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Add equalities relating input and output variables

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This commit is contained in:
Pieter Wuille 2014-11-13 07:00:44 -08:00
parent 5dd421bab5
commit fa0d620668
2 changed files with 194 additions and 0 deletions
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132
src/field_10x26_impl.h
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@ -248,6 +248,9 @@ SECP256K1_INLINE static void secp256k1_fe_add(secp256k1_fe_t *r, const secp256k1
SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uint32_t *b, uint32_t *r) {
const uint32_t M = 0x3FFFFFFUL, R0 = 0x3D10UL, R1 = 0x400UL;
// [... a b c] is a shorthand for ... + a<<52 + b<<26 + c<<0 mod n.
// px is a shorthand for sum(a[i]*b[x-i], i=0..x).
// Note that [x 0 0 0 0 0 0 0 0 0 0] = [x*R1 x*R0].
uint64_t c, d;
@ -261,9 +264,12 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[7] * b[2]
+ (uint64_t)a[8] * b[1]
+ (uint64_t)a[9] * b[0];
// [d 0 0 0 0 0 0 0 0 0] = [p9 0 0 0 0 0 0 0 0 0]
uint32_t t9 = d & M; d >>= 26;
// [d t9 0 0 0 0 0 0 0 0 0] = [p9 0 0 0 0 0 0 0 0 0]
c = (uint64_t)a[0] * b[0];
// [d t9 0 0 0 0 0 0 0 0 c] = [p9 0 0 0 0 0 0 0 0 p0]
d += (uint64_t)a[1] * b[9]
+ (uint64_t)a[2] * b[8]
+ (uint64_t)a[3] * b[7]
@ -273,11 +279,16 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[7] * b[3]
+ (uint64_t)a[8] * b[2]
+ (uint64_t)a[9] * b[1];
// [d t9 0 0 0 0 0 0 0 0 c] = [p10 p9 0 0 0 0 0 0 0 0 p0]
uint64_t u0 = d & M; d >>= 26; c += u0 * R0;
// [d u0 t9 0 0 0 0 0 0 0 0 c-u0*R0] = [p10 p9 0 0 0 0 0 0 0 0 p0]
uint32_t t0 = c & M; c >>= 26; c += u0 * R1;
// [d u0 t9 0 0 0 0 0 0 0 c-u0*R1 t0-u0*R0] = [p10 p9 0 0 0 0 0 0 0 0 p0]
// [d 0 t9 0 0 0 0 0 0 0 c t0] = [p10 p9 0 0 0 0 0 0 0 0 p0]
c += (uint64_t)a[0] * b[1]
+ (uint64_t)a[1] * b[0];
// [d 0 t9 0 0 0 0 0 0 0 c t0] = [p10 p9 0 0 0 0 0 0 0 p1 p0]
d += (uint64_t)a[2] * b[9]
+ (uint64_t)a[3] * b[8]
+ (uint64_t)a[4] * b[7]
@ -286,12 +297,17 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[7] * b[4]
+ (uint64_t)a[8] * b[3]
+ (uint64_t)a[9] * b[2];
// [d 0 t9 0 0 0 0 0 0 0 c t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
uint64_t u1 = d & M; d >>= 26; c += u1 * R0;
// [d u1 0 t9 0 0 0 0 0 0 0 c-u1*R0 t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
uint32_t t1 = c & M; c >>= 26; c += u1 * R1;
// [d u1 0 t9 0 0 0 0 0 0 c-u1*R1 t1-u1*R0 t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
// [d 0 0 t9 0 0 0 0 0 0 c t1 t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
c += (uint64_t)a[0] * b[2]
+ (uint64_t)a[1] * b[1]
+ (uint64_t)a[2] * b[0];
// [d 0 0 t9 0 0 0 0 0 0 c t1 t0] = [p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
d += (uint64_t)a[3] * b[9]
+ (uint64_t)a[4] * b[8]
+ (uint64_t)a[5] * b[7]
@ -299,34 +315,48 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[7] * b[5]
+ (uint64_t)a[8] * b[4]
+ (uint64_t)a[9] * b[3];
// [d 0 0 t9 0 0 0 0 0 0 c t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
uint64_t u2 = d & M; d >>= 26; c += u2 * R0;
// [d u2 0 0 t9 0 0 0 0 0 0 c-u2*R0 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
uint32_t t2 = c & M; c >>= 26; c += u2 * R1;
// [d u2 0 0 t9 0 0 0 0 0 c-u2*R1 t2-u2*R0 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
