_t)(a->d[7] >> 1) + (SECP256K1_N_H_7 & mask);
+#ifdef VERIFY
+    /* The line above only computed the bottom 32 bits of r->d[7]. Redo the computation
+     * in full 64 bits to make sure the top 32 bits are indeed zero. */
+    VERIFY_CHECK((t + (a->d[7] >> 1) + (SECP256K1_N_H_7 & mask)) >> 32 == 0);
+
+    secp256k1_scalar_verify(r);
+#endif
+}
+
 SECP256K1_INLINE static int secp256k1_scalar_is_one(const secp256k1_scalar *a) {
     secp256k1_scalar_verify(a);
 

src/scalar_low_impl.h

@@ -205,4 +205,12 @@ static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_sc
     secp256k1_scalar_verify(r);
 }
 
+static void secp256k1_scalar_half(secp256k1_scalar *r, const secp256k1_scalar *a) {
+    secp256k1_scalar_verify(a);
+
+    *r = (*a + ((-(uint32_t)(*a & 1)) & EXHAUSTIVE_TEST_ORDER)) >> 1;
+
+    secp256k1_scalar_verify(r);
+}
+
 #endif /* SECP256K1_SCALAR_REPR_IMPL_H */

src/tests.c

@@ -2285,6 +2285,13 @@ static void scalar_test(void) {
         CHECK(secp256k1_scalar_eq(&r1, &secp256k1_scalar_zero));
     }
 
+    {
+        /* Test halving. */
+        secp256k1_scalar r;
+        secp256k1_scalar_add(&r, &s, &s);
+        secp256k1_scalar_half(&r, &r);
+        CHECK(secp256k1_scalar_eq(&r, &s));
+    }
 }
 
 static void run_scalar_set_b32_seckey_tests(void) {
@@ -2337,6 +2344,38 @@ static void run_scalar_tests(void) {
         CHECK(secp256k1_scalar_is_zero(&o));
     }
 
+    {
+        /* Test that halving and doubling roundtrips on some fixed values. */
+        static const secp256k1_scalar HALF_TESTS[] = {
+            /* 0 */
+            SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0),
+            /* 1 */
+            SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 1),
+            /* -1 */
+            SECP256K1_SCALAR_CONST(0xfffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffeul, 0xbaaedce6ul, 0xaf48a03bul, 0xbfd25e8cul, 0xd0364140ul),
+            /* -2 (largest odd value) */
+            SECP256K1_SCALAR_CONST(0xfffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffeul, 0xbaaedce6ul, 0xaf48a03bul, 0xbfd25e8cul, 0xd036413Ful),
+            /* Half the secp256k1 order */
+            SECP256K1_SCALAR_CONST(0x7ffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful, 0x5d576e73ul, 0x57a4501dul, 0xdfe92f46ul, 0x681b20a0ul),
+            /* Half the secp256k1 order + 1 */
+            SECP256K1_SCALAR_CONST(0x7ffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful, 0x5d576e73ul, 0x57a4501dul, 0xdfe92f46ul, 0x681b20a1ul),
+            /* 2^255 */
+            SECP256K1_SCALAR_CONST(0x80000000ul, 0, 0, 0, 0, 0, 0, 0),
+            /* 2^255 - 1 */
+            SECP256K1_SCALAR_CONST(0x7ffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful, 0xfffffffful),
+        };
+        unsigned n;
+        for (n = 0; n < sizeof(HALF_TESTS) / sizeof(HALF_TESTS[0]); ++n) {
+            secp256k1_scalar s;
+            secp256k1_scalar_half(&s, &HALF_TESTS[n]);
+            secp256k1_scalar_add(&s, &s, &s);
+            CHECK(secp256k1_scalar_eq(&s, &HALF_TESTS[n]));
+            secp256k1_scalar_add(&s, &s, &s);
+            secp256k1_scalar_half(&s, &s);
+            CHECK(secp256k1_scalar_eq(&s, &HALF_TESTS[n]));
+        }
+    }
+
     {
         /* Does check_overflow check catch all ones? */
         static const secp256k1_scalar overflowed = SECP256K1_SCALAR_CONST(