/// <summary>
/// Случайная точка на Данной кривой.
/// </summary>
/// <returns></returns>
BigInteger[] GetPoint()
{
BigInteger rnd = Extension.Random(1, P);
while (Extension.Lezhandr(BigInteger.Remainder((rnd * rnd * rnd + A * rnd + B), P), P) != 1)
{
rnd = Extension.Random(1, P);
}
BigInteger xcoord = BigInteger.Remainder((rnd * rnd * rnd + A * rnd + B), P); // Правая часть уравнения кривой
return(new BigInteger[] { rnd, Extension.SquareRootModPrime(xcoord, P) });
}
void Roots(Polynom g)
{
if (g.Degree <= 2)
{
// Решение уравнений вида ax^2+bx+c = 0 mod p a,b,c>=0
if (g.coefficients[0].Degree == 1) // bx + c = 0
{
if (g.coefficients.Count == 1) // bx = 0
{
roots.Add(0);
return;
}
else // bx + c = 0
{
BigInteger b = g.coefficients[0].Coefficient;
BigInteger c = g.coefficients[1].Coefficient;
BigInteger x = BigInteger.Remainder(BigInteger.Negate(c) * Extension.Inverse(b, g.Fp), g.Fp);
while (x < 0)
{
x += g.Fp;
}
roots.Add(x);
return;
}
// ошибка
}
else // ax^2 + bx + c = 0
{
if (g.coefficients[1].Degree == 0) // ax^2 + c = 0
{
BigInteger a = g.coefficients[0].Coefficient;
BigInteger c = g.coefficients[1].Coefficient;
BigInteger ac = BigInteger.Remainder(BigInteger.Negate(c) * Extension.Inverse(a, g.Fp), g.Fp);
BigInteger r = Extension.SquareRootModPrime(ac, g.Fp);
roots.Add(r); // не проверенно
roots.Add(g.Fp - r);
return;
}
else // ax^2 + bx = 0 && ax^2 = 0 && ax^2 + bx + c = 0
{
if (g.coefficients.Count == 1)// ax ^ 2 = 0
{
roots.Add(0);
return;
}
else // ax^2 + bx = 0 && ax^2 + bx + c = 0
{
if (g.coefficients.Count == 2) // ax^2 + bx = 0
{
roots.Add(0);
Polynom div = null;
Polynom.Remainder(g, Parse("+x", this.Fp), out div);
Roots(div);
}
else // ax^2 + bx + c = 0
{
BigInteger a = g.coefficients[0].Coefficient;
BigInteger b = g.coefficients[1].Coefficient;
BigInteger c = g.coefficients[2].Coefficient;
BigInteger D = Extension.SquareRootModPrime(BigInteger.Remainder(b * b - 4 * a * c, g.Fp), g.Fp);
BigInteger x1 = BigInteger.Remainder((BigInteger.Negate(b) - D) * BigInteger.Remainder(Extension.Inverse(2 * a, g.Fp), g.Fp), g.Fp);
BigInteger x2 = BigInteger.Remainder((BigInteger.Negate(b) + D) * BigInteger.Remainder(Extension.Inverse(2 * a, g.Fp), g.Fp), g.Fp);
while (x1 < 0)
{
x1 += g.Fp;
}
while (x2 < 0)
{
x2 += g.Fp;
}
roots.Add(x1);
roots.Add(x2);
}
}
}
}
}
else
{
Polynom one = Parse("+1", g.Fp);
Polynom h = one;
while (h.Degree == 0 || h.Eguals(g))
{
BigInteger a = Extension.Random(1, g.Fp);
Polynom s = ModPow(Parse(string.Format("+x-{0}", a), g.Fp), (g.Fp - 1) / 2, g);
h = Polynom.GreatCommonDivisor(s - one, g);
}
Polynom div = null;
Remainder(g, h, out div);
Roots(h);
Roots(div);
}
}
/// <summary>
/// Явные Параметры кривых полученных комплексным умножением
/// </summary>
/// <param name="D">Фундаментальный дискриминант</param>
/// <returns></returns>
private static List <BigInteger[]> GetClearCurveComplexMultiply(int D)
{
Task <BigInteger> t1 = Task <BigInteger> .Factory.StartNew(delegate() { return(GetNonQuadr()); });
List <BigInteger[]> paramCurve = new List <BigInteger[]>();
switch (D)
{
case (-7):
BigInteger r = 125;
BigInteger s = 189;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r * BigInteger.ModPow(s, 3, Number), Number), BigInteger.Remainder(2 * r * BigInteger.ModPow(s, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r * BigInteger.ModPow(s, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r * BigInteger.ModPow(s, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-8):
BigInteger r0 = 125;
BigInteger s0 = 98;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r0 * BigInteger.ModPow(s0, 3, Number), Number), BigInteger.Remainder(2 * r0 * BigInteger.ModPow(s0, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r0 * BigInteger.ModPow(s0, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r0 * BigInteger.ModPow(s0, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-11):
BigInteger r1 = 512;
BigInteger s1 = 539;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r1 * BigInteger.ModPow(s1, 3, Number), Number), BigInteger.Remainder(2 * r1 * BigInteger.ModPow(s1, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r1 * BigInteger.ModPow(s1, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r1 * BigInteger.ModPow(s1, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-19):
BigInteger r2 = 512;
BigInteger s2 = 512;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r2 * BigInteger.ModPow(s2, 3, Number), Number), BigInteger.Remainder(2 * r2 * BigInteger.ModPow(s2, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r2 * BigInteger.ModPow(s2, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r2 * BigInteger.ModPow(s2, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-43):
BigInteger r3 = 512000;