// [d 0 0 0 t9 0 0 0 0 0 c t2 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
c += (uint64_t)a[0] * b[3]
+ (uint64_t)a[1] * b[2]
+ (uint64_t)a[2] * b[1]
+ (uint64_t)a[3] * b[0];
// [d 0 0 0 t9 0 0 0 0 0 c t2 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
d += (uint64_t)a[4] * b[9]
+ (uint64_t)a[5] * b[8]
+ (uint64_t)a[6] * b[7]
+ (uint64_t)a[7] * b[6]
+ (uint64_t)a[8] * b[5]
+ (uint64_t)a[9] * b[4];
// [d 0 0 0 t9 0 0 0 0 0 c t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
uint64_t u3 = d & M; d >>= 26; c += u3 * R0;
// [d u3 0 0 0 t9 0 0 0 0 0 c-u3*R0 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
uint32_t t3 = c & M; c >>= 26; c += u3 * R1;
// [d u3 0 0 0 t9 0 0 0 0 c-u3*R1 t3-u3*R0 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
// [d 0 0 0 0 t9 0 0 0 0 c t3 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
c += (uint64_t)a[0] * b[4]
+ (uint64_t)a[1] * b[3]
+ (uint64_t)a[2] * b[2]
+ (uint64_t)a[3] * b[1]
+ (uint64_t)a[4] * b[0];
// [d 0 0 0 0 t9 0 0 0 0 c t3 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
d += (uint64_t)a[5] * b[9]
+ (uint64_t)a[6] * b[8]
+ (uint64_t)a[7] * b[7]
+ (uint64_t)a[8] * b[6]
+ (uint64_t)a[9] * b[5];
// [d 0 0 0 0 t9 0 0 0 0 c t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
uint64_t u4 = d & M; d >>= 26; c += u4 * R0;
// [d u4 0 0 0 0 t9 0 0 0 0 c-u4*R0 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
uint32_t t4 = c & M; c >>= 26; c += u4 * R1;
// [d u4 0 0 0 0 t9 0 0 0 c-u4*R1 t4-u4*R0 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 t9 0 0 0 c t4 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
c += (uint64_t)a[0] * b[5]
+ (uint64_t)a[1] * b[4]
@ -334,12 +364,17 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[3] * b[2]
+ (uint64_t)a[4] * b[1]
+ (uint64_t)a[5] * b[0];
// [d 0 0 0 0 0 t9 0 0 0 c t4 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
d += (uint64_t)a[6] * b[9]
+ (uint64_t)a[7] * b[8]
+ (uint64_t)a[8] * b[7]
+ (uint64_t)a[9] * b[6];
// [d 0 0 0 0 0 t9 0 0 0 c t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
uint64_t u5 = d & M; d >>= 26; c += u5 * R0;
// [d u5 0 0 0 0 0 t9 0 0 0 c-u5*R0 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
uint32_t t5 = c & M; c >>= 26; c += u5 * R1;
// [d u5 0 0 0 0 0 t9 0 0 c-u5*R1 t5-u5*R0 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 t9 0 0 c t5 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
c += (uint64_t)a[0] * b[6]
+ (uint64_t)a[1] * b[5]
@ -348,11 +383,16 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[4] * b[2]
+ (uint64_t)a[5] * b[1]
+ (uint64_t)a[6] * b[0];
// [d 0 0 0 0 0 0 t9 0 0 c t5 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
d += (uint64_t)a[7] * b[9]
+ (uint64_t)a[8] * b[8]
+ (uint64_t)a[9] * b[7];
// [d 0 0 0 0 0 0 t9 0 0 c t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
uint64_t u6 = d & M; d >>= 26; c += u6 * R0;
// [d u6 0 0 0 0 0 0 t9 0 0 c-u6*R0 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
uint32_t t6 = c & M; c >>= 26; c += u6 * R1;
// [d u6 0 0 0 0 0 0 t9 0 c-u6*R1 t6-u6*R0 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 t9 0 c t6 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
c += (uint64_t)a[0] * b[7]
+ (uint64_t)a[1] * b[6]
@ -362,10 +402,15 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[5] * b[2]
