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#ifndef _SECP256K1_GROUP_
#define _SECP256K1_GROUP_
#include "field.h"
namespace secp256k1 {
class GroupElemJac;
/** Defines a point on the secp256k1 curve (y^2 = x^3 + 7) */
class GroupElem {
protected:
bool fInfinity;
FieldElem x;
FieldElem y;
public:
/** Creates the point at infinity */
GroupElem() {
fInfinity = true;
}
/** Creates the point with given affine coordinates */
GroupElem(const FieldElem &xin, const FieldElem &yin) {
fInfinity = false;
x = xin;
y = yin;
}
/** Checks whether this is the point at infinity */
bool IsInfinity() const {
return fInfinity;
}
void SetNeg(const GroupElem &p) {
*this = p;
y.Normalize();
y.SetNeg(y, 1);
}
void GetX(FieldElem &xout) const {
xout = x;
}
void GetY(FieldElem &yout) const {
yout = y;
}
std::string ToString() const {
if (fInfinity)
return "(inf)";
FieldElem xc = x, yc = y;
return "(" + xc.ToString() + "," + yc.ToString() + ")";
}
void SetJac(GroupElemJac &jac);
friend class GroupElemJac;
};
/** Represents a point on the secp256k1 curve, with jacobian coordinates */
class GroupElemJac : private GroupElem {
protected:
FieldElem z;
public:
/** Creates the point at infinity */
GroupElemJac() : GroupElem(), z(1) {}
/** Creates the point with given affine coordinates */
GroupElemJac(const FieldElem &xin, const FieldElem &yin) : GroupElem(xin,yin), z(1) {}
GroupElemJac(const GroupElem &in) : GroupElem(in), z(1) {}
void SetJac(GroupElemJac &jac) {
*this = jac;
}
/** Checks whether this is a non-infinite point on the curve */
bool IsValid() const {
if (IsInfinity())
return false;
// y^2 = x^3 + 7
// (Y/Z^3)^2 = (X/Z^2)^3 + 7
// Y^2 / Z^6 = X^3 / Z^6 + 7
// Y^2 = X^3 + 7*Z^6
FieldElem y2; y2.SetSquare(y);
FieldElem x3; x3.SetSquare(x); x3.SetMult(x3,x);
FieldElem z2; z2.SetSquare(z);
FieldElem z6; z6.SetSquare(z2); z6.SetMult(z6,z2);
z6 *= 7;
x3 += z6;
return y2 == x3;
}
/** Returns the affine coordinates of this point */
void GetAffine(GroupElem &aff) {
z.SetInverse(z);
FieldElem z2;
z2.SetSquare(z);
FieldElem z3;
z3.SetMult(z,z2);
x.SetMult(x,z2);
y.SetMult(y,z3);
z = FieldElem(1);
aff.fInfinity = fInfinity;
aff.x = x;
aff.y = y;
}
void GetX(FieldElem &xout) {
FieldElem zi;
zi.SetInverse(z);
zi.SetSquare(zi);
xout.SetMult(x, zi);
}
bool IsInfinity() const {
return fInfinity;
}
void GetY(FieldElem &yout) {
FieldElem zi;
zi.SetInverse(z);
FieldElem zi3; zi3.SetSquare(zi); zi3.SetMult(zi, zi3);
yout.SetMult(y, zi3);
}
void SetNeg(const GroupElemJac &p) {
*this = p;
y.Normalize();
y.SetNeg(y, 1);
}
/** Sets this point to have a given X coordinate & given Y oddness */
void SetCompressed(const FieldElem &xin, bool fOdd) {
x = xin;
FieldElem x2; x2.SetSquare(x);
FieldElem x3; x3.SetMult(x,x2);
fInfinity = false;
FieldElem c(7);
c += x3;
y.SetSquareRoot(c);
z = FieldElem(1);
if (y.IsOdd() != fOdd)
y.SetNeg(y,1);
}
/** Sets this point to be the EC double of another */
