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