// This is the standard modular power algorithm that
// you could find in any reference, but its use of
// the new modular reduction algorithm is new.
// The square and multiply method is in Wikipedia:
// https://en.wikipedia.org/wiki/Exponentiation_by_squaring
// x^n = (x^2)^((n - 1)/2) if n is odd.
// x^n = (x^2)^(n/2) if n is even.
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.Divide( Result, Modulus, Quotient, Remainder );
Result.Copy( Remainder );
}
if( Exponent.IsOne())
{
// Result stays the same.
return;
}
if( !UsePresetBaseArray )
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.Multiply( Result, XForModPower );
ModularReduction( TempForModPower, Result );
Result.Copy( TempForModPower );
}
ExponentCopy.ShiftRight( 1 ); // Divide by 2.
if( ExponentCopy.IsZero())
break;
// Square it.
IntMath.Multiply( XForModPower, XForModPower );
ModularReduction( 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() ));
if( HowBig > 2 )
throw( new Exception( "The never happens. Diff: " + HowBig.ToString() ));
ModularReduction( TempForModPower, Result );
Result.Copy( TempForModPower );
IntMath.Divide( Result, Modulus, Quotient, Remainder );
Result.Copy( Remainder );
if( Quotient.GetIndex() > 1 )
throw( new Exception( "This never happens. The quotient index is never more than 1." ));
}
internal bool FindTwoFactorsWithFermat( Integer Product, Integer P, Integer Q, ulong MinimumX )
{
ECTime StartTime = new ECTime();
StartTime.SetToNow();
Integer TestSqrt = new Integer();
Integer TestSquared = new Integer();
Integer SqrRoot = new Integer();
TestSquared.Copy( Product );
IntMath.Multiply( TestSquared, Product );
IntMath.SquareRoot( TestSquared, SqrRoot );
TestSqrt.Copy( SqrRoot );
IntMath.DoSquare( TestSqrt );
// IntMath.Multiply( TestSqrt, SqrRoot );
if( !TestSqrt.IsEqual( TestSquared ))
throw( new Exception( "The square test was bad." ));
// Some primes:
// 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
// 101, 103, 107
P.SetToZero();
Q.SetToZero();
Integer TestX = new Integer();
SetupQuadResArray( Product );
ulong BaseTo37 = QuadResBigBase * 29UL * 31UL * 37UL;
// ulong BaseTo31 = QuadResBigBase * 29UL * 31UL;
ulong ProdModTo37 = IntMath.GetMod64( Product, BaseTo37 );
// ulong ProdModTo31 = IntMath.GetMod64( Product, BaseTo31 );
for( ulong BaseCount = 0; BaseCount < (29 * 31 * 37); BaseCount++ )
{
if( (BaseCount & 0xF) == 0 )
Worker.ReportProgress( 0, "Find with Fermat BaseCount: " + BaseCount.ToString() );
if( Worker.CancellationPending )
return false;
ulong Base = (BaseCount + 1) * QuadResBigBase; // BaseCount times 223,092,870.
if( Base < MinimumX )
continue;
Base = BaseCount * QuadResBigBase; // BaseCount times 223,092,870.
for( uint Count = 0; Count < QuadResArrayLast; Count++ )
{
// The maximum CountPart can be is just under half the size of
// the Product. (Like if Y - X was equal to 1, and Y + X was
// equal to the Product.) If it got anywhere near that big it
// would be inefficient to try and find it this way.
