/// <summary>
/// Modulo Exponentiation
/// </summary>
/// <param name="exp"></param>
/// <param name="n"></param>
/// <returns></returns>
public BigInteger ModPow(BigInteger exp, BigInteger n)
{
if ((exp.data[maxLength - 1] & 0x80000000) != 0)
throw (new ArithmeticException("Positive exponents only."));
BigInteger resultNum = 1;
BigInteger tempNum;
bool thisNegative = false;
if ((data[maxLength - 1] & 0x80000000) != 0) // negative this
{
tempNum = -this%n;
thisNegative = true;
}
else
tempNum = this%n; // ensures (tempNum * tempNum) < b^(2k)
if ((n.data[maxLength - 1] & 0x80000000) != 0) // negative n
n = -n;
// calculate constant = b^(2k) / m
var constant = new BigInteger();
int i = n.dataLength << 1;
constant.data[i] = 0x00000001;
constant.dataLength = i + 1;
constant = constant/n;
int totalBits = exp.bitCount();
int count = 0;
// perform squaring and multiply exponentiation
for (int pos = 0; pos < exp.dataLength; pos++)
{
uint mask = 0x01;
//Console.WriteLine("pos = " + pos);
for (int index = 0; index < 32; index++)
{
if ((exp.data[pos] & mask) != 0)
resultNum = BarrettReduction(resultNum*tempNum, n, constant);
mask <<= 1;
tempNum = BarrettReduction(tempNum*tempNum, n, constant);
if (tempNum.dataLength == 1 && tempNum.data[0] == 1)
{
if (thisNegative && (exp.data[0] & 0x1) != 0) //odd exp
return -resultNum;
return resultNum;
}
count++;
if (count == totalBits)
break;
}
}
if (thisNegative && (exp.data[0] & 0x1) != 0) //odd exp
return -resultNum;
return resultNum;
}
/// <summary>
/// Overloading of unary operators
/// </summary>
/// <param name="bi1"></param>
/// <param name="shiftVal"></param>
/// <returns></returns>
public static BigInteger operator <<(BigInteger bi1, int shiftVal)
{
var result = new BigInteger(bi1);
result.dataLength = shiftLeft(result.data, shiftVal);
return result;
}
/// <summary>
/// Overloading of the NOT operator (1's complement)
/// </summary>
/// <param name="bi1"></param>
/// <returns></returns>
public static BigInteger operator ~(BigInteger bi1)
{
var result = new BigInteger(bi1);
for (int i = 0; i < maxLength; i++)
result.data[i] = (~(bi1.data[i]));
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
return result;
}
/// <summary>
/// Overloading of the unary ++ operator
/// </summary>
/// <param name="bi1"></param>
/// <returns></returns>
public static BigInteger operator ++(BigInteger bi1)
{
var result = new BigInteger(bi1);
long val, carry = 1;
int index = 0;
while (carry != 0 && index < maxLength)
{
val = (result.data[index]);
val++;
result.data[index] = (uint) (val & 0xFFFFFFFF);
carry = val >> 32;
index++;
}
if (index > result.dataLength)
result.dataLength = index;
else
{
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
}
// overflow check
int lastPos = maxLength - 1;
// overflow if initial value was +ve but ++ caused a sign
// change to negative.
if ((bi1.data[lastPos] & 0x80000000) == 0 &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException("Overflow in ++."));
}
return result;
}
/// <summary>
/// Overloading of the unary -- operator
/// </summary>
/// <param name="bi1"></param>
/// <returns></returns>
public static BigInteger operator --(BigInteger bi1)
{
var result = new BigInteger(bi1);
long val;
bool carryIn = true;
int index = 0;
while (carryIn && index < maxLength)
{
val = (result.data[index]);
val--;
result.data[index] = (uint) (val & 0xFFFFFFFF);
if (val >= 0)
carryIn = false;
index++;
}
if (index > result.dataLength)
result.dataLength = index;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check
int lastPos = maxLength - 1;
// overflow if initial value was -ve but -- caused a sign
// change to positive.
if ((bi1.data[lastPos] & 0x80000000) != 0 &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException("Underflow in --."));
}
return result;
}
/// <summary>
/// Tests the correct implementation of the modulo exponential function
/// using RSA encryption and decryption (using pre-computed encryption and
/// decryption keys).
/// </summary>
/// <param name="rounds"></param>
public static void RSATest(int rounds)
{
var rand = new Random(1);
var val = new byte[64];
// private and public key
var bi_e =
new BigInteger(
"a932b948feed4fb2b692609bd22164fc9edb59fae7880cc1eaff7b3c9626b7e5b241c27a974833b2622ebe09beb451917663d47232488f23a117fc97720f1e7",
16);
var bi_d =
new BigInteger(
"4adf2f7a89da93248509347d2ae506d683dd3a16357e859a980c4f77a4e2f7a01fae289f13a851df6e9db5adaa60bfd2b162bbbe31f7c8f828261a6839311929d2cef4f864dde65e556ce43c89bbbf9f1ac5511315847ce9cc8dc92470a747b8792d6a83b0092d2e5ebaf852c85cacf34278efa99160f2f8aa7ee7214de07b7",
16);
var bi_n =
new BigInteger(
"e8e77781f36a7b3188d711c2190b560f205a52391b3479cdb99fa010745cbeba5f2adc08e1de6bf38398a0487c4a73610d94ec36f17f3f46ad75e17bc1adfec99839589f45f95ccc94cb2a5c500b477eb3323d8cfab0c8458c96f0147a45d27e45a4d11d54d77684f65d48f15fafcc1ba208e71e921b9bd9017c16a5231af7f",
16);
Console.WriteLine("e =\n" + bi_e.ToString(10));
Console.WriteLine("\nd =\n" + bi_d.ToString(10));
Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");
for (int count = 0; count < rounds; count++)
{
// generate data of random length
int t1 = 0;
while (t1 == 0)
t1 = (int) (rand.NextDouble()*65);
bool done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t1)
val[i] = (byte) (rand.NextDouble()*256);
else
val[i] = 0;
if (val[i] != 0)
done = true;
}
}
while (val[0] == 0)
val[0] = (byte) (rand.NextDouble()*256);
Console.Write("Round = " + count);
// encrypt and decrypt data
var bi_data = new BigInteger(val, t1);
BigInteger bi_encrypted = bi_data.ModPow(bi_e, bi_n);
BigInteger bi_decrypted = bi_encrypted.ModPow(bi_d, bi_n);
// compare
if (bi_decrypted != bi_data)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(bi_data + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
/// <summary>
/// Tests the correct implementation of sqrt() method.
