internal void GreatestCommonDivisor(Integer X, Integer Y, Integer Gcd)
{
// This is the basic Euclidean Algorithm.
if (X.IsZero())
{
throw(new Exception("Doing GCD with a parameter that is zero."));
}
if (Y.IsZero())
{
throw(new Exception("Doing GCD with a parameter that is zero."));
}
if (X.IsEqual(Y))
{
Gcd.Copy(X);
return;
}
// Don't change the original numbers that came in as parameters.
if (X.ParamIsGreater(Y))
{
GcdX.Copy(Y);
GcdY.Copy(X);
}
else
{
GcdX.Copy(X);
GcdY.Copy(Y);
}
while (true)
{
IntMath.Divider.Divide(GcdX, GcdY, Quotient, Remainder);
if (Remainder.IsZero())
{
Gcd.Copy(GcdY); // It's the smaller one.
// It can't return from this loop until the remainder is zero.
return;
}
GcdX.Copy(GcdY);
GcdY.Copy(Remainder);
}
}
internal string ToString10(Integer From)
{
if (From.IsULong())
{
ulong N = From.GetAsULong();
if (From.IsNegative)
{
return("-" + N.ToString("N0"));
}
else
{
return(N.ToString("N0"));
}
}
string Result = "";
ToDivide.Copy(From);
int CommaCount = 0;
while (!ToDivide.IsZero())
{
uint Digit = (uint)Divider.ShortDivideRem(ToDivide, 10, Quotient);
ToDivide.Copy(Quotient);
if (((CommaCount % 3) == 0) && (CommaCount != 0))
{
Result = Digit.ToString() + "," + Result; // Or use a StringBuilder.
}
else
{
Result = Digit.ToString() + Result;
}
CommaCount++;
}
if (From.IsNegative)
{
return("-" + Result);
}
else
{
return(Result);
}
}
internal bool FindMultiplicativeInverseSmall(Integer ToFind, Integer KnownNumber, Integer Modulus)
{
// This method is for: KnownNumber * ToFind = 1 mod Modulus
// An example:
// PublicKeyExponent * X = 1 mod PhiN.
// PublicKeyExponent * X = 1 mod (P - 1)(Q - 1).
// This means that
// (PublicKeyExponent * X) = (Y * PhiN) + 1
// X is less than PhiN.
// So Y is less than PublicKExponent.
// Y can't be zero.
// If this equation can be solved then it can be solved modulo
// any number. So it has to be solvable mod PublicKExponent.
// See: Hasse Principle.
// This also depends on the idea that the KnownNumber is prime and
// that there is one unique modular inverse.
// if( !KnownNumber-is-a-prime )
// then it won't work.
if (!KnownNumber.IsULong())
{
throw(new Exception("FindMultiplicativeInverseSmall() was called with too big of a KnownNumber."));
}
ulong KnownNumberULong = KnownNumber.GetAsULong();
// 65537
if (KnownNumberULong > 1000000)
{
throw(new Exception("KnownNumberULong > 1000000. FindMultiplicativeInverseSmall() was called with too big of an exponent."));
}
// (Y * PhiN) + 1 mod PubKExponent has to be zero if Y is a solution.
ulong ModulusModKnown = Divider.GetMod32(Modulus, KnownNumberULong);
// Worker.ReportProgress( 0, "ModulusModExponent: " + ModulusModKnown.ToString( "N0" ));
// if( Worker.CancellationPending )
// return false;
// Y can't be zero.
// The exponent is a small number like 65537.
for (uint Y = 1; Y < (uint)KnownNumberULong; Y++)
{
ulong X = (ulong)Y * ModulusModKnown;
X++; // Add 1 to it for (Y * PhiN) + 1.
X = X % KnownNumberULong;
if (X == 0)
{
// if( Worker.CancellationPending )
// return false;
// What is PhiN mod 65537?
// That gives me Y.
// The private key exponent is X*65537 + ModPart
// The CipherText raised to that is the PlainText.
// P + zN = C^(X*65537 + ModPart)
// P + zN = C^(X*65537)(C^ModPart)
// P + zN = ((C^65537)^X)(C^ModPart)
// Worker.ReportProgress( 0, "Found Y at: " + Y.ToString( "N0" ));
ToFind.Copy(Modulus);
Multiplier.MultiplyULong(ToFind, Y);
ToFind.AddULong(1);
Divider.Divide(ToFind, KnownNumber, Quotient, Remainder);
if (!Remainder.IsZero())
{
throw(new Exception("This can't happen. !Remainder.IsZero()"));
}
ToFind.Copy(Quotient);
// Worker.ReportProgress( 0, "ToFind: " + ToString10( ToFind ));
break;
}
}
// if( Worker.CancellationPending )
// return false;
TestForModInverse1.Copy(ToFind);
Multiplier.MultiplyULong(TestForModInverse1, KnownNumberULong);
Divider.Divide(TestForModInverse1, Modulus, Quotient, Remainder);
if (!Remainder.IsOne())
{
// The definition is that it's congruent to 1 mod the modulus,
// so this has to be 1.
