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Pieter Wuille 2013-03-06 01:42:20 +01:00
parent af2348935e
commit 84fba6dde0
1 changed files with 162 additions and 126 deletions
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244
secp256k1.cpp
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@ -4,40 +4,42 @@
#include <gmpxx.h> #include <gmpxx.h>
#include <assert.h> #include <assert.h>
// #define VERIFY_BADNESS 1 // #define VERIFY_MAGNITUDE 1
namespace secp256k1 { namespace secp256k1 {
/** Implements arithmetic modulo FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F /** Implements arithmetic modulo FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F,
* A FieldElem has an implicit 'badness': * represented as 5 uint64_t's in base 2^52. The values are allowed to contain >52 each. In particular,
* - the output of SetSquare, SetMult and Normalize sets the badness to 1 * each FieldElem has a 'magnitude' associated with it. Internally, a magnitude M means each element
* - the inputs of SetSquare and SetMult cannot have badness above 8 * is at most M*(2^53-1), except the most significant one, which is limited to M*(2^49-1). All operations
* - adding two FieldElems adds the badness together * accept any input with magnitude at most M, and have different rules for propagating magnitude to their
* - multiplying a FieldElem with a constant multiplies its badness by that constant * output.
* - taking the negative requires specifying the badness of the input, the output having badness 1 higher
*/ */
class FieldElem { class FieldElem {
private: private:
// X = sum(i=0..4, elem[i]*2^52) // X = sum(i=0..4, elem[i]*2^52)
uint64_t n[5]; uint64_t n[5];
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
int badness; int magnitude;
#endif #endif
public: public:
/** Creates a constant field element. Magnitude=1 */
FieldElem(int x = 0) { FieldElem(int x = 0) {
n[0] = x; n[0] = x;
n[1] = n[2] = n[3] = n[4] = 0; n[1] = n[2] = n[3] = n[4] = 0;
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
badness = 1; magnitude = 1;
#endif #endif
} }
/** Normalizes the internal representation entries. Magnitude=1 */
void Normalize() { void Normalize() {
uint64_t c; uint64_t c;
if (n[0] > 0xFFFFFFFFFFFFFULL || n[1] > 0xFFFFFFFFFFFFFULL || n[2] > 0xFFFFFFFFFFFFFULL || n[3] > 0xFFFFFFFFFFFFFULL || n[4] > 0xFFFFFFFFFFFFULL) {
c = n[0]; c = n[0];
uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; uint64_t t0 = c & 0xFFFFFFFFFFFFFULL;
c = (c >> 52) + n[1]; c = (c >> 52) + n[1];
@ -62,16 +64,17 @@ public:
t4 = c & 0x0FFFFFFFFFFFFULL; t4 = c & 0x0FFFFFFFFFFFFULL;
c >>= 48; c >>= 48;
} }
if (t4 == 0xFFFFFFFFFFFFULL && t3 == 0xFFFFFFFFFFFFFULL && t2 == 0xFFFFFFFFFFFFFULL && t3 == 0xFFFFFFFFFFFFF && t4 >= 0xFFFFEFFFFFC2FULL) {
t4 = 0;
t3 = 0;
t2 = 0;
t1 = 0;
t0 -= 0xFFFFEFFFFFC2FULL;
}
n[0] = t0; n[1] = t1; n[2] = t2; n[3] = t3; n[4] = t4; n[0] = t0; n[1] = t1; n[2] = t2; n[3] = t3; n[4] = t4;
#ifdef VERIFY_BADNESS }
badness = 1; 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 #endif
} }
@ -100,26 +103,28 @@ public:
n[2] = ((in[1] >> 40) | (in[2] << 24)) & 0xFFFFFFFFFFFFFULL; n[2] = ((in[1] >> 40) | (in[2] << 24)) & 0xFFFFFFFFFFFFFULL;
n[3] = ((in[2] >> 28) | (in[3] << 36)) & 0xFFFFFFFFFFFFFULL; n[3] = ((in[2] >> 28) | (in[3] << 36)) & 0xFFFFFFFFFFFFFULL;
n[4] = (in[3] >> 16); n[4] = (in[3] >> 16);
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
badness = 1; magnitude = 1;
#endif #endif
} }