BigInteger s3 = 512001;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r3 * BigInteger.ModPow(s3, 3, Number), Number), BigInteger.Remainder(2 * r3 * BigInteger.ModPow(s3, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r3 * BigInteger.ModPow(s3, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r3 * BigInteger.ModPow(s3, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-67):
BigInteger r4 = 85184000;
BigInteger s4 = 85184001;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r4 * BigInteger.ModPow(s4, 3, Number), Number), BigInteger.Remainder(2 * r4 * BigInteger.ModPow(s4, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r4 * BigInteger.ModPow(s4, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r4 * BigInteger.ModPow(s4, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-163):
BigInteger r5 = 151931373056000;
BigInteger s5 = 151931373056001;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r5 * BigInteger.ModPow(s5, 3, Number), Number), BigInteger.Remainder(2 * r5 * BigInteger.ModPow(s5, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r5 * BigInteger.ModPow(s5, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r5 * BigInteger.ModPow(s5, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-15):
BigInteger r6 = 1225 - 2080 * Extension.SquareRootModPrime(5, Number);
BigInteger s6 = 5929;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r6 * BigInteger.ModPow(s6, 3, Number), Number), BigInteger.Remainder(2 * r6 * BigInteger.ModPow(s6, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r6 * BigInteger.ModPow(s6, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r6 * BigInteger.ModPow(s6, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-20):
BigInteger r7 = 108250 + 29835 * Extension.SquareRootModPrime(5, Number);
BigInteger s7 = 174724;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r7 * BigInteger.ModPow(s7, 3, Number), Number), BigInteger.Remainder(2 * r7 * BigInteger.ModPow(s7, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r7 * BigInteger.ModPow(s7, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r7 * BigInteger.ModPow(s7, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-24):
BigInteger r8 = 1757 - 494 * Extension.SquareRootModPrime(2, Number);
BigInteger s8 = 1058;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r8 * BigInteger.ModPow(s8, 3, Number), Number), BigInteger.Remainder(2 * r8 * BigInteger.ModPow(s8, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r8 * BigInteger.ModPow(s8, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r8 * BigInteger.ModPow(s8, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-35):
BigInteger r9 = -1126400 - 1589760 * Extension.SquareRootModPrime(5, Number);
BigInteger s9 = 2428447;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r9 * BigInteger.ModPow(s9, 3, Number), Number), BigInteger.Remainder(2 * r9 * BigInteger.ModPow(s9, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r9 * BigInteger.ModPow(s9, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r9 * BigInteger.ModPow(s9, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-40):
BigInteger r11 = 54175 - 1020 * Extension.SquareRootModPrime(5, Number);
BigInteger s11 = 51894;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r11 * BigInteger.ModPow(s11, 3, Number), Number), BigInteger.Remainder(2 * r11 * BigInteger.ModPow(s11, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r11 * BigInteger.ModPow(s11, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r11 * BigInteger.ModPow(s11, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-51):
BigInteger r12 = 75520 - 7936 * Extension.SquareRootModPrime(17, Number);
BigInteger s12 = 108241;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r12 * BigInteger.ModPow(s12, 3, Number), Number), BigInteger.Remainder(2 * r12 * BigInteger.ModPow(s12, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * r12 * BigInteger.ModPow(s12, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * r12 * BigInteger.ModPow(s12, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-52):
BigInteger rq = 1778750 + 5125 * Extension.SquareRootModPrime(13, Number);
BigInteger sq = 1797228;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rq * BigInteger.ModPow(sq, 3, Number), Number), BigInteger.Remainder(2 * rq * BigInteger.ModPow(sq, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rq * BigInteger.ModPow(sq, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rq * BigInteger.ModPow(sq, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-88):
BigInteger rw = 181713125 - 44250 * Extension.SquareRootModPrime(2, Number);
BigInteger sw = 181650546;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rw * BigInteger.ModPow(sw, 3, Number), Number), BigInteger.Remainder(2 * rw * BigInteger.ModPow(sw, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rw * BigInteger.ModPow(sw, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rw * BigInteger.ModPow(sw, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-91):
