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Add overflow analysis to field_5x52_int128_impl.h
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@sipa
sipa committed Nov 13, 2014
1 parent fa0d620 commit a518598
Showing 1 changed file with 97 additions and 0 deletions.
97 changes: 97 additions & 0 deletions src/field_5x52_int128_impl.h
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Original file line number Diff line number Diff line change
Expand Up @@ -7,7 +7,23 @@

#include <stdint.h>

#ifdef VERIFY
#define VERIFY_BITS(x, n) VERIFY_CHECK(((x) >> (n)) == 0)
#else
#define VERIFY_BITS(x, n) do { } while(0)
#endif

SECP256K1_INLINE static void secp256k1_fe_mul_inner(const uint64_t *a, const uint64_t *b, uint64_t *r) {
VERIFY_BITS(a[0], 56);
VERIFY_BITS(a[1], 56);
VERIFY_BITS(a[2], 56);
VERIFY_BITS(a[3], 56);
VERIFY_BITS(a[4], 52);
VERIFY_BITS(b[0], 56);
VERIFY_BITS(b[1], 56);
VERIFY_BITS(b[2], 56);
VERIFY_BITS(b[3], 56);
VERIFY_BITS(b[4], 52);

const uint64_t M = 0xFFFFFFFFFFFFFULL, R = 0x1000003D10ULL;
// [... a b c] is a shorthand for ... + a<<104 + b<<52 + c<<0 mod n.
Expand All @@ -20,83 +36,127 @@ 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];
VERIFY_BITS(d, 114);
// [d 0 0 0] = [p3 0 0 0]
c = (__int128)a[4] * b[4];
VERIFY_BITS(c, 112);
// [c 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
d += (c & M) * R; c >>= 52;
VERIFY_BITS(d, 115);
VERIFY_BITS(c, 60);
// [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;
VERIFY_BITS(t3, 52);
VERIFY_BITS(d, 63);
// [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];
VERIFY_BITS(d, 115);
// [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
d += c * R;
VERIFY_BITS(d, 116);
// [d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t t4 = d & M; d >>= 52;
VERIFY_BITS(t4, 52);
VERIFY_BITS(d, 64);
// [d t4 t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t tx = (t4 >> 48); t4 &= (M >> 4);
VERIFY_BITS(tx, 4);
VERIFY_BITS(t4, 48);
// [d t4+(tx<<48) t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]

c = (__int128)a[0] * b[0];
VERIFY_BITS(c, 112);
// [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];
VERIFY_BITS(d, 115);
// [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
uint64_t u0 = d & M; d >>= 52;
VERIFY_BITS(u0, 52);
VERIFY_BITS(d, 63);
// [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;
VERIFY_BITS(u0, 56);
// [d 0 t4+(u0<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
c += (__int128)u0 * (R >> 4);
VERIFY_BITS(c, 115);
// [d 0 t4 t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
uint64_t t0 = c & M; c >>= 52;
VERIFY_BITS(t0, 52);
VERIFY_BITS(c, 61);
// [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];
VERIFY_BITS(c, 114);
// [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];
VERIFY_BITS(d, 114);
// [d 0 t4 t3 0 c t0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
c += (d & M) * R; d >>= 52;
VERIFY_BITS(c, 115);
VERIFY_BITS(d, 62);
// [d 0 0 t4 t3 0 c t0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
uint64_t t1 = c & M; c >>= 52;
VERIFY_BITS(t1, 52);
VERIFY_BITS(c, 63);
// [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];
VERIFY_BITS(c, 114);
// [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];
VERIFY_BITS(d, 114);
// [d 0 0 t4 t3 c t1 t0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += (d & M) * R; d >>= 52;
VERIFY_BITS(c, 115);
VERIFY_BITS(d, 62);
// [d 0 0 0 t4 t3 c t1 t0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]

r[0] = t0;
VERIFY_BITS(r[0], 52);
// [d 0 0 0 t4 t3 c t1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[1] = t1;
VERIFY_BITS(r[1], 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;
VERIFY_BITS(r[2], 52);
VERIFY_BITS(c, 63);
// [d 0 0 0 t4 t3+c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += d * R + t3;;
VERIFY_BITS(c, 100);
// [t4 c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[3] = c & M; c >>= 52;
VERIFY_BITS(r[3], 52);
VERIFY_BITS(c, 48);
// [t4+c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += t4;
VERIFY_BITS(c, 49);
// [c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[4] = c;
VERIFY_BITS(r[4], 49);
// [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) {
VERIFY_BITS(a[0], 56);
VERIFY_BITS(a[1], 56);
VERIFY_BITS(a[2], 56);
VERIFY_BITS(a[3], 56);
VERIFY_BITS(a[4], 52);