void SetNeg(const FieldElem &a, int badnessIn) { /** Set a FieldElem to be the negative of another. Increases magnitude by one. */
#ifdef VERIFY_BADNESS void SetNeg(const FieldElem &a, int magnitudeIn) {
assert(a.badness <= badnessIn); #ifdef VERIFY_MAGNITUDE
badness = badnessIn + 1; assert(a.magnitude <= magnitudeIn);
magnitude = magnitudeIn + 1;
#endif #endif
n[0] = 0xFFFFEFFFFFC2FULL * (badnessIn + 1) - a.n[0]; n[0] = 0xFFFFEFFFFFC2FULL * (magnitudeIn + 1) - a.n[0];
n[1] = 0xFFFFFFFFFFFFFULL * (badnessIn + 1) - a.n[1]; n[1] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[1];
n[2] = 0xFFFFFFFFFFFFFULL * (badnessIn + 1) - a.n[2]; n[2] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[2];
n[3] = 0xFFFFFFFFFFFFFULL * (badnessIn + 1) - a.n[3]; n[3] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[3];
n[4] = 0x0FFFFFFFFFFFFULL * (badnessIn + 1) - a.n[4]; n[4] = 0x0FFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[4];
} }
/** Multiplies this FieldElem with an integer constant. Magnitude is multiplied by v */
void operator*=(int v) { void operator*=(int v) {
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
badness *= v; magnitude *= v;
#endif #endif
n[0] *= v; n[0] *= v;
n[1] *= v; n[1] *= v;
@ -129,8 +134,8 @@ public:
} }
void operator+=(const FieldElem &a) { void operator+=(const FieldElem &a) {
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
badness += a.badness; magnitude += a.magnitude;
#endif #endif
n[0] += a.n[0]; n[0] += a.n[0];
n[1] += a.n[1]; n[1] += a.n[1];
@ -139,12 +144,11 @@ public:
n[4] += a.n[4]; n[4] += a.n[4];
} }
// precondition: all values in a and b are at most 56 bits, except n[4] is at most 52 bits /** Set this FieldElem to be the multiplication of two others. Magnitude=1 */
// postcondition: all values are at most 53 bits, except n[4] is at most 49 bits
void SetMult(const FieldElem &a, const FieldElem &b) { void SetMult(const FieldElem &a, const FieldElem &b) {
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
assert(a.badness <= 8); assert(a.magnitude <= 8);
assert(b.badness <= 8); assert(b.magnitude <= 8);
#endif #endif
__int128 c = (__int128)a.n[0] * b.n[0]; __int128 c = (__int128)a.n[0] * b.n[0];
uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0 uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0
@ -194,16 +198,15 @@ public:
c = t0 + (__int128)c * 0x1000003D1ULL; c = t0 + (__int128)c * 0x1000003D1ULL;
n[0] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 1000008 n[0] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 1000008
n[1] = t1 + c; n[1] = t1 + c;
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
badness = 1; magnitude = 1;
#endif #endif
} }
// precondition: all values in a and b are at most 56 bits, except n[4] is at most 52 bits /** Set this FieldElem to be the square of another. Magnitude=1 */
// postcondition: all values are at most 53 bits, except n[4] is at most 49 bits
void SetSquare(const FieldElem &a) { void SetSquare(const FieldElem &a) {
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
assert(a.badness <= 8); assert(a.magnitude <= 8);
#endif #endif
__int128 c = (__int128)a.n[0] * a.n[0]; __int128 c = (__int128)a.n[0] * a.n[0];
uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0 uint64_t t0 = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 0FFFFFFFFFFFFFE0
@ -243,67 +246,82 @@ public:
c = t0 + (__int128)c * 0x1000003D1ULL; c = t0 + (__int128)c * 0x1000003D1ULL;