ulong CountPart = Base + QuadResArray[Count];
ulong Test = ProdModTo37 + (CountPart * CountPart);
// ulong Test = ProdModTo31 + (CountPart * CountPart);
Test = Test % BaseTo37;
// Test = Test % BaseTo31;
if( !IntegerMath.IsQuadResidue29( Test ))
continue;
if( !IntegerMath.IsQuadResidue31( Test ))
continue;
if( !IntegerMath.IsQuadResidue37( Test ))
continue;
ulong TestBytes = (CountPart & 0xFFFFF);
TestBytes *= (CountPart & 0xFFFFF);
ulong ProdBytes = Product.GetD( 1 );
ProdBytes <<= 8;
ProdBytes |= Product.GetD( 0 );
uint FirstBytes = (uint)(TestBytes + ProdBytes);
if( !IntegerMath.FirstBytesAreQuadRes( FirstBytes ))
{
// Worker.ReportProgress( 0, "First bytes aren't quad res." );
continue;
}
TestX.SetFromULong( CountPart );
IntMath.MultiplyULong( TestX, CountPart );
TestX.Add( Product );
// uint Mod37 = (uint)IntMath.GetMod32( TestX, 37 );
// if( !IntegerMath.IsQuadResidue37( Mod37 ))
// continue;
// Do more of these tests with 41, 43, 47...
// if( !IntegerMath.IsQuadResidue41( Mod37 ))
// continue;
// Avoid doing this square root at all costs.
if( IntMath.SquareRoot( TestX, SqrRoot ))
{
Worker.ReportProgress( 0, " " );
if( (CountPart & 1) == 0 )
Worker.ReportProgress( 0, "CountPart was even." );
else
Worker.ReportProgress( 0, "CountPart was odd." );
// Found an exact square root.
// P + (CountPart * CountPart) = Y*Y
// P = (Y + CountPart)Y - CountPart)
P.Copy( SqrRoot );
Integer ForSub = new Integer();
ForSub.SetFromULong( CountPart );
IntMath.Subtract( P, ForSub );
// Make Q the bigger one and put them in order.
Q.Copy( SqrRoot );
Q.AddULong( CountPart );
if( P.IsOne() || Q.IsOne())
{
// This happens when testing with small primes.
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "Went all the way to 1 in FindTwoFactorsWithFermat()." );
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, " " );
P.SetToZero(); // It has no factors.
Q.SetToZero();
return true; // Tested everything, so it's a prime.
}
Worker.ReportProgress( 0, "Found P: " + IntMath.ToString10( P ) );
Worker.ReportProgress( 0, "Found Q: " + IntMath.ToString10( Q ) );
Worker.ReportProgress( 0, "Seconds: " + StartTime.GetSecondsToNow().ToString( "N1" ));
Worker.ReportProgress( 0, " " );
throw( new Exception( "Testing this." ));
// return true; // With P and Q.
}
// else
// Worker.ReportProgress( 0, "It was not an exact square root." );
}
}
// P and Q would still be zero if it never found them.
return false;
}
internal void VerifyFactors()
{
Integer TestFactors = new Integer();
TestFactors.SetToOne();
for( int Count = 0; Count < FactorsArrayLast; Count++ )
IntMath.Multiply( TestFactors, FactorsArray[Count].Factor );
if( !TestFactors.IsEqual( OriginalFindFrom ))
throw( new Exception( "VerifyFactors didn't come out right." ));
}
/*
internal void ModularPowerOld( Integer Result, Integer Exponent, Integer ModN )
{
if( Result.IsZero())
return; // With Result still zero.
if( Result.IsEqual( ModN ))
{
// It is congruent to zero % ModN.
Result.SetToZero();
return;
}
// Result is not zero at this point.
if( Exponent.IsZero() )
{
Result.SetFromULong( 1 );
return;
}
if( ModN.ParamIsGreater( Result ))
{
Divide( Result, ModN, Quotient, Remainder );
Result.Copy( Remainder );
}
if( Exponent.IsEqualToULong( 1 ))
{
// Result stays the same.
return;
}
XForModPower.Copy( Result );
ExponentCopy.Copy( Exponent );
Result.SetFromULong( 1 );
while( !ExponentCopy.IsZero())
{
// If the bit is 1, then do a lot more work here.
if( (ExponentCopy.GetD( 0 ) & 1) == 1 )
{
// This is a multiplication for every _bit_. So a 1024-bit
// modulus means this gets called roughly 512 times.
