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bitcoin/field.h

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#ifndef _SECP256K1_FIELD_
#define _SECP256K1_FIELD_
using namespace std;
#include <assert.h>
#include <stdint.h>
#include <string>
#include "num.h"
// #define VERIFY_MAGNITUDE 1
namespace secp256k1 {
/** Implements arithmetic modulo FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F,
* represented as 5 uint64_t's in base 2^52. The values are allowed to contain >52 each. In particular,
* each FieldElem has a 'magnitude' associated with it. Internally, a magnitude M means each element
* is at most M*(2^53-1), except the most significant one, which is limited to M*(2^49-1). All operations
* accept any input with magnitude at most M, and have different rules for propagating magnitude to their
* output.
*/
class FieldElem {
private:
// X = sum(i=0..4, elem[i]*2^52)
uint64_t n[5];
#ifdef VERIFY_MAGNITUDE
int magnitude;
#endif
public:
/** Creates a constant field element. Magnitude=1 */
FieldElem(int x = 0) {
n[0] = x;
n[1] = n[2] = n[3] = n[4] = 0;
#ifdef VERIFY_MAGNITUDE
magnitude = 1;
#endif
}
FieldElem(const unsigned char *b32) {
SetBytes(b32);
}
/** Normalizes the internal representation entries. Magnitude=1 */
void Normalize() {
uint64_t c;
if (n[0] > 0xFFFFFFFFFFFFFULL || n[1] > 0xFFFFFFFFFFFFFULL || n[2] > 0xFFFFFFFFFFFFFULL || n[3] > 0xFFFFFFFFFFFFFULL || n[4] > 0xFFFFFFFFFFFFULL) {
c = n[0];
uint64_t t0 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + n[1];
uint64_t t1 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + n[2];
uint64_t t2 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + n[3];
uint64_t t3 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + n[4];
uint64_t t4 = c & 0x0FFFFFFFFFFFFULL;
c >>= 48;
if (c) {
c = c * 0x1000003D1ULL + t0;
t0 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + t1;
t1 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + t2;
t2 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + t3;
t3 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + t4;
t4 = c & 0x0FFFFFFFFFFFFULL;
c >>= 48;
}
n[0] = t0; n[1] = t1; n[2] = t2; n[3] = t3; n[4] = t4;
}
if (n[4] == 0xFFFFFFFFFFFFULL && n[3] == 0xFFFFFFFFFFFFFULL && n[2] == 0xFFFFFFFFFFFFFULL && n[1] == 0xFFFFFFFFFFFFF && n[0] >= 0xFFFFEFFFFFC2FULL) {
n[4] = 0;
n[3] = 0;
n[2] = 0;
n[1] = 0;
n[0] -= 0xFFFFEFFFFFC2FULL;
}
#ifdef VERIFY_MAGNITUDE
magnitude = 1;
#endif
}
bool IsZero() {
Normalize();
return (n[0] == 0 && n[1] == 0 && n[2] == 0 && n[3] == 0 && n[4] == 0);
}
bool friend operator==(FieldElem &a, FieldElem &b) {
a.Normalize();
b.Normalize();
return (a.n[0] == b.n[0] && a.n[1] == b.n[1] && a.n[2] == b.n[2] && a.n[3] == b.n[3] && a.n[4] == b.n[4]);
}
/** extract as 32-byte big endian array */
void GetBytes(unsigned char *o) {
Normalize();
for (int i=0; i<32; i++) {
int c = 0;
for (int j=0; j<2; j++) {
int limb = (8*i+4*j)/52;
int shift = (8*i+4*j)%52;
c |= ((n[limb] >> shift) & 0xF) << (4 * j);
}
o[31-i] = c;
}
}
/** set value of 32-byte big endian array */
void SetBytes(const unsigned char *in) {
n[0] = n[1] = n[2] = n[3] = n[4] = 0;
for (int i=0; i<32; i++) {
for (int j=0; j<2; j++) {
int limb = (8*i+4*j)/52;
int shift = (8*i+4*j)%52;
n[limb] |= (uint64_t)((in[31-i] >> (4*j)) & 0xF) << shift;
}
}
#ifdef VERIFY_MAGNITUDE
magnitude = 1;
#endif
}
/** Set a FieldElem to be the negative of another. Increases magnitude by one. */
void SetNeg(const FieldElem &a, int magnitudeIn) {
#ifdef VERIFY_MAGNITUDE
assert(a.magnitude <= magnitudeIn);
magnitude = magnitudeIn + 1;
#endif
n[0] = 0xFFFFEFFFFFC2FULL * (magnitudeIn + 1) - a.n[0];
n[1] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[1];
n[2] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[2];
n[3] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[3];
n[4] = 0x0FFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[4];
}
/** Multiplies this FieldElem with an integer constant. Magnitude is multiplied by v */
void operator*=(int v) {
#ifdef VERIFY_MAGNITUDE
magnitude *= v;
#endif