+ (uint64_t)a[6] * b[1]
+ (uint64_t)a[7] * b[0];
// [d 0 0 0 0 0 0 0 t9 0 c t6 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
d += (uint64_t)a[8] * b[9]
+ (uint64_t)a[9] * b[8];
// [d 0 0 0 0 0 0 0 t9 0 c t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
uint64_t u7 = d & M; d >>= 26; c += u7 * R0;
// [d u7 0 0 0 0 0 0 0 t9 0 c-u7*R0 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
uint32_t t7 = c & M; c >>= 26; c += u7 * R1;
// [d u7 0 0 0 0 0 0 0 t9 c-u7*R1 t7-u7*R0 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 0 t9 c t7 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
c += (uint64_t)a[0] * b[8]
+ (uint64_t)a[1] * b[7]
@ -376,30 +421,54 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint32_t *a, const uin
+ (uint64_t)a[6] * b[2]
+ (uint64_t)a[7] * b[1]
+ (uint64_t)a[8] * b[0];
// [d 0 0 0 0 0 0 0 0 t9 c t7 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d += (uint64_t)a[9] * b[9];
// [d 0 0 0 0 0 0 0 0 t9 c t7 t6 t5 t4 t3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
uint64_t u8 = d & M; d >>= 26; c += u8 * R0;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 t5 t4 t3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[3] = t3;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 t5 t4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[4] = t4;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 t5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[5] = t5;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[6] = t6;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[7] = t7;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[8] = c & M; c >>= 26; c += u8 * R1;
// [d u8 0 0 0 0 0 0 0 0 t9+c-u8*R1 r8-u8*R0 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 0 0 t9+c r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += d * R0 + t9;
// [d 0 0 0 0 0 0 0 0 0 c-d*R0 r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[9] = c & (M >> 4); c >>= 22; c += d * (R1 << 4);
// [d 0 0 0 0 0 0 0 0 r9+((c-d*R1<<4)<<22)-d*R0 r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 -d*R1 r9+(c<<22)-d*R0 r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d = c * (R0 >> 4) + t0;
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 t1 d-c*R0>>4] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[0] = d & M; d >>= 26;
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 t1+d r0-c*R0>>4] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d += c * (R1 >> 4) + t1;
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 d-c*R1>>4 r0-c*R0>>4] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [r9 r8 r7 r6 r5 r4 r3 t2 d r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[1] = d & M; d >>= 26;
// [r9 r8 r7 r6 r5 r4 r3 t2+d r1 r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d += t2;
// [r9 r8 r7 r6 r5 r4 r3 d r1 r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[2] = d;
// [r9 r8 r7 r6 r5 r4 r3 r2 r1 r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
}
SECP256K1_INLINE static void secp256k1_fe_sqr_inner(const uint32_t *a, uint32_t *r) {
const uint32_t M = 0x3FFFFFFUL, R0 = 0x3D10UL, R1 = 0x400UL;
// [... a b c] is a shorthand for ... + a<<52 + b<<26 + c<<0 mod n.
// px is a shorthand for sum(a[i]*a[x-i], i=0..x).
// Note that [x 0 0 0 0 0 0 0 0 0 0] = [x*R1 x*R0].