/// </summary>
/// <param name="rounds"></param>
public static void SqrtTest(int rounds)
{
var rand = new Random();
for (int count = 0; count < rounds; count++)
{
// generate data of random length
int t1 = 0;
while (t1 == 0)
t1 = (int) (rand.NextDouble()*1024);
Console.Write("Round = " + count);
var a = new BigInteger();
a.genRandomBits(t1, rand);
BigInteger b = a.sqrt();
BigInteger c = (b + 1)*(b + 1);
// check that b is the largest integer such that b*b <= a
if (c <= a)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(a + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
private bool LucasStrongTestHelper(BigInteger thisVal)
{
// Do the test (selects D based on Selfridge)
// Let D be the first element of the sequence
// 5, -7, 9, -11, 13, ... for which J(D,n) = -1
// Let P = 1, Q = (1-D) / 4
long D = 5, sign = -1, dCount = 0;
bool done = false;
while (!done)
{
int Jresult = Jacobi(D, thisVal);
if (Jresult == -1)
done = true; // J(D, this) = 1
else
{
if (Jresult == 0 && Math.Abs(D) < thisVal) // divisor found
return false;
if (dCount == 20)
{
// check for square
BigInteger root = thisVal.sqrt();
if (root*root == thisVal)
return false;
}
//Console.WriteLine(D);
D = (Math.Abs(D) + 2)*sign;
sign = -sign;
}
dCount++;
}
long Q = (1 - D) >> 2;
/*
Console.WriteLine("D = " + D);
Console.WriteLine("Q = " + Q);
Console.WriteLine("(n,D) = " + thisVal.gcd(D));
Console.WriteLine("(n,Q) = " + thisVal.gcd(Q));
Console.WriteLine("J(D|n) = " + BigInteger.Jacobi(D, thisVal));
*/
BigInteger p_add1 = thisVal + 1;
int s = 0;
for (int index = 0; index < p_add1.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((p_add1.data[index] & mask) != 0)
{
index = p_add1.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = p_add1 >> s;
// calculate constant = b^(2k) / m
// for Barrett Reduction
var constant = new BigInteger();
int nLen = thisVal.dataLength << 1;
constant.data[nLen] = 0x00000001;
constant.dataLength = nLen + 1;
constant = constant/thisVal;
BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0);
bool isPrime = false;
if ((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) ||
(lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
{
// u(t) = 0 or V(t) = 0
isPrime = true;
}
for (int i = 1; i < s; i++)
{
if (!isPrime)
{
// doubling of index
lucas[1] = thisVal.BarrettReduction(lucas[1]*lucas[1], thisVal, constant);
lucas[1] = (lucas[1] - (lucas[2] << 1))%thisVal;
//lucas[1] = ((lucas[1] * lucas[1]) - (lucas[2] << 1)) % thisVal;
if ((lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
isPrime = true;
}
lucas[2] = thisVal.BarrettReduction(lucas[2]*lucas[2], thisVal, constant); //Q^k
}
if (isPrime) // additional checks for composite numbers
{
// If n is prime and gcd(n, Q) == 1, then
// Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n
BigInteger g = thisVal.gcd(Q);
if (g.dataLength == 1 && g.data[0] == 1) // gcd(this, Q) == 1
{
if ((lucas[2].data[maxLength - 1] & 0x80000000) != 0)
lucas[2] += thisVal;
BigInteger temp = (Q*Jacobi(Q, thisVal))%thisVal;
if ((temp.data[maxLength - 1] & 0x80000000) != 0)
temp += thisVal;
if (lucas[2] != temp)
isPrime = false;
}
}
return isPrime;
}
/// <summary>
/// Constructor (Default value provided by BigInteger)
/// </summary>
/// <param name="bi"></param>
public BigInteger(BigInteger bi)
{
data = new uint[maxLength];
dataLength = bi.dataLength;
for (int i = 0; i < dataLength; i++)
data[i] = bi.data[i];
}
/// <summary>
/// Probabilistic prime test based on Rabin-Miller's
/// <![CDATA[
///
/// for any p > 0 with p - 1 = 2^s * t
///
/// p is probably prime (strong pseudoprime) if for any a < p,
/// 1) a^t mod p = 1 or
/// 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
///
/// Otherwise, p is composite.
///
/// Returns
/// -------
/// True if "this" is a strong pseudoprime to randomly chosen
/// bases. The number of chosen bases is given by the "confidence"
/// parameter.
///
/// False if "this" is definitely NOT prime.