n[0] *= v;
n[1] *= v;
n[2] *= v;
n[3] *= v;
n[4] *= v;
}
void operator+=(const FieldElem &a) {
#ifdef VERIFY_MAGNITUDE
magnitude += a.magnitude;
#endif
n[0] += a.n[0];
n[1] += a.n[1];
n[2] += a.n[2];
n[3] += a.n[3];
n[4] += a.n[4];
}
/** Set this FieldElem to be the multiplication of two others. Magnitude=1 */
void SetMult(const FieldElem &a, const FieldElem &b) {
#ifdef VERIFY_MAGNITUDE
assert(a.magnitude <= 8);
assert(b.magnitude <= 8);
#endif
__int128 c = (__int128)a.n[0] * b.n[0];
uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0
c = c + (__int128)a.n[0] * b.n[1] +
(__int128)a.n[1] * b.n[0];
uint64_t t1 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 20000000000000BF
c = c + (__int128)a.n[0] * b.n[2] +
(__int128)a.n[1] * b.n[1] +
(__int128)a.n[2] * b.n[0];
uint64_t t2 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 30000000000001A0
c = c + (__int128)a.n[0] * b.n[3] +
(__int128)a.n[1] * b.n[2] +
(__int128)a.n[2] * b.n[1] +
(__int128)a.n[3] * b.n[0];
uint64_t t3 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 4000000000000280
c = c + (__int128)a.n[0] * b.n[4] +
(__int128)a.n[1] * b.n[3] +
(__int128)a.n[2] * b.n[2] +
(__int128)a.n[3] * b.n[1] +
(__int128)a.n[4] * b.n[0];
uint64_t t4 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 320000000000037E
c = c + (__int128)a.n[1] * b.n[4] +
(__int128)a.n[2] * b.n[3] +
(__int128)a.n[3] * b.n[2] +
(__int128)a.n[4] * b.n[1];
uint64_t t5 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 22000000000002BE
c = c + (__int128)a.n[2] * b.n[4] +
(__int128)a.n[3] * b.n[3] +
(__int128)a.n[4] * b.n[2];
uint64_t t6 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 12000000000001DE
c = c + (__int128)a.n[3] * b.n[4] +
(__int128)a.n[4] * b.n[3];
uint64_t t7 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 02000000000000FE
c = c + (__int128)a.n[4] * b.n[4];
uint64_t t8 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 001000000000001E
uint64_t t9 = c;
c = t0 + (__int128)t5 * 0x1000003D10ULL;
t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t1 + (__int128)t6 * 0x1000003D10ULL;
t1 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t2 + (__int128)t7 * 0x1000003D10ULL;
n[2] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t3 + (__int128)t8 * 0x1000003D10ULL;
n[3] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t4 + (__int128)t9 * 0x1000003D10ULL;
n[4] = c & 0x0FFFFFFFFFFFFULL; c = c >> 48; // c max 000001000003D110
c = t0 + (__int128)c * 0x1000003D1ULL;
n[0] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 1000008
n[1] = t1 + c;
#ifdef VERIFY_MAGNITUDE
magnitude = 1;
#endif
}
/** Set this FieldElem to be the square of another. Magnitude=1 */
void SetSquare(const FieldElem &a) {
#ifdef VERIFY_MAGNITUDE
assert(a.magnitude <= 8);
#endif
__int128 c = (__int128)a.n[0] * a.n[0];
uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0
c = c + (__int128)(a.n[0]*2) * a.n[1];
uint64_t t1 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 20000000000000BF
c = c + (__int128)(a.n[0]*2) * a.n[2] +
(__int128)a.n[1] * a.n[1];
uint64_t t2 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 30000000000001A0
c = c + (__int128)(a.n[0]*2) * a.n[3] +
(__int128)(a.n[1]*2) * a.n[2];
uint64_t t3 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 4000000000000280
c = c + (__int128)(a.n[0]*2) * a.n[4] +
(__int128)(a.n[1]*2) * a.n[3] +
(__int128)a.n[2] * a.n[2];
uint64_t t4 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 320000000000037E
c = c + (__int128)(a.n[1]*2) * a.n[4] +
(__int128)(a.n[2]*2) * a.n[3];
uint64_t t5 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 22000000000002BE
c = c + (__int128)(a.n[2]*2) * a.n[4] +
(__int128)a.n[3] * a.n[3];
uint64_t t6 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 12000000000001DE
c = c + (__int128)(a.n[3]*2) * a.n[4];
uint64_t t7 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 02000000000000FE
c = c + (__int128)a.n[4] * a.n[4];
uint64_t t8 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 001000000000001E
uint64_t t9 = c;