n[0] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 1000008 n[0] = c & 0xFFFFFFFFFFFFFULL; c = c >> 52; // c max 1000008
n[1] = t1 + c; n[1] = t1 + c;
#ifdef VERIFY_BADNESS #ifdef VERIFY_MAGNITUDE
assert(a.badness <= 8); assert(a.magnitude <= 8);
#endif #endif
} }
/** Set this to be the (modular) square root of another FieldElem. Magnitude=1 */
void SetSquareRoot(const FieldElem &a) { void SetSquareRoot(const FieldElem &a) {
// calculate a^p, with p={12,15,24,30,31} // calculate a^p, with p={15,780,1022,1023}
FieldElem a2; a2.SetSquare(a); FieldElem a2; a2.SetSquare(a);
FieldElem a3; a3.SetMult(a2,a); FieldElem a3; a3.SetMult(a2,a);
FieldElem a6; a6.SetSquare(a3); FieldElem a6; a6.SetSquare(a3);
FieldElem a12; a12.SetSquare(a6); FieldElem a12; a12.SetSquare(a6);
FieldElem a15; a15.SetMult(a12,a3); FieldElem a15; a15.SetMult(a12,a3);
FieldElem a24; a24.SetSquare(a12);
FieldElem a30; a30.SetSquare(a15); FieldElem a30; a30.SetSquare(a15);
FieldElem a31; a31.SetMult(a30,a); 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; FieldElem x = a15;
for (int i=0; i<43; i++) { for (int i=0; i<21; i++) {
for (int j=0; j<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a31); x.SetMult(x,a1023);
} }
for (int j=0; j<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a30); x.SetMult(x,a1022);
for (int i=0; i<4; i++) { for (int i=0; i<2; i++) {
for (int j=0; j<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a31); x.SetMult(x,a1023);
} }
for (int j=0; j<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a24); SetMult(x,a780);
for (int j=0; j<5; j++) x.SetSquare(x);
SetMult(x,a12);
} }
// computes the modular inverse, by computing a^(p-2) bool IsOdd() {
// TODO: use extmod instead, much faster Normalize();
return n[0] & 1;
}
/** Set this to be the (modular) inverse of another FieldElem. Magnitude=1 */
void SetInverse(const FieldElem &a) { void SetInverse(const FieldElem &a) {
// calculare a^p, with p={13,27,31} // calculate a^p, with p={45,63,1019,1023}
FieldElem a2; a2.SetSquare(a); FieldElem a2; a2.SetSquare(a);
FieldElem a3; a3.SetMult(a2,a); FieldElem a3; a3.SetMult(a2,a);
FieldElem a6; a6.SetSquare(a3); FieldElem a4; a4.SetSquare(a);
FieldElem a7; a7.SetMult(a6,a); FieldElem a5; a5.SetMult(a4,a);
FieldElem a13; a13.SetMult(a6,a7); FieldElem a10; a10.SetSquare(a5);
FieldElem a26; a26.SetSquare(a13); FieldElem a11; a11.SetMult(a10,a);
FieldElem a27; a27.SetMult(a26,a); FieldElem a21; a21.SetMult(a11,a10);
FieldElem a30; a30.SetMult(a27,a3); FieldElem a42; a42.SetSquare(a21);
FieldElem a31; a31.SetMult(a30,a); FieldElem a45; a45.SetMult(a42,a3);
FieldElem x = a; FieldElem a63; a63.SetMult(a42,a21);
for (int i=0; i<44; i++) { FieldElem a126; a126.SetSquare(a63);
for (int j=0; j<5; j++) x.SetSquare(x); FieldElem a252; a252.SetSquare(a126);
x.SetMult(x,a31); 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<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a27); x.SetMult(x,a1019);
for (int i=0; i<4; i++) { for (int i=0; i<2; i++) {
for (int j=0; j<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a31); x.SetMult(x,a1023);
} }
for (int j=0; j<5; j++) x.SetSquare(x); for (int j=0; j<10; j++) x.SetSquare(x);
x.SetMult(x,a); SetMult(x,a45);
for (int j=0; j<5; j++) x.SetSquare(x);
SetMult(x,a13);
} }
std::string ToString() { std::string ToString() {
@ -352,9 +370,12 @@ private:
F z; F z;
public: public:
/** Creates the point at infinity */
GroupElem<F>() { GroupElem<F>() {
fInfinity = true;
} }