BigInteger re = 74752 - 36352 * Extension.SquareRootModPrime(13, Number);
BigInteger se = 205821;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * re * BigInteger.ModPow(se, 3, Number), Number), BigInteger.Remainder(2 * re * BigInteger.ModPow(se, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * re * BigInteger.ModPow(se, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * re * BigInteger.ModPow(se, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-115):
BigInteger ra = 269593600 - 89157120 * Extension.SquareRootModPrime(5, Number);
BigInteger sa = 468954981;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * ra * BigInteger.ModPow(sa, 3, Number), Number), BigInteger.Remainder(2 * ra * BigInteger.ModPow(sa, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * ra * BigInteger.ModPow(sa, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * ra * BigInteger.ModPow(sa, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-123):
BigInteger rs = 1025058304000 - 1248832000 * Extension.SquareRootModPrime(41, Number);
BigInteger ss = 1033054730449;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rs * BigInteger.ModPow(ss, 3, Number), Number), BigInteger.Remainder(2 * rs * BigInteger.ModPow(ss, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rs * BigInteger.ModPow(ss, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rs * BigInteger.ModPow(ss, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-148):
BigInteger rd = 499833128054750 + 356500625 * Extension.SquareRootModPrime(37, Number);
BigInteger sd = 499835296563372;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rd * BigInteger.ModPow(sd, 3, Number), Number), BigInteger.Remainder(2 * rd * BigInteger.ModPow(sd, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rd * BigInteger.ModPow(sd, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rd * BigInteger.ModPow(sd, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-187):
BigInteger rz = 91878880000 - 1074017568000 * Extension.SquareRootModPrime(17, Number);
BigInteger sz = 4520166756633;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rz * BigInteger.ModPow(sz, 3, Number), Number), BigInteger.Remainder(2 * rz * BigInteger.ModPow(sz, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rz * BigInteger.ModPow(sz, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rz * BigInteger.ModPow(sz, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-232):
BigInteger rx = 1728371226151263375 - 11276414500 * Extension.SquareRootModPrime(29, Number);
BigInteger sx = 1728371165425912854;
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rx * BigInteger.ModPow(sx, 3, Number), Number), BigInteger.Remainder(2 * rx * BigInteger.ModPow(sx, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rx * BigInteger.ModPow(sx, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rx * BigInteger.ModPow(sx, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-267):
BigInteger rc = BigInteger.Parse("3632253349307716000000") - 12320504793376000 * Extension.SquareRootModPrime(89, Number);
BigInteger sc = BigInteger.Parse("3632369580717474122449");
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rc * BigInteger.ModPow(sc, 3, Number), Number), BigInteger.Remainder(2 * rc * BigInteger.ModPow(sc, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rc * BigInteger.ModPow(sc, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rc * BigInteger.ModPow(sc, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-403):
BigInteger rcq = BigInteger.Parse("16416107434811840000") - 4799513373120384000 * Extension.SquareRootModPrime(13, Number);
BigInteger scq = BigInteger.Parse("33720998998872514077");
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rcq * BigInteger.ModPow(scq, 3, Number), Number), BigInteger.Remainder(2 * rcq * BigInteger.ModPow(scq, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rcq * BigInteger.ModPow(scq, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rcq * BigInteger.ModPow(scq, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
case (-427):
BigInteger rcqa = BigInteger.Parse("564510997315289728000") - 5784785611102784000 * Extension.SquareRootModPrime(61, Number);
BigInteger scqa = BigInteger.Parse("609691617259594724421");
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rcqa * BigInteger.ModPow(scqa, 3, Number), Number), BigInteger.Remainder(2 * rcqa * BigInteger.ModPow(scqa, 5, Number), Number) });
t1.Wait();
paramCurve.Add(new BigInteger[] { BigInteger.Remainder(-3 * rcqa * BigInteger.ModPow(scqa, 3, Number) * BigInteger.ModPow(t1.Result, 2, Number), Number), BigInteger.Remainder(2 * rcqa * BigInteger.ModPow(scqa, 5, Number) * BigInteger.ModPow(t1.Result, 3, Number), Number) });
break;
}
return(paramCurve);
}