/// ]]>
/// </summary>
/// <param name="confidence"></param>
/// <returns></returns>
public bool RabinMillerTest(int confidence)
{
BigInteger thisVal;
if ((data[maxLength - 1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if (thisVal.dataLength == 1)
{
// test small numbers
if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
return false;
else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
return true;
}
if ((thisVal.data[0] & 0x1) == 0) // even numbers
return false;
// calculate values of s and t
BigInteger p_sub1 = thisVal - (new BigInteger(1));
int s = 0;
for (int index = 0; index < p_sub1.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((p_sub1.data[index] & mask) != 0)
{
index = p_sub1.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = p_sub1 >> s;
int bits = thisVal.bitCount();
var a = new BigInteger();
var rand = new Random();
for (int round = 0; round < confidence; round++)
{
bool done = false;
while (!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while (testBits < 2)
testBits = (int) (rand.NextDouble()*bits);
a.genRandomBits(testBits, rand);
int byteLen = a.dataLength;
// make sure "a" is not 0
if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.gcd(thisVal);
if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
return false;
BigInteger b = a.ModPow(t, thisVal);
/*
Console.WriteLine("a = " + a.ToString(10));
Console.WriteLine("b = " + b.ToString(10));
Console.WriteLine("t = " + t.ToString(10));
Console.WriteLine("s = " + s);
*/
bool result = false;
if (b.dataLength == 1 && b.data[0] == 1) // a^t mod p = 1
result = true;
for (int j = 0; result == false && j < s; j++)
{
if (b == p_sub1) // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
{
result = true;
break;
}
b = (b*b)%thisVal;
}
if (result == false)
return false;
}
return true;
}
/// <summary>
/// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
/// <remarks>
/// <![CDATA[
/// p is probably prime if for any a < p (a is not multiple of p),
/// a^((p-1)/2) mod p = J(a, p)
///
/// where J is the Jacobi symbol.
///
/// Otherwise, p is composite.
///
/// Returns
/// -------
/// True if "this" is a Euler pseudoprime to randomly chosen
/// bases. The number of chosen bases is given by the "confidence"
/// parameter.
///
/// False if "this" is definitely NOT prime.
/// ]]></remarks>
/// </summary>
/// <param name="confidence"></param>
/// <returns></returns>
public bool SolovayStrassenTest(int confidence)
{
BigInteger thisVal;
if ((data[maxLength - 1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if (thisVal.dataLength == 1)
{
// test small numbers
if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
return false;
else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
return true;
}
if ((thisVal.data[0] & 0x1) == 0) // even numbers
return false;
int bits = thisVal.bitCount();
var a = new BigInteger();
BigInteger p_sub1 = thisVal - 1;
BigInteger p_sub1_shift = p_sub1 >> 1;
var rand = new Random();
for (int round = 0; round < confidence; round++)
{
bool done = false;
while (!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while (testBits < 2)
testBits = (int) (rand.NextDouble()*bits);
a.genRandomBits(testBits, rand);
int byteLen = a.dataLength;
// make sure "a" is not 0
if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.gcd(thisVal);
if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
return false;
// calculate a^((p-1)/2) mod p
BigInteger expResult = a.ModPow(p_sub1_shift, thisVal);
if (expResult == p_sub1)
expResult = -1;
// calculate Jacobi symbol
BigInteger jacob = Jacobi(a, thisVal);
//Console.WriteLine("a = " + a.ToString(10) + " b = " + thisVal.ToString(10));
//Console.WriteLine("expResult = " + expResult.ToString(10) + " Jacob = " + jacob.ToString(10));
// if they are different then it is not prime
if (expResult != jacob)
return false;
}
return true;
}
//***********************************************************************
// Probabilistic prime test based on Fermat's little theorem
//
//
//***********************************************************************
/// <summary>
/// Probabilistic prime test based on Fermat's little theorem
/// <example>
/// <![CDATA[
///
/// for any a < p (p does not divide a) if
/// a^(p-1) mod p != 1 then p is not prime.
///
/// Otherwise, p is probably prime (pseudoprime to the chosen base).
///
/// Returns
/// -------
/// True if "this" is a pseudoprime to randomly chosen
/// bases. The number of chosen bases is given by the "confidence"
/// parameter.
///
/// False if "this" is definitely NOT prime.
///
/// Note - this method is fast but fails for Carmichael numbers except
/// when the randomly chosen base is a factor of the number.
///
/// ]]>
/// </example>
/// </summary>
/// <param name="confidence"></param>
/// <returns></returns>
public bool FermatLittleTest(int confidence)
{
BigInteger thisVal;
if ((data[maxLength - 1] & 0x80000000) != 0) // negative
thisVal = -this;
else
thisVal = this;
if (thisVal.dataLength == 1)
{
// test small numbers
if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
return false;
else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
return true;
}
if ((thisVal.data[0] & 0x1) == 0) // even numbers
return false;
int bits = thisVal.bitCount();
var a = new BigInteger();
BigInteger p_sub1 = thisVal - (new BigInteger(1));
var rand = new Random();
for (int round = 0; round < confidence; round++)
{
bool done = false;
while (!done) // generate a < n
{
int testBits = 0;
// make sure "a" has at least 2 bits
while (testBits < 2)
testBits = (int) (rand.NextDouble()*bits);
a.genRandomBits(testBits, rand);
int byteLen = a.dataLength;
// make sure "a" is not 0
if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
done = true;
}
// check whether a factor exists (fix for version 1.03)
BigInteger gcdTest = a.gcd(thisVal);
if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
return false;
// calculate a^(p-1) mod p
BigInteger expResult = a.ModPow(p_sub1, thisVal);
int resultLen = expResult.dataLength;
// is NOT prime is a^(p-1) mod p != 1
if (resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1))
{
//Console.WriteLine("a = " + a.ToString());
return false;
}
}
return true;
}
/// <summary>
/// Returns gcd(this, bi)
/// </summary>
/// <param name="bi"></param>
/// <returns></returns>
public BigInteger gcd(BigInteger bi)
{
BigInteger x;
BigInteger y;
if ((data[maxLength - 1] & 0x80000000) != 0) // negative
x = -this;
else
x = this;
if ((bi.data[maxLength - 1] & 0x80000000) != 0) // negative
y = -bi;
else
y = bi;
BigInteger g = y;
while (x.dataLength > 1 || (x.dataLength == 1 && x.data[0] != 0))
{
g = x;
x = y%x;
y = g;
}
return g;
}
//***********************************************************************
// Fast calculation of modular reduction using Barrett's reduction.