// I've only seen this happen once. Were the primes P and Q not
// really primes?
throw(new Exception("Remainder has to be 1: " + ToString10(Remainder)));
}
return(true);
}
internal void Add(Integer Result, Integer ToAdd)
{
if (ToAdd.IsZero())
{
return;
}
// The most common form. They are both positive.
if (!Result.IsNegative && !ToAdd.IsNegative)
{
Result.Add(ToAdd);
return;
}
if (!Result.IsNegative && ToAdd.IsNegative)
{
TempAdd1.Copy(ToAdd);
TempAdd1.IsNegative = false;
if (TempAdd1.ParamIsGreater(Result))
{
Subtract(Result, TempAdd1);
return;
}
else
{
Subtract(TempAdd1, Result);
Result.Copy(TempAdd1);
Result.IsNegative = true;
return;
}
}
if (Result.IsNegative && !ToAdd.IsNegative)
{
TempAdd1.Copy(Result);
TempAdd1.IsNegative = false;
TempAdd2.Copy(ToAdd);
if (TempAdd1.ParamIsGreater(TempAdd2))
{
Subtract(TempAdd2, TempAdd1);
Result.Copy(TempAdd2);
return;
}
else
{
Subtract(TempAdd1, TempAdd2);
Result.Copy(TempAdd2);
Result.IsNegative = true;
return;
}
}
if (Result.IsNegative && ToAdd.IsNegative)
{
TempAdd1.Copy(Result);
TempAdd1.IsNegative = false;
TempAdd2.Copy(ToAdd);
TempAdd2.IsNegative = false;
TempAdd1.Add(TempAdd2);
Result.Copy(TempAdd1);
Result.IsNegative = true;
return;
}
}
internal void MakeRSAKeys()
{
int ShowBits = (PrimeIndex + 1) * 32;
// int TestLoops = 0;
Worker.ReportProgress(0, "Making RSA keys.");
Worker.ReportProgress(0, "Bits size is: " + ShowBits.ToString());
// ulong Loops = 0;
while (true)
{
if (Worker.CancellationPending)
{
return;
}
Thread.Sleep(1); // Let other things run.
// Make two prime factors.
// Normally you'd only make new primes when you pay the Certificate
// Authority for a new certificate.
if (!MakeAPrime(PrimeP, PrimeIndex, 20))
{
return;
}
if (Worker.CancellationPending)
{
return;
}
if (!MakeAPrime(PrimeQ, PrimeIndex, 20))
{
return;
}
if (Worker.CancellationPending)
{
return;
}
// This is extremely unlikely.
Integer Gcd = new Integer();
IntMath.GreatestCommonDivisor(PrimeP, PrimeQ, Gcd);
if (!Gcd.IsOne())
{
Worker.ReportProgress(0, "They had a GCD: " + IntMath.ToString10(Gcd));
continue;
}
if (Worker.CancellationPending)
{
return;
}
IntMath.GreatestCommonDivisor(PrimeP, PubKeyExponent, Gcd);
if (!Gcd.IsOne())
{
Worker.ReportProgress(0, "They had a GCD with PubKeyExponent: " + IntMath.ToString10(Gcd));
continue;
}
if (Worker.CancellationPending)
{
return;
}
IntMath.GreatestCommonDivisor(PrimeQ, PubKeyExponent, Gcd);
if (!Gcd.IsOne())
{
Worker.ReportProgress(0, "2) They had a GCD with PubKeyExponent: " + IntMath.ToString10(Gcd));
continue;
}
// For Modular Reduction. This only has to be done
// once, when P and Q are made.
IntMathForP.ModReduction.SetupGeneralBaseArray(PrimeP);
IntMathForQ.ModReduction.SetupGeneralBaseArray(PrimeQ);
PrimePMinus1.Copy(PrimeP);
IntMath.SubtractULong(PrimePMinus1, 1);
PrimeQMinus1.Copy(PrimeQ);
IntMath.SubtractULong(PrimeQMinus1, 1);
if (Worker.CancellationPending)
{
return;
}
// These checks should be more thorough to
// make sure the primes P and Q are numbers
// that can be used in a secure way.