/** Creates the point with given affine coordinates */
GroupElem<F>(const F &xin, const F &yin) { GroupElem<F>(const F &xin, const F &yin) {
fInfinity = false; fInfinity = false;
x = xin; x = xin;
@ -362,7 +383,15 @@ public:
z = F(1); z = F(1);
} }
/** Checks whether this is the point at infinity */
bool IsInfinity() const {
return fInfinity;
}
/** Checks whether this is a non-infinite point on the curve */
bool IsValid() { bool IsValid() {
if (IsInfinity())
return false;
// y^2 = x^3 + 7 // y^2 = x^3 + 7
// (Y/Z^3)^2 = (X/Z^2)^3 + 7 // (Y/Z^3)^2 = (X/Z^2)^3 + 7
// Y^2 / Z^6 = X^3 / Z^6 + 7 // Y^2 / Z^6 = X^3 / Z^6 + 7
@ -376,6 +405,7 @@ public:
return y2 == x3; return y2 == x3;
} }
/** Returns the affine coordinates of this point */
void GetAffine(F &xout, F &yout) { void GetAffine(F &xout, F &yout) {
z.SetInverse(z); z.SetInverse(z);
F z2; F z2;
@ -389,7 +419,8 @@ public:
yout = y; yout = y;
} }
bool SetCompressed(const F &xin) { /** Sets this point to have a given X coordinate & given Y oddness */
void SetCompressed(const F &xin, bool fOdd) {
x = xin; x = xin;
F x2; x2.SetSquare(x); F x2; x2.SetSquare(x);
F x3; x3.SetMult(x,x2); F x3; x3.SetMult(x,x2);
@ -398,16 +429,11 @@ public:
c += x3; c += x3;
y.SetSquareRoot(c); y.SetSquareRoot(c);
z = F(1); z = F(1);
if (y.IsOdd() != fOdd)
y.SetNeg(y,1);
} }
std::string ToString() { /** Sets this point to be the EC double of another */
F xt,yt;
if (fInfinity)
return "(inf)";
GetAffine(xt,yt);
return "(" + xt.ToString() + "," + yt.ToString() + ")";
}
void SetDouble(const GroupElem<F> &p) { void SetDouble(const GroupElem<F> &p) {
if (p.fInfinity || y.IsZero()) { if (p.fInfinity || y.IsZero()) {
fInfinity = true; fInfinity = true;
@ -437,6 +463,7 @@ public:
y += t2; // Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) y += t2; // Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4)
} }
/** Sets this point to be the EC addition of two others */
void SetAdd(const GroupElem<F> &p, const GroupElem<F> &q) { void SetAdd(const GroupElem<F> &p, const GroupElem<F> &q) {
if (p.fInfinity) { if (p.fInfinity) {
*this = q; *this = q;
@ -474,6 +501,15 @@ public:
h3.SetMult(h3,s1); h3.SetNeg(h3,1); h3.SetMult(h3,s1); h3.SetNeg(h3,1);
y += h3; y += h3;
} }
std::string ToString() {
F xt,yt;
if (fInfinity)
return "(inf)";
GetAffine(xt,yt);
return "(" + xt.ToString() + "," + yt.ToString() + ")";
}
}; };
} }
@ -484,16 +520,16 @@ int main() {
FieldElem f1,f2; FieldElem f1,f2;
f1.SetHex("8b30bbe9ae2a990696b22f670709dff3727fd8bc04d3362c6c7bf458e2846004"); f1.SetHex("8b30bbe9ae2a990696b22f670709dff3727fd8bc04d3362c6c7bf458e2846004");
// f2.SetHex("a357ae915c4a65281309edf20504740f0eb3343990216b4f81063cb65f2f7e0f"); // f2.SetHex("a357ae915c4a65281309edf20504740f0eb3343990216b4f81063cb65f2f7e0f");
GroupElem<FieldElem> g1; g1.SetCompressed(f1); GroupElem<FieldElem> g1;
printf("%s\n", g1.ToString().c_str()); printf("%s\n", g1.ToString().c_str());
GroupElem<FieldElem> p = g1; GroupElem<FieldElem> p = g1;
GroupElem<FieldElem> q = p; GroupElem<FieldElem> q = p;
//printf("ok %i\n", (int)p.IsValid());
p.SetCompressed(f1,false);
printf("ok %i\n", (int)p.IsValid()); printf("ok %i\n", (int)p.IsValid());
for (int i=0; i<1000000; i++) { printf("%s\n", p.ToString().c_str());
p.SetCompressed(f1); p.SetCompressed(f1,true);
f1.SetSquare(f1); printf("ok %i\n", (int)p.IsValid());
} printf("%s\n", p.ToString().c_str());
printf("ok %i\n", (int)q.IsValid());
printf("%s\n", q.ToString().c_str());
return 0; return 0;
} }