/* return sqrt(this) mod Modulus */
public FP sqrt()
{
reduce();
BIG b = new BIG(p);
if (ROM.MOD8 == 5)
{
b.dec(5);
b.norm();
b.shr(3);
FP i = new FP(this);
i.x.shl(1);
FP v = i.pow(b);
i.mul(v);
i.mul(v);
i.x.dec(1);
FP r = new FP(this);
r.mul(v);
r.mul(i);
r.reduce();
return(r);
}
else
{
b.inc(1);
b.norm();
b.shr(2);
return(pow(b));
}
}
/* r=x^n using XTR method on traces of FP12s */
public FP4 xtr_pow(BIG n)
{
FP4 a = new FP4(3);
FP4 b = new FP4(this);
FP4 c = new FP4(b);
c.xtr_D();
FP4 t = new FP4(0);
FP4 r = new FP4(0);
n.norm();
int par = n.parity();
BIG v = new BIG(n);
v.fshr(1);
if (par == 0)
{
v.dec(1);
v.norm();
}
int nb = v.nbits();
for (int i = nb - 1; i >= 0; i--)
{
if (v.bit(i) != 1)
{
t.copy(b);
conj();
c.conj();
b.xtr_A(a, this, c);
conj();
c.copy(t);
c.xtr_D();
a.xtr_D();
}
else
{
t.copy(a);
t.conj();
a.copy(b);
a.xtr_D();
b.xtr_A(c, this, t);
c.xtr_D();
}
}
if (par == 0)
{
r.copy(c);
}
else
{
r.copy(b);
}
r.reduce();
return(r);
}
/* Optimal R-ate pairing */
public static FP12 ate(ECP2 P, ECP Q)
{
FP2 f = new FP2(new BIG(ROM.CURVE_Fra), new BIG(ROM.CURVE_Frb));
BIG x = new BIG(ROM.CURVE_Bnx);
BIG n = new BIG(x);
ECP2 K = new ECP2();
FP12 lv;
n.pmul(6);
n.dec(2);
n.norm();
P.affine();
Q.affine();
FP Qx = new FP(Q.getx());
FP Qy = new FP(Q.gety());
ECP2 A = new ECP2();
FP12 r = new FP12(1);
A.copy(P);
int nb = n.nbits();
for (int i = nb - 2; i >= 1; i--)
{
lv = line(A, A, Qx, Qy);
r.smul(lv);
if (n.bit(i) == 1)
{
lv = line(A, P, Qx, Qy);
r.smul(lv);
}
r.sqr();
}
lv = line(A, A, Qx, Qy);
r.smul(lv);
/* R-ate fixup */
r.conj();
K.copy(P);
K.frob(f);
A.neg();
lv = line(A, K, Qx, Qy);
r.smul(lv);
K.frob(f);
K.neg();
lv = line(A, K, Qx, Qy);
r.smul(lv);
return(r);
}
/* returns u derived from P. Random value in range 1 to return value should then be added to u */
public static int unmap(BIG u, ECP P)
{
int s = P.S;
ECP R;
int r = 0;
BIG x = P.X;
u.copy(x);
while (true)
{
u.dec(1);
u.norm();
r++;
R = new ECP(u, s);
if (!R.is_infinity())
{
break;
}
}
return(r);
}
/* return e.this */
public ECP mul(BIG e)
{
if (e.iszilch() || is_infinity())
{
return(new ECP());
}
ECP P = new ECP();
if (ROM.CURVETYPE == ROM.MONTGOMERY)
{
/* use Ladder */
int nb, i, b;
ECP D = new ECP();
ECP R0 = new ECP();
R0.copy(this);
ECP R1 = new ECP();
R1.copy(this);
R1.dbl();
D.copy(this);
D.affine();
nb = e.nbits();
for (i = nb - 2; i >= 0; i--)
{
b = e.bit(i);
P.copy(R1);
P.dadd(R0, D);
R0.cswap(R1, b);
R1.copy(P);
R0.dbl();
R0.cswap(R1, b);
}
P.copy(R0);
}
else
{
// fixed size windows
int i, b, nb, m, s, ns;
BIG mt = new BIG();
BIG t = new BIG();
ECP Q = new ECP();
ECP C = new ECP();
ECP[] W = new ECP[8];
sbyte[] w = new sbyte[1 + (ROM.NLEN * ROM.BASEBITS + 3) / 4];
affine();
// precompute table
Q.copy(this);
Q.dbl();
W[0] = new ECP();
W[0].copy(this);
for (i = 1; i < 8; i++)
{
W[i] = new ECP();
W[i].copy(W[i - 1]);
W[i].add(Q);
}
// convert the table to affine
if (ROM.CURVETYPE == ROM.WEIERSTRASS)
{
multiaffine(8, W);
}
// make exponent odd - add 2P if even, P if odd
t.copy(e);
s = t.parity();
t.inc(1);
t.norm();
ns = t.parity();
mt.copy(t);
mt.inc(1);
mt.norm();
t.cmove(mt, s);
Q.cmove(this, ns);
C.copy(Q);
nb = 1 + (t.nbits() + 3) / 4;