Worker.ReportProgress(0, "The Index of Prime P is: " + PrimeP.GetIndex().ToString());
Worker.ReportProgress(0, "Prime P:");
Worker.ReportProgress(0, IntMath.ToString10(PrimeP));
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "Prime Q:");
Worker.ReportProgress(0, IntMath.ToString10(PrimeQ));
Worker.ReportProgress(0, " ");
PubKeyN.Copy(PrimeP);
IntMath.Multiply(PubKeyN, PrimeQ);
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "PubKeyN:");
Worker.ReportProgress(0, IntMath.ToString10(PubKeyN));
Worker.ReportProgress(0, " ");
// Test Division:
Integer QuotientTest = new Integer();
Integer RemainderTest = new Integer();
IntMath.Divider.Divide(PubKeyN, PrimeP, QuotientTest, RemainderTest);
if (!RemainderTest.IsZero())
{
throw(new Exception("RemainderTest should be zero after divide by PrimeP."));
}
IntMath.Multiply(QuotientTest, PrimeP);
if (!QuotientTest.IsEqual(PubKeyN))
{
throw(new Exception("QuotientTest didn't come out right."));
}
// Euler's Theorem:
// https://en.wikipedia.org/wiki/Euler's_theorem
// ==========
// Work on the Least Common Multiple thing for
// P - 1 and Q - 1.
// =====
IntMath.GreatestCommonDivisor(PrimePMinus1, PrimeQMinus1, Gcd);
Worker.ReportProgress(0, "GCD of PrimePMinus1, PrimeQMinus1 is: " + IntMath.ToString10(Gcd));
if (!Gcd.IsULong())
{
Worker.ReportProgress(0, "This GCD number is too big: " + IntMath.ToString10(Gcd));
continue;
}
else
{
ulong TooBig = Gcd.GetAsULong();
// How big of a GCD is too big?
// ==============
if (TooBig > 1234567)
{
// (P - 1)(Q - 1) + (P - 1) + (Q - 1) = PQ - 1
Worker.ReportProgress(0, "This GCD number is bigger than 1234567: " + IntMath.ToString10(Gcd));
continue;
}
}
Integer Temp1 = new Integer();
PhiN.Copy(PrimePMinus1);
Temp1.Copy(PrimeQMinus1);
IntMath.Multiply(PhiN, Temp1);
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "PhiN:");
Worker.ReportProgress(0, IntMath.ToString10(PhiN));
Worker.ReportProgress(0, " ");
if (Worker.CancellationPending)
{
return;
}
// In RFC 2437 there are commonly used letters/symbols to represent
// the numbers used. So the number e is the public exponent.
// The number e that is used here is called PubKeyExponentUint = 65537.
// In the RFC the private key d is the multiplicative inverse of
// e mod PhiN. Which is mod (P - 1)(Q - 1). It's called
// PrivKInverseExponent here.
if (!IntMath.FindMultiplicativeInverseSmall(PrivKInverseExponent, PubKeyExponent, PhiN, Worker))
{
return;
}
if (PrivKInverseExponent.IsZero())
{
continue;
}
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "PrivKInverseExponent: " + IntMath.ToString10(PrivKInverseExponent));
if (Worker.CancellationPending)
{
return;
}
// In RFC 2437 it defines a number dP which is the multiplicative
// inverse, mod (P - 1) of e. That dP is named PrivKInverseExponentDP here.
Worker.ReportProgress(0, " ");
if (!IntMath.FindMultiplicativeInverseSmall(PrivKInverseExponentDP, PubKeyExponent, PrimePMinus1, Worker))
{
return;
}
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "PrivKInverseExponentDP: " + IntMath.ToString10(PrivKInverseExponentDP));
if (PrivKInverseExponentDP.IsZero())
{
continue;
}
// PrivKInverseExponentDP is PrivKInverseExponent mod PrimePMinus1.
Integer Test1 = new Integer();
Test1.Copy(PrivKInverseExponent);
IntMath.Divider.Divide(Test1, PrimePMinus1, Quotient, Remainder);
Test1.Copy(Remainder);
if (!Test1.IsEqual(PrivKInverseExponentDP))
{
throw(new Exception("This does not match the definition of PrivKInverseExponentDP."));
}
if (Worker.CancellationPending)
{
return;
}
// In RFC 2437 it defines a number dQ which is the multiplicative
// inverse, mod (Q - 1) of e. That dQ is named PrivKInverseExponentDQ here.
Worker.ReportProgress(0, " ");
if (!IntMath.FindMultiplicativeInverseSmall(PrivKInverseExponentDQ, PubKeyExponent, PrimeQMinus1, Worker))
{
return;
}
if (PrivKInverseExponentDQ.IsZero())
{
continue;
}
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "PrivKInverseExponentDQ: " + IntMath.ToString10(PrivKInverseExponentDQ));
if (Worker.CancellationPending)
{
return;
}
Test1.Copy(PrivKInverseExponent);
IntMath.Divider.Divide(Test1, PrimeQMinus1, Quotient, Remainder);
Test1.Copy(Remainder);
if (!Test1.IsEqual(PrivKInverseExponentDQ))
{
throw(new Exception("This does not match the definition of PrivKInverseExponentDQ."));
}
// Make a random number to test encryption/decryption.