void SetDouble(const GroupElemJac &p) {
FieldElem t5 = p.y;
if (p.fInfinity || t5.IsZero()) {
fInfinity = true;
return;
}
FieldElem t1,t2,t3,t4;
z.SetMult(t5,p.z);
z *= 2; // Z' = 2*Y*Z (2)
t1.SetSquare(p.x);
t1 *= 3; // T1 = 3*X^2 (3)
t2.SetSquare(t1); // T2 = 9*X^4 (1)
t3.SetSquare(t5);
t3 *= 2; // T3 = 2*Y^2 (2)
t4.SetSquare(t3);
t4 *= 2; // T4 = 8*Y^4 (2)
t3.SetMult(p.x,t3); // T3 = 2*X*Y^2 (1)
x = t3;
x *= 4; // X' = 8*X*Y^2 (4)
x.SetNeg(x,4); // X' = -8*X*Y^2 (5)
x += t2; // X' = 9*X^4 - 8*X*Y^2 (6)
t2.SetNeg(t2,1); // T2 = -9*X^4 (2)
t3 *= 6; // T3 = 12*X*Y^2 (6)
t3 += t2; // T3 = 12*X*Y^2 - 9*X^4 (8)
y.SetMult(t1,t3); // Y' = 36*X^3*Y^2 - 27*X^6 (1)
t2.SetNeg(t4,2); // T2 = -8*Y^4 (3)
y += t2; // Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4)
fInfinity = false;
}
/** Sets this point to be the EC addition of two others */
void SetAdd(const GroupElemJac &p, const GroupElemJac &q) {
if (p.fInfinity) {
*this = q;
return;
}
if (q.fInfinity) {
*this = p;
return;
}
fInfinity = false;
const FieldElem &x1 = p.x, &y1 = p.y, &z1 = p.z, &x2 = q.x, &y2 = q.y, &z2 = q.z;
FieldElem z22; z22.SetSquare(z2);
FieldElem z12; z12.SetSquare(z1);
FieldElem u1; u1.SetMult(x1, z22);
FieldElem u2; u2.SetMult(x2, z12);
FieldElem s1; s1.SetMult(y1, z22); s1.SetMult(s1, z2);
FieldElem s2; s2.SetMult(y2, z12); s2.SetMult(s2, z1);
if (u1 == u2) {
if (s1 == s2) {
SetDouble(p);
} else {
fInfinity = true;
}
return;
}
FieldElem h; h.SetNeg(u1,1); h += u2;
FieldElem r; r.SetNeg(s1,1); r += s2;
FieldElem r2; r2.SetSquare(r);
FieldElem h2; h2.SetSquare(h);
FieldElem h3; h3.SetMult(h,h2);
z.SetMult(z1,z2); z.SetMult(z, h);
FieldElem t; t.SetMult(u1,h2);
x = t; x *= 2; x += h3; x.SetNeg(x,3); x += r2;
y.SetNeg(x,5); y += t; y.SetMult(y,r);
h3.SetMult(h3,s1); h3.SetNeg(h3,1);
y += h3;
}
/** Sets this point to be the EC addition of two others (one of which is in affine coordinates) */
void SetAdd(const GroupElemJac &p, const GroupElem &q) {
if (p.fInfinity) {
x = q.x;
y = q.y;
fInfinity = q.fInfinity;
z = FieldElem(1);
return;
}
if (q.fInfinity) {
*this = p;
return;
}
fInfinity = false;
const FieldElem &x1 = p.x, &y1 = p.y, &z1 = p.z, &x2 = q.x, &y2 = q.y;
FieldElem z12; z12.SetSquare(z1);
FieldElem u1 = x1; u1.Normalize();
FieldElem u2; u2.SetMult(x2, z12);
FieldElem s1 = y1; s1.Normalize();
FieldElem s2; s2.SetMult(y2, z12); s2.SetMult(s2, z1);
if (u1 == u2) {
if (s1 == s2) {
SetDouble(p);
} else {
fInfinity = true;
}
return;
}
FieldElem h; h.SetNeg(u1,1); h += u2;
FieldElem r; r.SetNeg(s1,1); r += s2;
FieldElem r2; r2.SetSquare(r);
FieldElem h2; h2.SetSquare(h);
FieldElem h3; h3.SetMult(h,h2);
z = p.z; z.SetMult(z, h);
FieldElem t; t.SetMult(u1,h2);
x = t; x *= 2; x += h3; x.SetNeg(x,3); x += r2;
y.SetNeg(x,5); y += t; y.SetMult(y,r);
h3.SetMult(h3,s1); h3.SetNeg(h3,1);
y += h3;
}
std::string ToString() const {
GroupElemJac cop = *this;