// Requires x < b^(2k), where b is the base. In this case, base is
// 2^32 (uint).
//
// Reference [4]
//***********************************************************************
private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
{
int k = n.dataLength,
kPlusOne = k + 1,
kMinusOne = k - 1;
var q1 = new BigInteger();
// q1 = x / b^(k-1)
for (int i = kMinusOne, j = 0; i < x.dataLength; i++, j++)
q1.data[j] = x.data[i];
q1.dataLength = x.dataLength - kMinusOne;
if (q1.dataLength <= 0)
q1.dataLength = 1;
BigInteger q2 = q1*constant;
var q3 = new BigInteger();
// q3 = q2 / b^(k+1)
for (int i = kPlusOne, j = 0; i < q2.dataLength; i++, j++)
q3.data[j] = q2.data[i];
q3.dataLength = q2.dataLength - kPlusOne;
if (q3.dataLength <= 0)
q3.dataLength = 1;
// r1 = x mod b^(k+1)
// i.e. keep the lowest (k+1) words
var r1 = new BigInteger();
int lengthToCopy = (x.dataLength > kPlusOne) ? kPlusOne : x.dataLength;
for (int i = 0; i < lengthToCopy; i++)
r1.data[i] = x.data[i];
r1.dataLength = lengthToCopy;
// r2 = (q3 * n) mod b^(k+1)
// partial multiplication of q3 and n
var r2 = new BigInteger();
for (int i = 0; i < q3.dataLength; i++)
{
if (q3.data[i] == 0) continue;
ulong mcarry = 0;
int t = i;
for (int j = 0; j < n.dataLength && t < kPlusOne; j++, t++)
{
// t = i + j
ulong val = (q3.data[i]*(ulong) n.data[j]) +
r2.data[t] + mcarry;
r2.data[t] = (uint) (val & 0xFFFFFFFF);
mcarry = (val >> 32);
}
if (t < kPlusOne)
r2.data[t] = (uint) mcarry;
}
r2.dataLength = kPlusOne;
while (r2.dataLength > 1 && r2.data[r2.dataLength - 1] == 0)
r2.dataLength--;
r1 -= r2;
if ((r1.data[maxLength - 1] & 0x80000000) != 0) // negative
{
var val = new BigInteger();
val.data[kPlusOne] = 0x00000001;
val.dataLength = kPlusOne + 1;
r1 += val;
}
while (r1 >= n)
r1 -= n;
return r1;
}
//***********************************************************************
// Performs the calculation of the kth term in the Lucas Sequence.
// For details of the algorithm, see reference [9].
//
// k must be odd. i.e LSB == 1
//***********************************************************************
private static BigInteger[] LucasSequenceHelper(BigInteger P, BigInteger Q,
BigInteger k, BigInteger n,
BigInteger constant, int s)
{
var result = new BigInteger[3];
if ((k.data[0] & 0x00000001) == 0)
throw (new ArgumentException("Argument k must be odd."));
int numbits = k.bitCount();
uint mask = (uint) 0x1 << ((numbits & 0x1F) - 1);
// v = v0, v1 = v1, u1 = u1, Q_k = Q^0
BigInteger v = 2%n,
Q_k = 1%n,
v1 = P%n,
u1 = Q_k;
bool flag = true;
for (int i = k.dataLength - 1; i >= 0; i--) // iterate on the binary expansion of k
{
//Console.WriteLine("round");
while (mask != 0)
{
if (i == 0 && mask == 0x00000001) // last bit
break;
if ((k.data[i] & mask) != 0) // bit is set
{
// index doubling with addition
u1 = (u1*v1)%n;
v = ((v*v1) - (P*Q_k))%n;
v1 = n.BarrettReduction(v1*v1, n, constant);
v1 = (v1 - ((Q_k*Q) << 1))%n;
if (flag)
flag = false;
else
Q_k = n.BarrettReduction(Q_k*Q_k, n, constant);
Q_k = (Q_k*Q)%n;
}
else
{
// index doubling
u1 = ((u1*v) - Q_k)%n;
v1 = ((v*v1) - (P*Q_k))%n;
v = n.BarrettReduction(v*v, n, constant);
v = (v - (Q_k << 1))%n;
if (flag)
{
Q_k = Q%n;
flag = false;
}
else
Q_k = n.BarrettReduction(Q_k*Q_k, n, constant);
}
mask >>= 1;
}
mask = 0x80000000;
}
// at this point u1 = u(n+1) and v = v(n)
// since the last bit always 1, we need to transform u1 to u(2n+1) and v to v(2n+1)
u1 = ((u1*v) - Q_k)%n;
v = ((v*v1) - (P*Q_k))%n;
if (flag)
flag = false;
else
Q_k = n.BarrettReduction(Q_k*Q_k, n, constant);
Q_k = (Q_k*Q)%n;
for (int i = 0; i < s; i++)
{
// index doubling
u1 = (u1*v)%n;
v = ((v*v) - (Q_k << 1))%n;
if (flag)
{
Q_k = Q%n;
flag = false;
}
else
Q_k = n.BarrettReduction(Q_k*Q_k, n, constant);
}
result[0] = u1;
result[1] = v;
result[2] = Q_k;
return result;
}
/// <summary>
/// Computes the Jacobi Symbol for a and b.