Integer ToEncrypt = new Integer();
int HowManyBytes = PrimeIndex * 4;
byte[] RandBytes = MakeRandomBytes(HowManyBytes);
if (RandBytes == null)
{
Worker.ReportProgress(0, "Error making random bytes in MakeRSAKeys().");
return;
}
if (!ToEncrypt.MakeRandomOdd(PrimeIndex - 1, RandBytes))
{
Worker.ReportProgress(0, "Error making random number ToEncrypt.");
return;
}
Integer PlainTextNumber = new Integer();
PlainTextNumber.Copy(ToEncrypt);
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "Before encrypting number: " + IntMath.ToString10(ToEncrypt));
Worker.ReportProgress(0, " ");
IntMath.ModReduction.ModularPower(ToEncrypt, PubKeyExponent, PubKeyN, false);
if (Worker.CancellationPending)
{
return;
}
// Worker.ReportProgress( 0, IntMath.GetStatusString() );
Integer CipherTextNumber = new Integer();
CipherTextNumber.Copy(ToEncrypt);
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "Encrypted number: " + IntMath.ToString10(CipherTextNumber));
Worker.ReportProgress(0, " ");
ECTime DecryptTime = new ECTime();
DecryptTime.SetToNow();
IntMath.ModReduction.ModularPower(ToEncrypt, PrivKInverseExponent, PubKeyN, false);
Worker.ReportProgress(0, "Decrypted number: " + IntMath.ToString10(ToEncrypt));
if (!PlainTextNumber.IsEqual(ToEncrypt))
{
throw(new Exception("PlainTextNumber not equal to unencrypted value."));
// Because P or Q wasn't really a prime?
// Worker.ReportProgress( 0, "PlainTextNumber not equal to unencrypted value." );
// continue;
}
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "Decrypt time seconds: " + DecryptTime.GetSecondsToNow().ToString("N2"));
Worker.ReportProgress(0, " ");
if (Worker.CancellationPending)
{
return;
}
// Test the standard optimized way of decrypting:
if (!ToEncrypt.MakeRandomOdd(PrimeIndex - 1, RandBytes))
{
Worker.ReportProgress(0, "Error making random number in MakeRSAKeys().");
return;
}
PlainTextNumber.Copy(ToEncrypt);
IntMath.ModReduction.ModularPower(ToEncrypt, PubKeyExponent, PubKeyN, false);
if (Worker.CancellationPending)
{
return;
}
CipherTextNumber.Copy(ToEncrypt);
// QInv is the multiplicative inverse of PrimeQ mod PrimeP.
if (!IntMath.MultiplicativeInverse(PrimeQ, PrimeP, QInv, Worker))
{
throw(new Exception("MultiplicativeInverse() returned false."));
}
if (QInv.IsNegative)
{
throw(new Exception("QInv is negative."));
}
Worker.ReportProgress(0, "QInv is: " + IntMath.ToString10(QInv));
DecryptWithQInverse(CipherTextNumber,
ToEncrypt, // Decrypt it to this.
PlainTextNumber, // Test it against this.
PubKeyN,
PrivKInverseExponentDP,
PrivKInverseExponentDQ,
PrimeP,
PrimeQ,
Worker);
Worker.ReportProgress(0, " ");
Worker.ReportProgress(0, "Found the values:");
Worker.ReportProgress(0, "Seconds: " + StartTime.GetSecondsToNow().ToString("N0"));
Worker.ReportProgress(0, " ");
Worker.ReportProgress(1, "Prime1: " + IntMath.ToString10(PrimeP));
Worker.ReportProgress(0, " ");
Worker.ReportProgress(1, "Prime2: " + IntMath.ToString10(PrimeQ));
Worker.ReportProgress(0, " ");
Worker.ReportProgress(1, "PubKeyN: " + IntMath.ToString10(PubKeyN));
Worker.ReportProgress(0, " ");
Worker.ReportProgress(1, "PrivKInverseExponent: " + IntMath.ToString10(PrivKInverseExponent));
// return; // Comment this out to just leave it while( true ) for testing.
}
}
// This works like LongDivide1 except that it
// estimates the maximum value for the digit and
// the for-loop for bit testing is called
// as a separate function.
private bool LongDivide2(Integer ToDivide,
Integer DivideBy,
Integer Quotient,
Integer Remainder)
{
Integer Test1 = new Integer();
int TestIndex = ToDivide.GetIndex() - DivideBy.GetIndex();
// See if TestIndex is too high.
if (TestIndex != 0)
{
// Is 1 too high?