uint64_t c, d;
@ -408,100 +477,163 @@ SECP256K1_INLINE static void secp256k1_fe_sqr_inner(const uint32_t *a, uint32_t
+ (uint64_t)(a[2]*2) * a[7]
+ (uint64_t)(a[3]*2) * a[6]
+ (uint64_t)(a[4]*2) * a[5];
// [d 0 0 0 0 0 0 0 0 0] = [p9 0 0 0 0 0 0 0 0 0]
uint32_t t9 = d & M; d >>= 26;
// [d t9 0 0 0 0 0 0 0 0 0] = [p9 0 0 0 0 0 0 0 0 0]
c = (uint64_t)a[0] * a[0];
// [d t9 0 0 0 0 0 0 0 0 c] = [p9 0 0 0 0 0 0 0 0 p0]
d += (uint64_t)(a[1]*2) * a[9]
+ (uint64_t)(a[2]*2) * a[8]
+ (uint64_t)(a[3]*2) * a[7]
+ (uint64_t)(a[4]*2) * a[6]
+ (uint64_t)a[5] * a[5];
// [d t9 0 0 0 0 0 0 0 0 c] = [p10 p9 0 0 0 0 0 0 0 0 p0]
uint64_t u0 = d & M; d >>= 26; c += u0 * R0;
// [d u0 t9 0 0 0 0 0 0 0 0 c-u0*R0] = [p10 p9 0 0 0 0 0 0 0 0 p0]
uint32_t t0 = c & M; c >>= 26; c += u0 * R1;
// [d u0 t9 0 0 0 0 0 0 0 c-u0*R1 t0-u0*R0] = [p10 p9 0 0 0 0 0 0 0 0 p0]
// [d 0 t9 0 0 0 0 0 0 0 c t0] = [p10 p9 0 0 0 0 0 0 0 0 p0]
c += (uint64_t)(a[0]*2) * a[1];
// [d 0 t9 0 0 0 0 0 0 0 c t0] = [p10 p9 0 0 0 0 0 0 0 p1 p0]
d += (uint64_t)(a[2]*2) * a[9]
+ (uint64_t)(a[3]*2) * a[8]
+ (uint64_t)(a[4]*2) * a[7]
+ (uint64_t)(a[5]*2) * a[6];
// [d 0 t9 0 0 0 0 0 0 0 c t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
uint64_t u1 = d & M; d >>= 26; c += u1 * R0;
// [d u1 0 t9 0 0 0 0 0 0 0 c-u1*R0 t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
uint32_t t1 = c & M; c >>= 26; c += u1 * R1;
// [d u1 0 t9 0 0 0 0 0 0 c-u1*R1 t1-u1*R0 t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
// [d 0 0 t9 0 0 0 0 0 0 c t1 t0] = [p11 p10 p9 0 0 0 0 0 0 0 p1 p0]
c += (uint64_t)(a[0]*2) * a[2]
+ (uint64_t)a[1] * a[1];
// [d 0 0 t9 0 0 0 0 0 0 c t1 t0] = [p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
d += (uint64_t)(a[3]*2) * a[9]
+ (uint64_t)(a[4]*2) * a[8]
+ (uint64_t)(a[5]*2) * a[7]
+ (uint64_t)a[6] * a[6];
// [d 0 0 t9 0 0 0 0 0 0 c t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
uint64_t u2 = d & M; d >>= 26; c += u2 * R0;
// [d u2 0 0 t9 0 0 0 0 0 0 c-u2*R0 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
uint32_t t2 = c & M; c >>= 26; c += u2 * R1;
// [d u2 0 0 t9 0 0 0 0 0 c-u2*R1 t2-u2*R0 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
// [d 0 0 0 t9 0 0 0 0 0 c t2 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 0 p2 p1 p0]
c += (uint64_t)(a[0]*2) * a[3]
+ (uint64_t)(a[1]*2) * a[2];
// [d 0 0 0 t9 0 0 0 0 0 c t2 t1 t0] = [p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
d += (uint64_t)(a[4]*2) * a[9]
+ (uint64_t)(a[5]*2) * a[8]
+ (uint64_t)(a[6]*2) * a[7];
// [d 0 0 0 t9 0 0 0 0 0 c t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
uint64_t u3 = d & M; d >>= 26; c += u3 * R0;
// [d u3 0 0 0 t9 0 0 0 0 0 c-u3*R0 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
uint32_t t3 = c & M; c >>= 26; c += u3 * R1;
// [d u3 0 0 0 t9 0 0 0 0 c-u3*R1 t3-u3*R0 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
// [d 0 0 0 0 t9 0 0 0 0 c t3 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 0 p3 p2 p1 p0]
c += (uint64_t)(a[0]*2) * a[4]
+ (uint64_t)(a[1]*2) * a[3]
+ (uint64_t)a[2] * a[2];
// [d 0 0 0 0 t9 0 0 0 0 c t3 t2 t1 t0] = [p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
d += (uint64_t)(a[5]*2) * a[9]
+ (uint64_t)(a[6]*2) * a[8]
+ (uint64_t)a[7] * a[7];
// [d 0 0 0 0 t9 0 0 0 0 c t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
uint64_t u4 = d & M; d >>= 26; c += u4 * R0;
// [d u4 0 0 0 0 t9 0 0 0 0 c-u4*R0 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
uint32_t t4 = c & M; c >>= 26; c += u4 * R1;
// [d u4 0 0 0 0 t9 0 0 0 c-u4*R1 t4-u4*R0 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 t9 0 0 0 c t4 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 0 p4 p3 p2 p1 p0]