c = t0 + (__int128)t5 * 0x1000003D10ULL;
t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t1 + (__int128)t6 * 0x1000003D10ULL;
t1 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t2 + (__int128)t7 * 0x1000003D10ULL;
n[2] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t3 + (__int128)t8 * 0x1000003D10ULL;
n[3] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0000001000003D10
c = c + t4 + (__int128)t9 * 0x1000003D10ULL;
n[4] = c & 0x0FFFFFFFFFFFFULL; c = c >> 48; // c max 000001000003D110
c = t0 + (__int128)c * 0x1000003D1ULL;
n[0] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 1000008
n[1] = t1 + c;
#ifdef VERIFY_MAGNITUDE
assert(a.magnitude <= 8);
#endif
}
/** Set this to be the (modular) square root of another FieldElem. Magnitude=1 */
void SetSquareRoot(const FieldElem &a) {
// calculate a^p, with p={15,780,1022,1023}
FieldElem a2; a2.SetSquare(a);
FieldElem a3; a3.SetMult(a2,a);
FieldElem a6; a6.SetSquare(a3);
FieldElem a12; a12.SetSquare(a6);
FieldElem a15; a15.SetMult(a12,a3);
FieldElem a30; a30.SetSquare(a15);
FieldElem a60; a60.SetSquare(a30);
FieldElem a120; a120.SetSquare(a60);
FieldElem a240; a240.SetSquare(a120);
FieldElem a255; a255.SetMult(a240,a15);
FieldElem a510; a510.SetSquare(a255);
FieldElem a750; a750.SetMult(a510,a240);
FieldElem a780; a780.SetMult(a750,a30);
FieldElem a1020; a1020.SetSquare(a510);
FieldElem a1022; a1022.SetMult(a1020,a2);
FieldElem a1023; a1023.SetMult(a1022,a);
FieldElem x = a15;
for (int i=0; i<21; i++) {
for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a1023);
}
for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a1022);
for (int i=0; i<2; i++) {
for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a1023);
}
for (int j=0; j<10; j++) x.SetSquare(x);
SetMult(x,a780);
}
bool IsOdd() {
Normalize();
return n[0] & 1;
}
/** Set this to be the (modular) inverse of another FieldElem. Magnitude=1 */
void SetInverse(const FieldElem &a) {
// calculate a^p, with p={45,63,1019,1023}
FieldElem a2; a2.SetSquare(a);
FieldElem a3; a3.SetMult(a2,a);
FieldElem a4; a4.SetSquare(a2);
FieldElem a5; a5.SetMult(a4,a);
FieldElem a10; a10.SetSquare(a5);
FieldElem a11; a11.SetMult(a10,a);
FieldElem a21; a21.SetMult(a11,a10);
FieldElem a42; a42.SetSquare(a21);
FieldElem a45; a45.SetMult(a42,a3);
FieldElem a63; a63.SetMult(a42,a21);
FieldElem a126; a126.SetSquare(a63);
FieldElem a252; a252.SetSquare(a126);
FieldElem a504; a504.SetSquare(a252);
FieldElem a1008; a1008.SetSquare(a504);
FieldElem a1019; a1019.SetMult(a1008,a11);
FieldElem a1023; a1023.SetMult(a1019,a4);
FieldElem x = a63;
for (int i=0; i<21; i++) {
for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a1023);
}
for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a1019);
for (int i=0; i<2; i++) {
for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a1023);
}
for (int j=0; j<10; j++) x.SetSquare(x);
SetMult(x,a45);
}
std::string ToString() {
unsigned char tmp[32];
GetBytes(tmp);
std::string ret;
for (int i=0; i<32; i++) {
static const char *c = "0123456789ABCDEF";
ret += c[(tmp[i] >> 4) & 0xF];
ret += c[(tmp[i]) & 0xF];
}
return ret;
}
void SetHex(const std::string &str) {
unsigned char tmp[32] = {};
static const int cvt[256] = {0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 1, 2, 3, 4, 5, 6,7,8,9,0,0,0,0,0,0,
0,10,11,12,13,14,15,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0,10,11,12,13,14,15,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,
0, 0, 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0};
for (int i=0; i<32; i++) {
if (str.length() > i*2)
tmp[i] = (cvt[(unsigned char)str[2*i]] << 4) + cvt[(unsigned char)str[2*i+1]];
}
SetBytes(tmp);
}
};
static const unsigned char field_p_[] = {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F};
class FieldConstants {
private:
Context ctx;
public:
const Number field_p;
FieldConstants() : field_p(ctx, field_p_, sizeof(field_p_)) {}
};
const FieldConstants &GetFieldConst() {
static const FieldConstants field_const;
return field_const;
}
}
#endif
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