Test1.SetDigitAndClear(TestIndex, 1);
IntMath.MultiplyTopOne(Test1, DivideBy);
if (ToDivide.ParamIsGreater(Test1))
{
TestIndex--;
}
}
// If you were multiplying 99 times 97 you'd get
// 9,603 and the upper two digits [96] are used
// to find the MaxValue. But if you multiply
// 12 * 13 you'd have 156 and only the upper one
// digit is used to find the MaxValue.
// Here it checks if it should use one digit or
// two:
ulong MaxValue;
if ((ToDivide.GetIndex() - 1) > (DivideBy.GetIndex() + TestIndex))
{
MaxValue = ToDivide.GetD(ToDivide.GetIndex());
}
else
{
MaxValue = ToDivide.GetD(ToDivide.GetIndex()) << 32;
MaxValue |= ToDivide.GetD(ToDivide.GetIndex() - 1);
}
MaxValue = MaxValue / DivideBy.GetD(DivideBy.GetIndex());
Quotient.SetDigitAndClear(TestIndex, 1);
Quotient.SetD(TestIndex, 0);
TestDivideBits(MaxValue,
true,
TestIndex,
ToDivide,
DivideBy,
Quotient,
Remainder);
if (TestIndex == 0)
{
Test1.Copy(Quotient);
IntMath.Multiply(Test1, DivideBy);
Remainder.Copy(ToDivide);
IntMath.Subtract(Remainder, Test1);
///////////////
if (DivideBy.ParamIsGreater(Remainder))
{
throw(new Exception("Remainder > DivideBy in LongDivide2()."));
}
//////////////
if (Remainder.IsZero())
{
return(true);
}
else
{
return(false);
}
}
TestIndex--;
while (true)
{
// This remainder is used the same way you do
// long division with paper and pen and you
// keep working with a remainder until the
// remainder is reduced to something smaller
// than DivideBy. You look at the remainder
// to estimate your next quotient digit.
Test1.Copy(Quotient);
IntMath.Multiply(Test1, DivideBy);
Remainder.Copy(ToDivide);
IntMath.Subtract(Remainder, Test1);
MaxValue = Remainder.GetD(Remainder.GetIndex()) << 32;
MaxValue |= Remainder.GetD(Remainder.GetIndex() - 1);
MaxValue = MaxValue / DivideBy.GetD(DivideBy.GetIndex());
TestDivideBits(MaxValue,
false,
TestIndex,
ToDivide,
DivideBy,
Quotient,
Remainder);
if (TestIndex == 0)
{
break;
}
TestIndex--;
}
Test1.Copy(Quotient);
IntMath.Multiply(Test1, DivideBy);
Remainder.Copy(ToDivide);
IntMath.Subtract(Remainder, Test1);
//////////////////////////////
if (DivideBy.ParamIsGreater(Remainder))
{
throw(new Exception("Remainder > DivideBy in LongDivide2()."));
}
////////////////////////////////
if (Remainder.IsZero())
{
return(true);
}
else
{
return(false);
}
}
internal void Divide(Integer ToDivideOriginal,
Integer DivideByOriginal,
Integer Quotient,
Integer Remainder)
{
if (ToDivideOriginal.IsNegative)
{
throw(new Exception("Divide() can't be called with negative numbers."));
}
if (DivideByOriginal.IsNegative)
{
throw(new Exception("Divide() can't be called with negative numbers."));
}
// Returns true if it divides exactly with zero remainder.
// This first checks for some basics before trying to divide it:
if (DivideByOriginal.IsZero())
{
throw(new Exception("Divide() dividing by zero."));
}
ToDivide.Copy(ToDivideOriginal);
DivideBy.Copy(DivideByOriginal);
if (ToDivide.ParamIsGreater(DivideBy))
{
Quotient.SetToZero();
Remainder.Copy(ToDivide);
return; // false;
}
if (ToDivide.IsEqual(DivideBy))
{
Quotient.SetFromULong(1);
Remainder.SetToZero();
return; // true;
}
// At this point DivideBy is smaller than ToDivide.