/// Algorithm adapted from [3] and [4] with some optimizations
/// </summary>
/// <param name="a"></param>
/// <param name="b"></param>
/// <returns></returns>
public static int Jacobi(BigInteger a, BigInteger b)
{
// Jacobi defined only for odd integers
if ((b.data[0] & 0x1) == 0)
throw (new ArgumentException("Jacobi defined only for odd integers."));
if (a >= b) a %= b;
if (a.dataLength == 1 && a.data[0] == 0) return 0; // a == 0
if (a.dataLength == 1 && a.data[0] == 1) return 1; // a == 1
if (a < 0)
{
if ((((b - 1).data[0]) & 0x2) == 0) //if( (((b-1) >> 1).data[0] & 0x1) == 0)
return Jacobi(-a, b);
else
return -Jacobi(-a, b);
}
int e = 0;
for (int index = 0; index < a.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((a.data[index] & mask) != 0)
{
index = a.dataLength; // to break the outer loop
break;
}
mask <<= 1;
e++;
}
}
BigInteger a1 = a >> e;
int s = 1;
if ((e & 0x1) != 0 && ((b.data[0] & 0x7) == 3 || (b.data[0] & 0x7) == 5))
s = -1;
if ((b.data[0] & 0x3) == 3 && (a1.data[0] & 0x3) == 3)
s = -s;
if (a1.dataLength == 1 && a1.data[0] == 1)
return s;
else
return (s*Jacobi(b%a1, a1));
}
/// <summary>
/// Tests the correct implementation of the /, %, * and + operators
/// </summary>
/// <param name="rounds"></param>
public static void MulDivTest(int rounds)
{
var rand = new Random();
var val = new byte[64];
var val2 = new byte[64];
for (int count = 0; count < rounds; count++)
{
// generate 2 numbers of random length
int t1 = 0;
while (t1 == 0)
t1 = (int) (rand.NextDouble()*65);
int t2 = 0;
while (t2 == 0)
t2 = (int) (rand.NextDouble()*65);
bool done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t1)
val[i] = (byte) (rand.NextDouble()*256);
else
val[i] = 0;
if (val[i] != 0)
done = true;
}
}
done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t2)
val2[i] = (byte) (rand.NextDouble()*256);
else
val2[i] = 0;
if (val2[i] != 0)
done = true;
}
}
while (val[0] == 0)
val[0] = (byte) (rand.NextDouble()*256);
while (val2[0] == 0)
val2[0] = (byte) (rand.NextDouble()*256);
Console.WriteLine(count);
var bn1 = new BigInteger(val, t1);
var bn2 = new BigInteger(val2, t2);
// Determine the quotient and remainder by dividing
// the first number by the second.
BigInteger bn3 = bn1/bn2;
BigInteger bn4 = bn1%bn2;
// Recalculate the number
BigInteger bn5 = (bn3*bn2) + bn4;
// Make sure they're the same
if (bn5 != bn1)
{
Console.WriteLine("Error at " + count);
Console.WriteLine(bn1 + "\n");
Console.WriteLine(bn2 + "\n");
Console.WriteLine(bn3 + "\n");
Console.WriteLine(bn4 + "\n");
Console.WriteLine(bn5 + "\n");
return;
}
}
}
/// <summary>
/// Generates a positive BigInteger that is probably prime.
/// </summary>
/// <param name="bits"></param>
/// <param name="confidence"></param>
/// <param name="rand"></param>
/// <returns></returns>
public static BigInteger genPseudoPrime(int bits, int confidence, Random rand)
{
var result = new BigInteger();
bool done = false;
while (!done)
{
result.genRandomBits(bits, rand);
result.data[0] |= 0x01; // make it odd
// prime test
done = result.isProbablePrime(confidence);
}
return result;
}
/// <summary>
/// Tests the correct implementation of the modulo exponential and
/// inverse modulo functions using RSA encryption and decryption. The two
/// pseudoprimes p and q are fixed, but the two RSA keys are generated
/// for each round of testing.
/// </summary>
/// <param name="rounds"></param>
public static void RSATest2(int rounds)
{
var rand = new Random();
var val = new byte[64];
byte[] pseudoPrime1 =
{
0x85, 0x84, 0x64, 0xFD, 0x70, 0x6A,
0x9F, 0xF0, 0x94, 0x0C, 0x3E, 0x2C,
0x74, 0x34, 0x05, 0xC9, 0x55, 0xB3,
0x85, 0x32, 0x98, 0x71, 0xF9, 0x41,
0x21, 0x5F, 0x02, 0x9E, 0xEA, 0x56,
0x8D, 0x8C, 0x44, 0xCC, 0xEE, 0xEE,
0x3D, 0x2C, 0x9D, 0x2C, 0x12, 0x41,
0x1E, 0xF1, 0xC5, 0x32, 0xC3, 0xAA,
0x31, 0x4A, 0x52, 0xD8, 0xE8, 0xAF,
0x42, 0xF4, 0x72, 0xA1, 0x2A, 0x0D,
0x97, 0xB1, 0x31, 0xB3,
};
byte[] pseudoPrime2 =
{
0x99, 0x98, 0xCA, 0xB8, 0x5E, 0xD7,
0xE5, 0xDC, 0x28, 0x5C, 0x6F, 0x0E,
0x15, 0x09, 0x59, 0x6E, 0x84, 0xF3,
0x81, 0xCD, 0xDE, 0x42, 0xDC, 0x93,
0xC2, 0x7A, 0x62, 0xAC, 0x6C, 0xAF,
0xDE, 0x74, 0xE3, 0xCB, 0x60, 0x20,
0x38, 0x9C, 0x21, 0xC3, 0xDC, 0xC8,
0xA2, 0x4D, 0xC6, 0x2A, 0x35, 0x7F,
0xF3, 0xA9, 0xE8, 0x1D, 0x7B, 0x2C,
0x78, 0xFA, 0xB8, 0x02, 0x55, 0x80,
0x9B, 0xC2, 0xA5, 0xCB,
};
var bi_p = new BigInteger(pseudoPrime1);
var bi_q = new BigInteger(pseudoPrime2);
BigInteger bi_pq = (bi_p - 1)*(bi_q - 1);
BigInteger bi_n = bi_p*bi_q;
for (int count = 0; count < rounds; count++)
{
// generate private and public key
BigInteger bi_e = bi_pq.genCoPrime(512, rand);
BigInteger bi_d = bi_e.modInverse(bi_pq);