c += (uint64_t)(a[0]*2) * a[5]
+ (uint64_t)(a[1]*2) * a[4]
+ (uint64_t)(a[2]*2) * a[3];
// [d 0 0 0 0 0 t9 0 0 0 c t4 t3 t2 t1 t0] = [p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
d += (uint64_t)(a[6]*2) * a[9]
+ (uint64_t)(a[7]*2) * a[8];
// [d 0 0 0 0 0 t9 0 0 0 c t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
uint64_t u5 = d & M; d >>= 26; c += u5 * R0;
// [d u5 0 0 0 0 0 t9 0 0 0 c-u5*R0 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
uint32_t t5 = c & M; c >>= 26; c += u5 * R1;
// [d u5 0 0 0 0 0 t9 0 0 c-u5*R1 t5-u5*R0 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 t9 0 0 c t5 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 0 p5 p4 p3 p2 p1 p0]
c += (uint64_t)(a[0]*2) * a[6]
+ (uint64_t)(a[1]*2) * a[5]
+ (uint64_t)(a[2]*2) * a[4]
+ (uint64_t)a[3] * a[3];
// [d 0 0 0 0 0 0 t9 0 0 c t5 t4 t3 t2 t1 t0] = [p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
d += (uint64_t)(a[7]*2) * a[9]
+ (uint64_t)a[8] * a[8];
// [d 0 0 0 0 0 0 t9 0 0 c t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
uint64_t u6 = d & M; d >>= 26; c += u6 * R0;
// [d u6 0 0 0 0 0 0 t9 0 0 c-u6*R0 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
uint32_t t6 = c & M; c >>= 26; c += u6 * R1;
// [d u6 0 0 0 0 0 0 t9 0 c-u6*R1 t6-u6*R0 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 t9 0 c t6 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 0 p6 p5 p4 p3 p2 p1 p0]
c += (uint64_t)(a[0]*2) * a[7]
+ (uint64_t)(a[1]*2) * a[6]
+ (uint64_t)(a[2]*2) * a[5]
+ (uint64_t)(a[3]*2) * a[4];
// [d 0 0 0 0 0 0 0 t9 0 c t6 t5 t4 t3 t2 t1 t0] = [p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
d += (uint64_t)(a[8]*2) * a[9];
// [d 0 0 0 0 0 0 0 t9 0 c t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
uint64_t u7 = d & M; d >>= 26; c += u7 * R0;
// [d u7 0 0 0 0 0 0 0 t9 0 c-u7*R0 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
uint32_t t7 = c & M; c >>= 26; c += u7 * R1;
// [d u7 0 0 0 0 0 0 0 t9 c-u7*R1 t7-u7*R0 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 0 t9 c t7 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 0 p7 p6 p5 p4 p3 p2 p1 p0]
c += (uint64_t)(a[0]*2) * a[8]
+ (uint64_t)(a[1]*2) * a[7]
+ (uint64_t)(a[2]*2) * a[6]
+ (uint64_t)(a[3]*2) * a[5]
+ (uint64_t)a[4] * a[4];
// [d 0 0 0 0 0 0 0 0 t9 c t7 t6 t5 t4 t3 t2 t1 t0] = [p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d += (uint64_t)a[9] * a[9];
// [d 0 0 0 0 0 0 0 0 t9 c t7 t6 t5 t4 t3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
uint64_t u8 = d & M; d >>= 26; c += u8 * R0;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 t5 t4 t3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[3] = t3;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 t5 t4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[4] = t4;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 t5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[5] = t5;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 t6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[6] = t6;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 t7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[7] = t7;
// [d u8 0 0 0 0 0 0 0 0 t9 c-u8*R0 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[8] = c & M; c >>= 26; c += u8 * R1;
// [d u8 0 0 0 0 0 0 0 0 t9+c-u8*R1 r8-u8*R0 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 0 0 t9+c r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += d * R0 + t9;
// [d 0 0 0 0 0 0 0 0 0 c-d*R0 r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[9] = c & (M >> 4); c >>= 22; c += d * (R1 << 4);