if (ToDivide.IsULong())
{
ulong ToDivideU = ToDivide.GetAsULong();
ulong DivideByU = DivideBy.GetAsULong();
ulong QuotientU = ToDivideU / DivideByU;
ulong RemainderU = ToDivideU % DivideByU;
Quotient.SetFromULong(QuotientU);
Remainder.SetFromULong(RemainderU);
// if( RemainderU == 0 )
return; // true;
// else
// return false;
}
if (DivideBy.GetIndex() == 0)
{
ShortDivide(ToDivide, DivideBy, Quotient, Remainder);
return;
}
Integer ToDivideTest2 = new Integer();
Integer DivideByTest2 = new Integer();
Integer QuotientTest2 = new Integer();
Integer RemainderTest2 = new Integer();
Integer ToDivideTest3 = new Integer();
Integer DivideByTest3 = new Integer();
Integer QuotientTest3 = new Integer();
Integer RemainderTest3 = new Integer();
ToDivideTest2.Copy(ToDivide);
ToDivideTest3.Copy(ToDivide);
DivideByTest2.Copy(DivideBy);
DivideByTest3.Copy(DivideBy);
LongDivide1(ToDivideTest2, DivideByTest2, QuotientTest2, RemainderTest2);
LongDivide2(ToDivideTest3, DivideByTest3, QuotientTest3, RemainderTest3);
LongDivide3(ToDivide, DivideBy, Quotient, Remainder);
if (!Quotient.IsEqual(QuotientTest2))
{
throw(new Exception("!Quotient.IsEqual( QuotientTest2 )"));
}
if (!Quotient.IsEqual(QuotientTest3))
{
throw(new Exception("!Quotient.IsEqual( QuotientTest3 )"));
}
if (!Remainder.IsEqual(RemainderTest2))
{
throw(new Exception("!Remainder.IsEqual( RemainderTest2 )"));
}
if (!Remainder.IsEqual(RemainderTest3))
{
throw(new Exception("!Remainder.IsEqual( RemainderTest3 )"));
}
}
// This is the standard modular power algorithm that
// you could find in any reference, but its use of
// my modular reduction algorithm in it is new (in 2015).
// (I mean as opposed to using some other modular reduction
// algorithm.)
// The square and multiply method is in Wikipedia:
// https://en.wikipedia.org/wiki/Exponentiation_by_squaring
internal void ModularPower(Integer Result, Integer Exponent, Integer Modulus, bool UsePresetBaseArray)
{
if (Result.IsZero())
{
return; // With Result still zero.
}
if (Result.IsEqual(Modulus))
{
// It is congruent to zero % ModN.
Result.SetToZero();
return;
}
// Result is not zero at this point.
if (Exponent.IsZero())
{
Result.SetFromULong(1);
return;
}
if (Modulus.ParamIsGreater(Result))
{
// throw( new Exception( "This is not supposed to be input for RSA plain text." ));
IntMath.Divider.Divide(Result, Modulus, Quotient, Remainder);
Result.Copy(Remainder);
}
if (Exponent.IsOne())
{
// Result stays the same.
return;
}
if (!UsePresetBaseArray)
{
IntMath.ModReduction.SetupGeneralBaseArray(Modulus);
}
XForModPower.Copy(Result);
ExponentCopy.Copy(Exponent);
// int TestIndex = 0;
Result.SetFromULong(1);
while (true)
{
if ((ExponentCopy.GetD(0) & 1) == 1) // If the bottom bit is 1.
{
IntMath.Multiplier.Multiply(Result, XForModPower);
// if( Result.ParamIsGreater( CurrentModReductionBase ))
// TestForModReduction2.Copy( Result );
IntMath.ModReduction.Reduce(TempForModPower, Result);
// ModularReduction2( TestForModReduction2ForModPower, TestForModReduction2 );
// if( !TestForModReduction2ForModPower.IsEqual( TempForModPower ))
// {
// throw( new Exception( "Mod Reduction 2 is not right." ));
// }
Result.Copy(TempForModPower);
}
ExponentCopy.ShiftRight(1); // Divide by 2.
if (ExponentCopy.IsZero())
{
break;
}
// Square it.
IntMath.Multiplier.Multiply(XForModPower, XForModPower);
// if( XForModPower.ParamIsGreater( CurrentModReductionBase ))
IntMath.ModReduction.Reduce(TempForModPower, XForModPower);
XForModPower.Copy(TempForModPower);
}
// When ModularReduction() gets called it multiplies a base number
// by a uint sized digit. So that can make the result one digit bigger
// than GeneralBase. Then when they are added up you can get carry
// bits that can make it a little bigger.
int HowBig = Result.GetIndex() - Modulus.GetIndex();
// if( HowBig > 1 )
// throw( new Exception( "This does happen. Diff: " + HowBig.ToString() ));
// Do a proof for how big this can be.
if (HowBig > 2)
{
throw(new Exception("This never happens. Diff: " + HowBig.ToString()));
}
IntMath.ModReduction.Reduce(TempForModPower, Result);
Result.Copy(TempForModPower);
// Notice that this Divide() is done once. Not
// a thousand or two thousand times.