Console.WriteLine("\ne =\n" + bi_e.ToString(10));
Console.WriteLine("\nd =\n" + bi_d.ToString(10));
Console.WriteLine("\nn =\n" + bi_n.ToString(10) + "\n");
// generate data of random length
int t1 = 0;
while (t1 == 0)
t1 = (int) (rand.NextDouble()*65);
bool done = false;
while (!done)
{
for (int i = 0; i < 64; i++)
{
if (i < t1)
val[i] = (byte) (rand.NextDouble()*256);
else
val[i] = 0;
if (val[i] != 0)
done = true;
}
}
while (val[0] == 0)
val[0] = (byte) (rand.NextDouble()*256);
Console.Write("Round = " + count);
// encrypt and decrypt data
var bi_data = new BigInteger(val, t1);
BigInteger bi_encrypted = bi_data.ModPow(bi_e, bi_n);
BigInteger bi_decrypted = bi_encrypted.ModPow(bi_d, bi_n);
// compare
if (bi_decrypted != bi_data)
{
Console.WriteLine("\nError at round " + count);
Console.WriteLine(bi_data + "\n");
return;
}
Console.WriteLine(" <PASSED>.");
}
}
/// <summary>
/// Generates a random number with the specified number of bits such
/// that gcd(number, this) = 1
/// </summary>
/// <param name="bits"></param>
/// <param name="rand"></param>
/// <returns></returns>
public BigInteger genCoPrime(int bits, Random rand)
{
bool done = false;
var result = new BigInteger();
while (!done)
{
result.genRandomBits(bits, rand);
//Console.WriteLine(result.ToString(16));
// gcd test
BigInteger g = result.gcd(this);
if (g.dataLength == 1 && g.data[0] == 1)
done = true;
}
return result;
}
/// <summary>
/// Overloading of addition operator
/// </summary>
/// <param name="bi1"></param>
/// <param name="bi2"></param>
/// <returns></returns>
public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
{
var result = new BigInteger();
result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
long carry = 0;
for (int i = 0; i < result.dataLength; i++)
{
long sum = bi1.data[i] + (long) bi2.data[i] + carry;
carry = sum >> 32;
result.data[i] = (uint) (sum & 0xFFFFFFFF);
}
if (carry != 0 && result.dataLength < maxLength)
{
result.data[result.dataLength] = (uint) (carry);
result.dataLength++;
}
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check
int lastPos = maxLength - 1;
if ((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException());
}
return result;
}
/// <summary>
/// Returns the modulo inverse of this. Throws ArithmeticException if
/// the inverse does not exist. (i.e. gcd(this, modulus) != 1)
/// </summary>
/// <param name="modulus"></param>
/// <returns></returns>
public BigInteger modInverse(BigInteger modulus)
{
BigInteger[] p = {0, 1};
var q = new BigInteger[2]; // quotients
BigInteger[] r = {0, 0}; // remainders
int step = 0;
BigInteger a = modulus;
BigInteger b = this;
while (b.dataLength > 1 || (b.dataLength == 1 && b.data[0] != 0))
{
var quotient = new BigInteger();
var remainder = new BigInteger();
if (step > 1)
{
BigInteger pval = (p[0] - (p[1]*q[0]))%modulus;
p[0] = p[1];
p[1] = pval;
}
if (b.dataLength == 1)
singleByteDivide(a, b, quotient, remainder);
else
multiByteDivide(a, b, quotient, remainder);
/*
Console.WriteLine(quotient.dataLength);
Console.WriteLine("{0} = {1}({2}) + {3} p = {4}", a.ToString(10),
b.ToString(10), quotient.ToString(10), remainder.ToString(10),
p[1].ToString(10));
*/
q[0] = q[1];
r[0] = r[1];
q[1] = quotient;
r[1] = remainder;
a = b;
b = remainder;
step++;
}
if (r[0].dataLength > 1 || (r[0].dataLength == 1 && r[0].data[0] != 1))
throw (new ArithmeticException("No inverse!"));
BigInteger result = ((p[0] - (p[1]*q[0]))%modulus);
if ((result.data[maxLength - 1] & 0x80000000) != 0)
result += modulus; // get the least positive modulus
return result;
}
/// <summary>
/// Overloading of subtraction operator
/// </summary>
/// <param name="bi1"></param>
/// <param name="bi2"></param>
/// <returns></returns>
public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
{
var result = new BigInteger();
result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
long carryIn = 0;
for (int i = 0; i < result.dataLength; i++)
{
long diff;
diff = bi1.data[i] - (long) bi2.data[i] - carryIn;
result.data[i] = (uint) (diff & 0xFFFFFFFF);
if (diff < 0)
carryIn = 1;
else
carryIn = 0;
}
// roll over to negative
if (carryIn != 0)
{
for (int i = result.dataLength; i < maxLength; i++)
result.data[i] = 0xFFFFFFFF;
result.dataLength = maxLength;
}
// fixed in v1.03 to give correct datalength for a - (-b)
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check
int lastPos = maxLength - 1;
if ((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) &&
(result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
{
throw (new ArithmeticException());
}
return result;
}
/// <summary>
/// Returns a value that is equivalent to the integer square root
/// of the BigInteger.