// [d 0 0 0 0 0 0 0 0 r9+((c-d*R1<<4)<<22)-d*R0 r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [d 0 0 0 0 0 0 0 -d*R1 r9+(c<<22)-d*R0 r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 t1 t0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d = c * (R0 >> 4) + t0;
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 t1 d-c*R0>>4] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[0] = d & M; d >>= 26;
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 t1+d r0-c*R0>>4] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d += c * (R1 >> 4) + t1;
// [r9+(c<<22) r8 r7 r6 r5 r4 r3 t2 d-c*R1>>4 r0-c*R0>>4] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
// [r9 r8 r7 r6 r5 r4 r3 t2 d r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[1] = d & M; d >>= 26;
// [r9 r8 r7 r6 r5 r4 r3 t2+d r1 r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
d += t2;
// [r9 r8 r7 r6 r5 r4 r3 d r1 r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[2] = d;
// [r9 r8 r7 r6 r5 r4 r3 r2 r1 r0] = [p18 p17 p16 p15 p14 p13 p12 p11 p10 p9 p8 p7 p6 p5 p4 p3 p2 p1 p0]
}

62
src/field_5x52_int128_impl.h
View file

@ -10,6 +10,9 @@
SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uint64_t *b, uint64_t *r) {
const uint64_t M = 0xFFFFFFFFFFFFFULL, R = 0x1000003D10ULL;
// [... a b c] is a shorthand for ... + a<<104 + b<<52 + c<<0 mod n.
// px is a shorthand for sum(a[i]*b[x-i], i=0..x).
// Note that [x 0 0 0 0 0] = [x*R].
__int128 c, d;
@ -17,56 +20,88 @@ SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uin
+ (__int128)a[1] * b[2]
+ (__int128)a[2] * b[1]
+ (__int128)a[3] * b[0];
// [d 0 0 0] = [p3 0 0 0]
c = (__int128)a[4] * b[4];
// [c 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
d += (c & M) * R; c >>= 52;
// [c 0 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
uint64_t t3 = d & M; d >>= 52;
// [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
d += (__int128)a[0] * b[4]
+ (__int128)a[1] * b[3]
+ (__int128)a[2] * b[2]
+ (__int128)a[3] * b[1]
+ (__int128)a[4] * b[0];
// [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
d += c * R;
// [d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t t4 = d & M; d >>= 52;
// [d t4 t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t tx = (t4 >> 48); t4 &= (M >> 4);
// [d t4+(tx<<48) t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
c = (__int128)a[0] * b[0];
// [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 0 p4 p3 0 0 p0]
d += (__int128)a[1] * b[4]
+ (__int128)a[2] * b[3]
+ (__int128)a[3] * b[2]
+ (__int128)a[4] * b[1];
// [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
uint64_t u0 = d & M; d >>= 52;
// [d u0 t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
// [d 0 t4+(tx<<48)+(u0<<52) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
u0 = (u0 << 4) | tx;
// [d 0 t4+(u0<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
c += (__int128)u0 * (R >> 4);
// [d 0 t4 t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
uint64_t t0 = c & M; c >>= 52;
// [d 0 t4 t3 0 c t0] = [p8 0 0 p5 p4 p3 0 0 p0]
c += (__int128)a[0] * b[1]
+ (__int128)a[1] * b[0];
// [d 0 t4 t3 0 c t0] = [p8 0 0 p5 p4 p3 0 p1 p0]
d += (__int128)a[2] * b[4]
+ (__int128)a[3] * b[3]
+ (__int128)a[4] * b[2];
// [d 0 t4 t3 0 c t0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
c += (d & M) * R; d >>= 52;
// [d 0 0 t4 t3 0 c t0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
uint64_t t1 = c & M; c >>= 52;
// [d 0 0 t4 t3 c t1 t0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
c += (__int128)a[0] * b[2]
+ (__int128)a[1] * b[1]
+ (__int128)a[2] * b[0];
// [d 0 0 t4 t3 c t1 t0] = [p8 0 p6 p5 p4 p3 p2 p1 p0]
d += (__int128)a[3] * b[4]
