/// <summary>
/// M.
/// Computes x to the power of y at mod z.
/// </summary>
/// <returns></returns>
public static BigNum Power(BigNum x, BigNum y, BigNum z)
{
BigNum result = new BigNum();
if (y.CompNumWithoutPrint(GlobalVariables.zero))
{
result = new BigNum(GlobalVariables.one);
}
else
{
if (y.IsEven())
{
// x^y = [x^(y/2)]^2 mod z.
BigNum yDividedBy2 = new BigNum();
DivNum(y, GlobalVariables.two, yDividedBy2, result); // y = y/2.
result = Power(x, yDividedBy2, z);
result.MultNum(new BigNum(result));
DivNum(result, z, new BigNum(), result);
}
else
{
BigNum yMinus1DividedBy2 = new BigNum(y);
yMinus1DividedBy2.SubNum(GlobalVariables.one); // y = y-1.
DivNum(y, GlobalVariables.two, yMinus1DividedBy2, result); // y = y-1/2.
result = Power(x, yMinus1DividedBy2, z);
result.MultNum(new BigNum(result));
DivNum(result, z, new BigNum(), result);
result.MultNum(x);
DivNum(result, z, new BigNum(), result);
}
}
return(result);
}
/// <summary>
/// Gets BigNum <paramref name="b"/>.
/// Generates p and q, prime numbers.
/// While b is not inverse number to p-1 and q-1, p and q re-generates.
/// After finding good p and q, calculate a that is inverse of b mod ((p-1)(q-1)).
/// n is calculated as p * q.
/// Returns by reference the BigNum a, p, q, and n.
/// </summary>
/// <param name="b"></param>
/// <param name="a"></param>
/// <param name="q"></param>
/// <param name="p"></param>
/// <param name="nByRef"></param>
public static void GenRSA(BigNum b, BigNum a, BigNum q, BigNum p, BigNum nByRef)
{
bool isBInverseOfPMinus1AndQMinus1 = false;
BigNum n = new BigNum();
while (!isBInverseOfPMinus1AndQMinus1)
{
p.Set(GenPrime());
q.Set(GenPrime());
BigNum pMinus1 = new BigNum(p);
pMinus1.SubNum(GlobalVariables.one);
BigNum qMinus1 = new BigNum(q);
qMinus1.SubNum(GlobalVariables.one);
BigNum fiOfN = new BigNum(pMinus1);
fiOfN.MultNum(qMinus1);
BigNum gcd = new BigNum();
ExtendedGCD(b, fiOfN, new BigNum(), new BigNum(), gcd);
if (gcd.CompNumWithoutPrint(GlobalVariables.one))
{
isBInverseOfPMinus1AndQMinus1 = true;
a.Set(Inverse(b, fiOfN));
if (a.IsNegative)
{
a.AddNum(fiOfN);
}
n = new BigNum(p);
n.MultNum(q);
}
}
nByRef.Set(n);
}
/// <summary>
/// Overloading the ">=" operator.
/// If num1 >= num2, num1-num2 should be positive => subResult.IsNegative = false.
/// It should return true, therefore !subResult.IsNegative.
/// </summary>
/// <param name="num1"></param>
/// <param name="num2"></param>
/// <returns></returns>
public static bool operator >=(BigNum num1, BigNum num2)
{
BigNum subResult = new BigNum(num1);
subResult.SubNum(num2);
return(!subResult.IsNegative);
}
/// <summary>
/// K.
/// Gets two BigNum type numbers a and m and returns by reference (d, x, y), also BigNum type numbers.
/// d represents GCD(m,a). (Greatest common divider).
/// Note: If d = 1, it means that x is the opposite number of m(mod a) and
/// y is the opposite number of a(mod m).
/// Proofe for Note: 1 = [m*x + a*y](mod m) ==> 1 = a*y.