/// <remarks>
/// <![CDATA[
/// The integer square root of "this" is defined as the largest integer n
/// such that (n * n) <= this
///
/// ]]>
/// </remarks>
/// </summary>
/// <returns></returns>
public BigInteger sqrt()
{
var numBits = (uint) bitCount();
if ((numBits & 0x1) != 0) // odd number of bits
numBits = (numBits >> 1) + 1;
else
numBits = (numBits >> 1);
uint bytePos = numBits >> 5;
var bitPos = (byte) (numBits & 0x1F);
uint mask;
var result = new BigInteger();
if (bitPos == 0)
mask = 0x80000000;
else
{
mask = (uint) 1 << bitPos;
bytePos++;
}
result.dataLength = (int) bytePos;
for (int i = (int) bytePos - 1; i >= 0; i--)
{
while (mask != 0)
{
// guess
result.data[i] ^= mask;
// undo the guess if its square is larger than this
if ((result*result) > this)
result.data[i] ^= mask;
mask >>= 1;
}
mask = 0x80000000;
}
return result;
}
/// <summary>
/// Overloading of multiplication operator
/// </summary>
/// <param name="bi1"></param>
/// <param name="bi2"></param>
/// <returns></returns>
public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
{
int lastPos = maxLength - 1;
bool bi1Neg = false, bi2Neg = false;
// take the absolute value of the inputs
try
{
if ((bi1.data[lastPos] & 0x80000000) != 0) // bi1 negative
{
bi1Neg = true;
bi1 = -bi1;
}
if ((bi2.data[lastPos] & 0x80000000) != 0) // bi2 negative
{
bi2Neg = true;
bi2 = -bi2;
}
}
catch (Exception)
{
}
var result = new BigInteger();
// multiply the absolute values
try
{
for (int i = 0; i < bi1.dataLength; i++)
{
if (bi1.data[i] == 0) continue;
ulong mcarry = 0;
for (int j = 0, k = i; j < bi2.dataLength; j++, k++)
{
// k = i + j
ulong val = (bi1.data[i]*(ulong) bi2.data[j]) +
result.data[k] + mcarry;
result.data[k] = (uint) (val & 0xFFFFFFFF);
mcarry = (val >> 32);
}
if (mcarry != 0)
result.data[i + bi2.dataLength] = (uint) mcarry;
}
}
catch (Exception)
{
throw (new ArithmeticException("Multiplication overflow."));
}
result.dataLength = bi1.dataLength + bi2.dataLength;
if (result.dataLength > maxLength)
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
// overflow check (result is -ve)
if ((result.data[lastPos] & 0x80000000) != 0)
{
if (bi1Neg != bi2Neg && result.data[lastPos] == 0x80000000) // different sign
{
// handle the special case where multiplication produces
// a max negative number in 2's complement.
if (result.dataLength == 1)
return result;
else
{
bool isMaxNeg = true;
for (int i = 0; i < result.dataLength - 1 && isMaxNeg; i++)
{
if (result.data[i] != 0)
isMaxNeg = false;
}
if (isMaxNeg)
return result;
}
}
throw (new ArithmeticException("Multiplication overflow."));
}
// if input has different signs, then result is -ve
if (bi1Neg != bi2Neg)
return -result;
return result;
}
/// <summary>
/// Constructor (Default value provided by a string of digits of the specified base)
/// <example>
/// Example (base 10)
/// -----------------
/// To initialize "a" with the default value of 1234 in base 10
/// BigInteger a = new BigInteger("1234", 10)
///
/// To initialize "a" with the default value of -1234
/// BigInteger a = new BigInteger("-1234", 10)
///
/// Example (base 16)
/// -----------------
/// To initialize "a" with the default value of 0x1D4F in base 16
/// BigInteger a = new BigInteger("1D4F", 16)
///
/// To initialize "a" with the default value of -0x1D4F
/// BigInteger a = new BigInteger("-1D4F", 16)
///
/// Note that string values are specified in the £¨sign£¬magnitude£©
/// format.
/// </example>
/// </summary>
/// <param name="value"></param>
/// <param name="radix"></param>
public BigInteger(string value, int radix)
{
var multiplier = new BigInteger(1);
var result = new BigInteger();
value = (value.ToUpper()).Trim();
int limit = 0;
if (value[0] == '-')
limit = 1;
for (int i = value.Length - 1; i >= limit; i--)
{
int posVal = value[i];
if (posVal >= '0' && posVal <= '9')
posVal -= '0';
else if (posVal >= 'A' && posVal <= 'Z')
posVal = (posVal - 'A') + 10;
else
posVal = 9999999; // arbitrary large
if (posVal >= radix)
throw (new ArithmeticException("Invalid string in constructor."));
else
{
if (value[0] == '-')
posVal = -posVal;
result = result + (multiplier*posVal);
if ((i - 1) >= limit)
multiplier = multiplier*radix;
}
}
if (value[0] == '-') // negative values
{
if ((result.data[maxLength - 1] & 0x80000000) == 0)
throw (new ArithmeticException("Negative underflow in constructor."));
}
else // positive values
{
if ((result.data[maxLength - 1] & 0x80000000) != 0)
throw (new ArithmeticException("Positive overflow in constructor."));
}
data = new uint[maxLength];
for (int i = 0; i < result.dataLength; i++)
data[i] = result.data[i];
dataLength = result.dataLength;
}
/// <summary>
/// Overloading of unary >> operators
/// </summary>
/// <param name="bi1"></param>
/// <param name="shiftVal"></param>
/// <returns></returns>
public static BigInteger operator >>(BigInteger bi1, int shiftVal)
{
var result = new BigInteger(bi1);
result.dataLength = shiftRight(result.data, shiftVal);
if ((bi1.data[maxLength - 1] & 0x80000000) != 0) // negative
{
for (int i = maxLength - 1; i >= result.dataLength; i--)
result.data[i] = 0xFFFFFFFF;
uint mask = 0x80000000;
for (int i = 0; i < 32; i++)
{
if ((result.data[result.dataLength - 1] & mask) != 0)
break;
result.data[result.dataLength - 1] |= mask;
mask >>= 1;
}
result.dataLength = maxLength;
}
return result;
}
/// <summary>
/// <![CDATA[
/// Returns the k_th number in the Lucas Sequence reduced modulo n.