/*
* Integer ResultTest = new Integer();
* Integer ModulusTest = new Integer();
* Integer QuotientTest = new Integer();
* Integer RemainderTest = new Integer();
*
* ResultTest.Copy( Result );
* ModulusTest.Copy( Modulus );
* IntMath.Divider.DivideForSmallQuotient( ResultTest,
* ModulusTest,
* QuotientTest,
* RemainderTest );
*
*/
IntMath.Divider.Divide(Result, Modulus, Quotient, Remainder);
// if( !RemainderTest.IsEqual( Remainder ))
// throw( new Exception( "DivideForSmallQuotient() !RemainderTest.IsEqual( Remainder )." ));
// if( !QuotientTest.IsEqual( Quotient ))
// throw( new Exception( "DivideForSmallQuotient() !QuotientTest.IsEqual( Quotient )." ));
Result.Copy(Remainder);
if (Quotient.GetIndex() > 1)
{
throw(new Exception("This never happens. The quotient index is never more than 1."));
}
}
internal bool MultiplicativeInverse(Integer X, Integer Modulus, Integer MultInverse)
{
// This is the extended Euclidean Algorithm.
// A*X + B*Y = Gcd
// A*X + B*Y = 1 If there's a multiplicative inverse.
// A*X = 1 - B*Y so A is the multiplicative inverse of X mod Y.
if (X.IsZero())
{
throw(new Exception("Doing Multiplicative Inverse with a parameter that is zero."));
}
if (Modulus.IsZero())
{
throw(new Exception("Doing Multiplicative Inverse with a parameter that is zero."));
}
// This happens sometimes:
// if( Modulus.ParamIsGreaterOrEq( X ))
// throw( new Exception( "Modulus.ParamIsGreaterOrEq( X ) for Euclid." ));
// Worker.ReportProgress( 0, " " );
// Worker.ReportProgress( 0, " " );
// Worker.ReportProgress( 0, "Top of mod inverse." );
// U is the old part to keep.
U0.SetToZero();
U1.SetToOne();
U2.Copy(Modulus); // Don't change the original numbers that came in as parameters.
// V is the new part.
V0.SetToOne();
V1.SetToZero();
V2.Copy(X);
T0.SetToZero();
T1.SetToZero();
T2.SetToZero();
Quotient.SetToZero();
// while( not forever if there's a problem )
for (int Count = 0; Count < 10000; Count++)
{
if (U2.IsNegative)
{
throw(new Exception("U2 was negative."));
}
if (V2.IsNegative)
{
throw(new Exception("V2 was negative."));
}
IntMath.Divider.Divide(U2, V2, Quotient, Remainder);
if (Remainder.IsZero())
{
// Worker.ReportProgress( 0, "Remainder is zero. No multiplicative-inverse." );
return(false);
}
TempEuclid1.Copy(U0);
TempEuclid2.Copy(V0);
IntMath.Multiplier.Multiply(TempEuclid2, Quotient);
IntMath.Subtract(TempEuclid1, TempEuclid2);
T0.Copy(TempEuclid1);
TempEuclid1.Copy(U1);
TempEuclid2.Copy(V1);
IntMath.Multiplier.Multiply(TempEuclid2, Quotient);
IntMath.Subtract(TempEuclid1, TempEuclid2);
T1.Copy(TempEuclid1);
TempEuclid1.Copy(U2);
TempEuclid2.Copy(V2);
IntMath.Multiplier.Multiply(TempEuclid2, Quotient);
IntMath.Subtract(TempEuclid1, TempEuclid2);
T2.Copy(TempEuclid1);
U0.Copy(V0);
U1.Copy(V1);
U2.Copy(V2);
V0.Copy(T0);
V1.Copy(T1);
V2.Copy(T2);
if (Remainder.IsOne())
{
// Worker.ReportProgress( 0, " " );
// Worker.ReportProgress( 0, "Remainder is 1. There is a multiplicative-inverse." );
break;
}
}
MultInverse.Copy(T0);
if (MultInverse.IsNegative)
{
IntMath.Add(MultInverse, Modulus);
}
// Worker.ReportProgress( 0, "MultInverse: " + ToString10( MultInverse ));
TestForModInverse1.Copy(MultInverse);
TestForModInverse2.Copy(X);
IntMath.Multiplier.Multiply(TestForModInverse1, TestForModInverse2);
IntMath.Divider.Divide(TestForModInverse1, Modulus, Quotient, Remainder);
if (!Remainder.IsOne()) // By the definition of Multiplicative inverse:
{
throw(new Exception("MultInverse is wrong: " + IntMath.ToString10(Remainder)));
}
// Worker.ReportProgress( 0, "MultInverse is the right number: " + ToString10( MultInverse ));
return(true);
}
internal void Multiply(Integer Result, Integer ToMul)
{
// try
// {
if (Result.IsZero())
{
return;
}
if (ToMul.IsULong())
{
MultiplyULong(Result, ToMul.GetAsULong());
SetMultiplySign(Result, ToMul);
return;
}
// It could never get here if ToMul is zero because GetIsULong()
// would be true for zero.