// convert exponent to signed 4-bit window
for (i = 0; i < nb; i++)
{
w[i] = (sbyte)(t.lastbits(5) - 16);
t.dec(w[i]);
t.norm();
t.fshr(4);
}
w[nb] = (sbyte)t.lastbits(5);
P.copy(W[(w[nb] - 1) / 2]);
for (i = nb - 1; i >= 0; i--)
{
Q.select(W, w[i]);
P.dbl();
P.dbl();
P.dbl();
P.dbl();
P.add(Q);
}
P.sub(C); // apply correction
}
P.affine();
return(P);
}
/* Return e.this+f.Q */
public ECP mul2(BIG e, ECP Q, BIG f)
{
BIG te = new BIG();
BIG tf = new BIG();
BIG mt = new BIG();
ECP S = new ECP();
ECP T = new ECP();
ECP C = new ECP();
ECP[] W = new ECP[8];
sbyte[] w = new sbyte[1 + (ROM.NLEN * ROM.BASEBITS + 1) / 2];
int i, s, ns, nb;
sbyte a, b;
affine();
Q.affine();
te.copy(e);
tf.copy(f);
// precompute table
W[1] = new ECP();
W[1].copy(this);
W[1].sub(Q);
W[2] = new ECP();
W[2].copy(this);
W[2].add(Q);
S.copy(Q);
S.dbl();
W[0] = new ECP();
W[0].copy(W[1]);
W[0].sub(S);
W[3] = new ECP();
W[3].copy(W[2]);
W[3].add(S);
T.copy(this);
T.dbl();
W[5] = new ECP();
W[5].copy(W[1]);
W[5].add(T);
W[6] = new ECP();
W[6].copy(W[2]);
W[6].add(T);
W[4] = new ECP();
W[4].copy(W[5]);
W[4].sub(S);
W[7] = new ECP();
W[7].copy(W[6]);
W[7].add(S);
// convert the table to affine
if (ROM.CURVETYPE == ROM.WEIERSTRASS)
{
multiaffine(8, W);
}
// if multiplier is odd, add 2, else add 1 to multiplier, and add 2P or P to correction
s = te.parity();
te.inc(1);
te.norm();
ns = te.parity();
mt.copy(te);
mt.inc(1);
mt.norm();
te.cmove(mt, s);
T.cmove(this, ns);
C.copy(T);
s = tf.parity();
tf.inc(1);
tf.norm();
ns = tf.parity();
mt.copy(tf);
mt.inc(1);
mt.norm();
tf.cmove(mt, s);
S.cmove(Q, ns);
C.add(S);
mt.copy(te);
mt.add(tf);
mt.norm();
nb = 1 + (mt.nbits() + 1) / 2;
// convert exponent to signed 2-bit window
for (i = 0; i < nb; i++)
{
a = (sbyte)(te.lastbits(3) - 4);
te.dec(a);
te.norm();
te.fshr(2);
b = (sbyte)(tf.lastbits(3) - 4);
tf.dec(b);
tf.norm();
tf.fshr(2);
w[i] = (sbyte)(4 * a + b);
}
w[nb] = (sbyte)(4 * te.lastbits(3) + tf.lastbits(3));
S.copy(W[(w[nb] - 1) / 2]);
for (i = nb - 1; i >= 0; i--)
{
T.select(W, w[i]);
S.dbl();
S.dbl();
S.add(T);
}
S.sub(C); // apply correction
S.affine();
return(S);
}
/* P*=e */
public ECP2 mul(BIG e)
{
/* fixed size windows */
int i, b, nb, m, s, ns;
BIG mt = new BIG();
BIG t = new BIG();
ECP2 P = new ECP2();
ECP2 Q = new ECP2();
ECP2 C = new ECP2();
ECP2[] W = new ECP2[8];
sbyte[] w = new sbyte[1 + (ROM.NLEN * ROM.BASEBITS + 3) / 4];
if (is_infinity())
{
return(new ECP2());
}
affine();
/* precompute table */
Q.copy(this);
Q.dbl();
W[0] = new ECP2();
W[0].copy(this);
for (i = 1; i < 8; i++)
{
W[i] = new ECP2();
W[i].copy(W[i - 1]);
W[i].add(Q);
}
/* convert the table to affine */
multiaffine(8, W);
/* make exponent odd - add 2P if even, P if odd */
t.copy(e);
s = t.parity();
t.inc(1);
t.norm();
ns = t.parity();
mt.copy(t);
mt.inc(1);
mt.norm();
t.cmove(mt, s);
Q.cmove(this, ns);
C.copy(Q);
nb = 1 + (t.nbits() + 3) / 4;
/* convert exponent to signed 4-bit window */
for (i = 0; i < nb; i++)
{
w[i] = (sbyte)(t.lastbits(5) - 16);
t.dec(w[i]);
t.norm();
t.fshr(4);
}
w[nb] = (sbyte)t.lastbits(5);
P.copy(W[(w[nb] - 1) / 2]);
for (i = nb - 1; i >= 0; i--)
{
Q.select(W, w[i]);
P.dbl();
P.dbl();
P.dbl();
P.dbl();
P.add(Q);
}
P.sub(C);
P.affine();
return(P);
}