/// </summary>
/// <param name="a"></param>
/// <param name="m"></param>
/// <param name="x"></param>
/// <param name="y"></param>
/// <param name="d"></param>
public static void ExtendedGCD(BigNum a, BigNum m, BigNum x, BigNum y, BigNum d)
{
BigNum r0 = new BigNum(m);
BigNum x0 = new BigNum(GlobalVariables.one);
BigNum y0 = new BigNum(GlobalVariables.zero);
BigNum r1 = new BigNum(a);
BigNum x1 = new BigNum(GlobalVariables.zero);
BigNum y1 = new BigNum(GlobalVariables.one);
BigNum r = new BigNum(GlobalVariables.one); // Just to be able to enter the while loop.
BigNum q = new BigNum();
DivNum(r0, r1, q, r);
while (!r.CompNumWithoutPrint(GlobalVariables.zero))
{
// x = x0 - qx1.
BigNum qx1 = new BigNum(q);
qx1.MultNum(x1);
x.Set(x0);
x.SubNum(qx1);
// y = y0 - qy1.
BigNum qy1 = new BigNum(q);
qy1.MultNum(y1);
y.Set(y0);
y.SubNum(qy1);
r0.Set(r1);
x0.Set(x1);
y0.Set(y1);
r1.Set(r);
x1.Set(x);
y1.Set(y);
DivNum(r0, r1, q, r);
}
d.Set(r1);
x.Set(x1);
y.Set(y1);
}
public static void I()
{
Console.WriteLine("(I) Using SubNum method.");
Console.WriteLine("18 - 5 = 13");
BigNum bigNumI1 = new BigNum(18L);
BigNum bigNumI2 = new BigNum(5L);
bigNumI1.SubNum(bigNumI2);
Console.WriteLine(bigNumI1.WriteNum());
Console.WriteLine("375683 - 12345 = 363338 (1011000101101001010 in binary.)");
BigNum bigNumI3 = new BigNum(375683L);
BigNum bigNumI4 = new BigNum(12345L);
bigNumI3.SubNum(bigNumI4);
Console.WriteLine(bigNumI3.WriteNum());
Console.WriteLine();
}
/// <summary>
/// N.
/// Runs Miller-Rabin algorithm k times.
/// </summary>
/// <param name="n"></param>
/// <param name="k"></param>
/// <returns></returns>
public static bool IsPrime(BigNum n, int k)
{
bool isPrimeNumber = true;
BigNum t = new BigNum();
BigNum u = new BigNum();
BigNum nMinusOne = new BigNum(n);
nMinusOne.SubNum(GlobalVariables.one);
FindExpressionOfGivenNumberByPowerOfTwoTimesOddNumber(nMinusOne, t, u);
for (int i = 0; i < k; ++i)
{
if (!MillerRabinAlgorithm(n, new BigNum(t), u))
{
isPrimeNumber = false;
break;
}
}
return(isPrimeNumber);
}
/// <summary>
/// J.
/// Gets two BigNum type numbers, divided and divider.
/// Returns by reference the result - quotient and reminder.
/// </summary>
/// <param name="divider"></param>
public static void DivNum(BigNum divided, BigNum divider, BigNum quotientRetrunByRef, BigNum reminderReturnByRef)
{
List <int> positionsOf1s = new List <int>();
if (divided.MSB1Location.HasValue)
{
int indexPos = divided.MSB1Location.Value + divider.Length - 1;
BigNum subResult = new BigNum(divided);
while (indexPos < GlobalVariables.MAX_NUM)
{
BigNum numberFromLeftToIndex = GetSubBigNumFromLeftToIndex(subResult, indexPos);
if (numberFromLeftToIndex >= divider)
{
// Saving quotient's info.
positionsOf1s.Add(indexPos);
// Subtract.
int numberOfShifts = GlobalVariables.MAX_NUM - indexPos - 1;
BigNum numberToSubtract = new BigNum(divider);
numberToSubtract.ShiftLeft(numberOfShifts);
subResult.SubNum(numberToSubtract);
}
indexPos++;
}
quotientRetrunByRef.ReadNum(positionsOf1s);
reminderReturnByRef.Set(subResult);
}
else
{
Console.WriteLine("Cannot divide by zero");
}
}