///
/// Uses index doubling to speed up the process. For example, to calculate V(k),
/// we maintain two numbers in the sequence V(n) and V(n+1).
///
/// To obtain V(2n), we use the identity
/// V(2n) = (V(n) * V(n)) - (2 * Q^n)
/// To obtain V(2n+1), we first write it as
/// V(2n+1) = V((n+1) + n)
/// and use the identity
/// V(m+n) = V(m) * V(n) - Q * V(m-n)
/// Hence,
/// V((n+1) + n) = V(n+1) * V(n) - Q^n * V((n+1) - n)
/// = V(n+1) * V(n) - Q^n * V(1)
/// = V(n+1) * V(n) - Q^n * P
///
/// We use k in its binary expansion and perform index doubling for each
/// bit position. For each bit position that is set, we perform an
/// index doubling followed by an index addition. This means that for V(n),
/// we need to update it to V(2n+1). For V(n+1), we need to update it to
/// V((2n+1)+1) = V(2*(n+1))
///
/// This function returns
/// [0] = U(k)
/// [1] = V(k)
/// [2] = Q^n
///
/// Where U(0) = 0 % n, U(1) = 1 % n
/// V(0) = 2 % n, V(1) = P % n
/// ]]>
/// </summary>
/// <param name="P"></param>
/// <param name="Q"></param>
/// <param name="k"></param>
/// <param name="n"></param>
/// <returns></returns>
public static BigInteger[] LucasSequence(BigInteger P, BigInteger Q,
BigInteger k, BigInteger n)
{
if (k.dataLength == 1 && k.data[0] == 0)
{
var result = new BigInteger[3];
result[0] = 0;
result[1] = 2%n;
result[2] = 1%n;
return result;
}
// calculate constant = b^(2k) / m
// for Barrett Reduction
var constant = new BigInteger();
int nLen = n.dataLength << 1;
constant.data[nLen] = 0x00000001;
constant.dataLength = nLen + 1;
constant = constant/n;
// calculate values of s and t
int s = 0;
for (int index = 0; index < k.dataLength; index++)
{
uint mask = 0x01;
for (int i = 0; i < 32; i++)
{
if ((k.data[index] & mask) != 0)
{
index = k.dataLength; // to break the outer loop
break;
}
mask <<= 1;
s++;
}
}
BigInteger t = k >> s;
//Console.WriteLine("s = " + s + " t = " + t);
return LucasSequenceHelper(P, Q, t, n, constant, s);
}
/// <summary>
/// Overloading of the NEGATE operator (2's complement)
/// </summary>
/// <param name="bi1"></param>
/// <returns></returns>
public static BigInteger operator -(BigInteger bi1)
{
// handle neg of zero separately since it'll cause an overflow
// if we proceed.
if (bi1.dataLength == 1 && bi1.data[0] == 0)
return (new BigInteger());
var result = new BigInteger(bi1);
// 1's complement
for (int i = 0; i < maxLength; i++)
result.data[i] = (~(bi1.data[i]));
// add one to result of 1's complement
long val, carry = 1;
int index = 0;
while (carry != 0 && index < maxLength)
{
val = (result.data[index]);
val++;
result.data[index] = (uint) (val & 0xFFFFFFFF);
carry = val >> 32;
index++;
}
if ((bi1.data[maxLength - 1] & 0x80000000) == (result.data[maxLength - 1] & 0x80000000))
throw (new ArithmeticException("Overflow in negation.\n"));
result.dataLength = maxLength;
while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
result.dataLength--;
return result;
}
//***********************************************************************
// Returns a string representing the BigInteger in sign-and-magnitude
// format in the specified radix.
//
// Example
// -------
// If the value of BigInteger is -255 in base 10, then
// ToString(16) returns "-FF"
//
//***********************************************************************
/// <summary>
/// Returns a string representing the BigInteger in sign-and-magnitude
/// format in the specified radix.
/// <example>
/// // If the value of BigInteger is -255 in base 10, then
/// ToString(16) returns "-FF"
/// </example>
/// </summary>
/// <param name="radix"></param>
/// <returns></returns>
public string ToString(int radix)
{
if (radix < 2 || radix > 36)
throw (new ArgumentException("Radix must be >= 2 and <= 36"));
string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
string result = "";
BigInteger a = this;
bool negative = false;
if ((a.data[maxLength - 1] & 0x80000000) != 0)
{
negative = true;
try
{
a = -a;
}
catch (Exception)
{
}
}
var quotient = new BigInteger();
var remainder = new BigInteger();
var biRadix = new BigInteger(radix);
if (a.dataLength == 1 && a.data[0] == 0)
result = "0";
else
{
while (a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0))
{
singleByteDivide(a, biRadix, quotient, remainder);
if (remainder.data[0] < 10)
result = remainder.data[0] + result;
else
result = charSet[(int) remainder.data[0] - 10] + result;
a = quotient;
}
if (negative)
result = "-" + result;
}
return result;
}