+ (__int128)a[4] * b[3];
// [d 0 0 t4 t3 c t1 t0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += (d & M) * R; d >>= 52;
// [d 0 0 0 t4 t3 c t1 t0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[0] = t0;
// [d 0 0 0 t4 t3 c t1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[1] = t1;
// [d 0 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[2] = c & M; c >>= 52;
// [d 0 0 0 t4 t3+c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += d * R + t3;;
// [t4 c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[3] = c & M; c >>= 52;
// [t4+c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += t4;
// [c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[4] = c;
// [r4 r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
}
SECP256K1_INLINE static void secp256k1_fe_sqr_inner(const uint64_t *a, uint64_t *r) {
const uint64_t M = 0xFFFFFFFFFFFFFULL, R = 0x1000003D10ULL;
// [... a b c] is a shorthand for ... + a<<104 + b<<52 + c<<0 mod n.
// px is a shorthand for sum(a[i]*a[x-i], i=0..x).
// Note that [x 0 0 0 0 0] = [x*R].
__int128 c, d;
@ -74,43 +109,70 @@ SECP256K1_INLINE static void secp256k1_fe_sqr_inner(const uint64_t *a, uint64_t
d = (__int128)(a0*2) * a3
+ (__int128)(a1*2) * a2;
// [d 0 0 0] = [p3 0 0 0]
c = (__int128)a4 * a4;
// [c 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
d += (c & M) * R; c >>= 52;
// [c 0 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
uint64_t t3 = d & M; d >>= 52;
// [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
a4 *= 2;
d += (__int128)a0 * a4
+ (__int128)(a1*2) * a3
+ (__int128)a2 * a2;
// [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
d += c * R;
// [d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t t4 = d & M; d >>= 52;
// [d t4 t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t tx = (t4 >> 48); t4 &= (M >> 4);
// [d t4+(tx<<48) t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
c = (__int128)a0 * a0;
// [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 0 p4 p3 0 0 p0]
d += (__int128)a1 * a4
+ (__int128)(a2*2) * a3;
// [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
uint64_t u0 = d & M; d >>= 52;
// [d u0 t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
// [d 0 t4+(tx<<48)+(u0<<52) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
u0 = (u0 << 4) | tx;
// [d 0 t4+(u0<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
c += (__int128)u0 * (R >> 4);
// [d 0 t4 t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
r[0] = c & M; c >>= 52;
// [d 0 t4 t3 0 c r0] = [p8 0 0 p5 p4 p3 0 0 p0]
a0 *= 2;
c += (__int128)a0 * a1;
// [d 0 t4 t3 0 c r0] = [p8 0 0 p5 p4 p3 0 p1 p0]
d += (__int128)a2 * a4
+ (__int128)a3 * a3;
// [d 0 t4 t3 0 c r0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
c += (d & M) * R; d >>= 52;
// [d 0 0 t4 t3 0 c r0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
r[1] = c & M; c >>= 52;
// [d 0 0 t4 t3 c r1 r0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
c += (__int128)a0 * a2
+ (__int128)a1 * a1;
// [d 0 0 t4 t3 c r1 r0] = [p8 0 p6 p5 p4 p3 p2 p1 p0]
d += (__int128)a3 * a4;
// [d 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += (d & M) * R; d >>= 52;
// [d 0 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[2] = c & M; c >>= 52;
// [d 0 0 0 t4 t3+c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += d * R + t3;;
// [t4 c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[3] = c & M; c >>= 52;
// [t4+c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += t4;
// [c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[4] = c;
// [r4 r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
}
#endif