// if( ToMul.IsZero())
int TotalIndex = Result.GetIndex() + ToMul.GetIndex();
if (TotalIndex >= Integer.DigitArraySize)
{
throw(new Exception("Multiply() overflow."));
}
int CountTo = ToMul.GetIndex();
for (int Row = 0; Row <= CountTo; Row++)
{
if (ToMul.GetD(Row) == 0)
{
int CountZeros = Result.GetIndex();
for (int Column = 0; Column <= CountZeros; Column++)
{
M[Column + Row, Row] = 0;
}
}
else
{
int CountMult = Result.GetIndex();
for (int Column = 0; Column <= CountMult; Column++)
{
M[Column + Row, Row] = ToMul.GetD(Row) * Result.GetD(Column);
}
}
}
// Add the columns up with a carry.
Result.SetD(0, M[0, 0] & 0xFFFFFFFF);
ulong Carry = M[0, 0] >> 32;
int ResultIndex = Result.GetIndex();
int MulIndex = ToMul.GetIndex();
for (int Column = 1; Column <= TotalIndex; Column++)
{
ulong TotalLeft = 0;
ulong TotalRight = 0;
for (int Row = 0; Row <= MulIndex; Row++)
{
if (Row > Column)
{
break;
}
if (Column > (ResultIndex + Row))
{
continue;
}
// Split the ulongs into right and left sides
// so that they don't overflow.
TotalRight += M[Column, Row] & 0xFFFFFFFF;
TotalLeft += M[Column, Row] >> 32;
}
TotalRight += Carry;
Result.SetD(Column, TotalRight & 0xFFFFFFFF);
Carry = TotalRight >> 32;
Carry += TotalLeft;
}
Result.SetIndex(TotalIndex);
if (Carry != 0)
{
Result.IncrementIndex(); // This can throw an exception if it overflowed the index.
Result.SetD(Result.GetIndex(), Carry);
}
SetMultiplySign(Result, ToMul);
}
internal void MultiplyULong(Integer Result, ulong ToMul)
{
// Using compile-time checks, this one overflows:
// const ulong Test = ((ulong)0xFFFFFFFF + 1) * ((ulong)0xFFFFFFFF + 1);
// This one doesn't:
// const ulong Test = (ulong)0xFFFFFFFF * ((ulong)0xFFFFFFFF + 1);
if (Result.IsZero())
{
return; // Then the answer is zero, which it already is.
}
if (ToMul == 0)
{
Result.SetToZero();
return;
}
ulong B0 = ToMul & 0xFFFFFFFF;
ulong B1 = ToMul >> 32;
if (B1 == 0)
{
MultiplyUInt(Result, (uint)B0);
return;
}
// Since B1 is not zero:
if ((Result.GetIndex() + 1) >= Integer.DigitArraySize)
{
throw(new Exception("Overflow in MultiplyULong."));
}
int CountTo = Result.GetIndex();
for (int Column = 0; Column <= CountTo; Column++)
{
ulong Digit = Result.GetD(Column);
M[Column, 0] = B0 * Digit;
// Column + 1 and Row is 1, so it's just like pen and paper.
M[Column + 1, 1] = B1 * Digit;
}
// Since B1 is not zero, the index is set one higher.
Result.IncrementIndex(); // Might throw an exception if it goes out of range.
M[Result.GetIndex(), 0] = 0; // Otherwise it would be undefined
// when it's added up below.
// Add these up with a carry.
Result.SetD(0, M[0, 0] & 0xFFFFFFFF);
ulong Carry = M[0, 0] >> 32;
CountTo = Result.GetIndex();
for (int Column = 1; Column <= CountTo; Column++)
{
// This does overflow:
// const ulong Test = ((ulong)0xFFFFFFFF * (ulong)(0xFFFFFFFF))
// + ((ulong)0xFFFFFFFF * (ulong)(0xFFFFFFFF));
// Split the ulongs into right and left sides
// so that they don't overflow.
ulong TotalLeft = 0;
ulong TotalRight = 0;
// There's only the two rows for this.
for (int Row = 0; Row <= 1; Row++)
{
ulong MValue = M[Column, Row];
TotalRight += MValue & 0xFFFFFFFF;
TotalLeft += MValue >> 32;
}
TotalRight += Carry;
Result.SetD(Column, TotalRight & 0xFFFFFFFF);
Carry = TotalRight >> 32;
Carry += TotalLeft;
}
if (Carry != 0)
{
Result.IncrementIndex(); // This can throw an exception.
Result.SetD(Result.GetIndex(), Carry);
}
}