GroupElem aff;
cop.GetAffine(aff);
return aff.ToString();
}
void SetMulLambda(const GroupElemJac &p);
};
void GroupElem::SetJac(GroupElemJac &jac) {
jac.GetAffine(*this);
}
static const unsigned char order_[] = {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,
0xBA,0xAE,0xDC,0xE6,0xAF,0x48,0xA0,0x3B,
0xBF,0xD2,0x5E,0x8C,0xD0,0x36,0x41,0x41};
static const unsigned char g_x_[] = {0x79,0xBE,0x66,0x7E,0xF9,0xDC,0xBB,0xAC,
0x55,0xA0,0x62,0x95,0xCE,0x87,0x0B,0x07,
0x02,0x9B,0xFC,0xDB,0x2D,0xCE,0x28,0xD9,
0x59,0xF2,0x81,0x5B,0x16,0xF8,0x17,0x98};
static const unsigned char g_y_[] = {0x48,0x3A,0xDA,0x77,0x26,0xA3,0xC4,0x65,
0x5D,0xA4,0xFB,0xFC,0x0E,0x11,0x08,0xA8,
0xFD,0x17,0xB4,0x48,0xA6,0x85,0x54,0x19,
0x9C,0x47,0xD0,0x8F,0xFB,0x10,0xD4,0xB8};
// properties of secp256k1's efficiently computable endomorphism
static const unsigned char lambda_[] = {0x53,0x63,0xad,0x4c,0xc0,0x5c,0x30,0xe0,
0xa5,0x26,0x1c,0x02,0x88,0x12,0x64,0x5a,
0x12,0x2e,0x22,0xea,0x20,0x81,0x66,0x78,
0xdf,0x02,0x96,0x7c,0x1b,0x23,0xbd,0x72};
static const unsigned char beta_[] = {0x7a,0xe9,0x6a,0x2b,0x65,0x7c,0x07,0x10,
0x6e,0x64,0x47,0x9e,0xac,0x34,0x34,0xe9,
0x9c,0xf0,0x49,0x75,0x12,0xf5,0x89,0x95,
0xc1,0x39,0x6c,0x28,0x71,0x95,0x01,0xee};
static const unsigned char a1b2_[] = {0x30,0x86,0xd2,0x21,0xa7,0xd4,0x6b,0xcd,
0xe8,0x6c,0x90,0xe4,0x92,0x84,0xeb,0x15};
static const unsigned char b1_[] = {0xe4,0x43,0x7e,0xd6,0x01,0x0e,0x88,0x28,
0x6f,0x54,0x7f,0xa9,0x0a,0xbf,0xe4,0xc3};
static const unsigned char a2_[] = {0x01,
0x14,0xca,0x50,0xf7,0xa8,0xe2,0xf3,0xf6,
0x57,0xc1,0x10,0x8d,0x9d,0x44,0xcf,0xd8};
class GroupConstants {
private:
Context ctx;
const FieldElem g_x;
const FieldElem g_y;
public:
const Number order;
const GroupElem g;
const FieldElem beta;
const Number lambda, a1b2, b1, a2;
GroupConstants() : order(ctx, order_, sizeof(order_)),
g_x(g_x_), g_y(g_y_), g(g_x,g_y),
beta(beta_),
lambda(ctx, lambda_, sizeof(lambda_)),
a1b2(ctx, a1b2_, sizeof(a1b2_)),
b1(ctx, b1_, sizeof(b1_)),
a2(ctx, a2_, sizeof(a2_)) {}
};
const GroupConstants &GetGroupConst() {
static const GroupConstants group_const;
return group_const;
}
void GroupElemJac::SetMulLambda(const GroupElemJac &p) {
FieldElem beta = GetGroupConst().beta;
*this = p;
x.SetMult(x, beta);
}
void SplitExp(Context &ctx, const Number &exp, Number &exp1, Number &exp2) {
const GroupConstants &c = GetGroupConst();
Context ct(ctx);
Number bnc1(ct), bnc2(ct), bnt1(ct), bnt2(ct), bnn2(ct);
bnn2.SetNumber(c.order);
bnn2.Shift1();
bnc1.SetMult(ct, exp, c.a1b2);
bnc1.SetAdd(ct, bnc1, bnn2);
bnc1.SetDiv(ct, bnc1, c.order);
bnc2.SetMult(ct, exp, c.b1);
bnc2.SetAdd(ct, bnc2, bnn2);
bnc2.SetDiv(ct, bnc2, c.order);
bnt1.SetMult(ct, bnc1, c.a1b2);
bnt2.SetMult(ct, bnc2, c.a2);
bnt1.SetAdd(ct, bnt1, bnt2);
exp1.SetSub(ct, exp, bnt1);
bnt1.SetMult(ct, bnc1, c.b1);
bnt2.SetMult(ct, bnc2, c.a1b2);
exp2.SetSub(ct, bnt1, bnt2);
}
}
#endif