const uint64_t M = 0xFFFFFFFFFFFFFULL, R = 0x1000003D10ULL;
// [... a b c] is a shorthand for ... + a<<104 + b<<52 + c<<0 mod n.
Expand All @@ -109,69 +169,106 @@ SECP256K1_INLINE static void secp256k1_fe_sqr_inner(const uint64_t *a, uint64_t

d = (__int128)(a0*2) * a3
+ (__int128)(a1*2) * a2;
VERIFY_BITS(d, 114);
// [d 0 0 0] = [p3 0 0 0]
c = (__int128)a4 * a4;
VERIFY_BITS(c, 112);
// [c 0 0 0 0 d 0 0 0] = [p8 0 0 0 0 p3 0 0 0]
d += (c & M) * R; c >>= 52;
VERIFY_BITS(d, 115);
VERIFY_BITS(c, 60);
// [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;
VERIFY_BITS(t3, 52);
VERIFY_BITS(d, 63);
// [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;
VERIFY_BITS(d, 115);
// [c 0 0 0 0 d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
d += c * R;
VERIFY_BITS(d, 116);
// [d t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t t4 = d & M; d >>= 52;
VERIFY_BITS(t4, 52);
VERIFY_BITS(d, 64);
// [d t4 t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]
uint64_t tx = (t4 >> 48); t4 &= (M >> 4);
VERIFY_BITS(tx, 4);
VERIFY_BITS(t4, 48);
// [d t4+(tx<<48) t3 0 0 0] = [p8 0 0 0 p4 p3 0 0 0]

c = (__int128)a0 * a0;
VERIFY_BITS(c, 112);
// [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;
VERIFY_BITS(d, 114);
// [d t4+(tx<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
uint64_t u0 = d & M; d >>= 52;
VERIFY_BITS(u0, 52);
VERIFY_BITS(d, 62);
// [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;
VERIFY_BITS(u0, 56);
// [d 0 t4+(u0<<48) t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
c += (__int128)u0 * (R >> 4);
VERIFY_BITS(c, 113);
// [d 0 t4 t3 0 0 c] = [p8 0 0 p5 p4 p3 0 0 p0]
r[0] = c & M; c >>= 52;
VERIFY_BITS(r[0], 52);
VERIFY_BITS(c, 61);
// [d 0 t4 t3 0 c r0] = [p8 0 0 p5 p4 p3 0 0 p0]

a0 *= 2;
c += (__int128)a0 * a1;
VERIFY_BITS(c, 114);
// [d 0 t4 t3 0 c r0] = [p8 0 0 p5 p4 p3 0 p1 p0]
d += (__int128)a2 * a4
+ (__int128)a3 * a3;
VERIFY_BITS(d, 114);
// [d 0 t4 t3 0 c r0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
c += (d & M) * R; d >>= 52;
VERIFY_BITS(c, 115);
VERIFY_BITS(d, 62);
// [d 0 0 t4 t3 0 c r0] = [p8 0 p6 p5 p4 p3 0 p1 p0]
r[1] = c & M; c >>= 52;
VERIFY_BITS(r[1], 52);
VERIFY_BITS(c, 63);
// [d 0 0 t4 t3 c r1 r0] = [p8 0 p6 p5 p4 p3 0 p1 p0]

c += (__int128)a0 * a2
+ (__int128)a1 * a1;
VERIFY_BITS(c, 114);
// [d 0 0 t4 t3 c r1 r0] = [p8 0 p6 p5 p4 p3 p2 p1 p0]
d += (__int128)a3 * a4;
VERIFY_BITS(d, 114);
// [d 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += (d & M) * R; d >>= 52;
VERIFY_BITS(c, 115);
VERIFY_BITS(d, 62);
// [d 0 0 0 t4 t3 c r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[2] = c & M; c >>= 52;
VERIFY_BITS(r[2], 52);
VERIFY_BITS(c, 63);
// [d 0 0 0 t4 t3+c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]

c += d * R + t3;;
VERIFY_BITS(c, 100);
// [t4 c r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[3] = c & M; c >>= 52;
VERIFY_BITS(r[3], 52);
VERIFY_BITS(c, 48);
// [t4+c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
c += t4;
VERIFY_BITS(c, 49);
// [c r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
r[4] = c;
VERIFY_BITS(r[4], 49);
// [r4 r3 r2 r1 r0] = [p8 p7 p6 p5 p4 p3 p2 p1 p0]
}

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