// The commonly used public exponent is 65537, which has
// only two bits set to 1, the rest are all zeros. But the
// private key exponents are long randomish numbers.
// (See: Hamming Weight.)
Multiply( Result, XForModPower );
SubtractULong( ExponentCopy, 1 );
// Usually it's true that the Result is greater than ModN.
if( ModN.ParamIsGreater( Result ))
{
// Here is where that really long division algorithm gets used a
// lot in a loop. And this Divide() gets called roughly about
// 512 times.
Divide( Result, ModN, Quotient, Remainder );
Result.Copy( Remainder );
}
}
// Square it.
// This is a multiplication for every _bit_. So a 1024-bit
// modulus means this gets called 1024 times.
Multiply( XForModPower, XForModPower );
ExponentCopy.ShiftRight( 1 ); // Divide by 2.
if( ModN.ParamIsGreater( XForModPower ))
{
// And this Divide() gets called about 1024 times.
Divide( XForModPower, ModN, Quotient, Remainder );
XForModPower.Copy( Remainder );
}
}
}
*/
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 )
{
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 void MakeBaseNumbers()
{
try
{
MakeYBaseToPrimesArray();
if( Worker.CancellationPending )
return;
Integer YTop = new Integer();
Integer Y = new Integer();
Integer XSquared = new Integer();
Integer Test = new Integer();
YTop.SetToZero();
uint XSquaredBitLength = 1;
ExponentVectorNumber ExpNumber = new ExponentVectorNumber( IntMath );
uint Loops = 0;
uint BSmoothCount = 0;
uint BSmoothTestsCount = 0;
uint IncrementYBy = 0;
while( true )
{
if( Worker.CancellationPending )
return;
Loops++;
if( (Loops & 0xF) == 0 )
{
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "Loops: " + Loops.ToString( "N0" ));
Worker.ReportProgress( 0, "BSmoothCount: " + BSmoothCount.ToString( "N0" ));
Worker.ReportProgress( 0, "BSmoothTestsCount: " + BSmoothTestsCount.ToString( "N0" ));
if( BSmoothTestsCount != 0 )
{
double TestsRatio = (double)BSmoothCount / (double)BSmoothTestsCount;
Worker.ReportProgress( 0, "TestsRatio: " + TestsRatio.ToString( "N3" ));
}
}
/*
if( (Loops & 0xFFFFF) == 0 )
{
// Use Task Manager to tweak the CPU Utilization if you want
// it be below 100 percent.
Thread.Sleep( 1 );
}
*/
// About 98 percent of the time it is running IncrementBy().
IncrementYBy += IncrementConst;
uint BitLength = IncrementBy();
const uint SomeOptimumBitLength = 2;
if( BitLength < SomeOptimumBitLength )
continue;
// This BitLength has to do with how many small factors you want
// in the number. But it doesn't limit your factor base at all.
// You can still have any size prime in your factor base (up to
// IntegerMath.PrimeArrayLength). Compare the size of
// YBaseToPrimesArrayLast to IntegerMath.PrimeArrayLength.
BSmoothTestsCount++;
YTop.AddULong( IncrementYBy );
IncrementYBy = 0;
Y.Copy( ProductSqrRoot );
Y.Add( YTop );
XSquared.Copy( Y );
IntMath.DoSquare( XSquared );
if( XSquared.ParamIsGreater( Product ))
throw( new Exception( "Bug. XSquared.ParamIsGreater( Product )." ));
IntMath.Subtract( XSquared, Product );
XSquaredBitLength = (uint)(XSquared.GetIndex() * 32);
uint TopDigit = (uint)XSquared.GetD( XSquared.GetIndex());
uint TopLength = GetBitLength( TopDigit );
XSquaredBitLength += TopLength;
if( XSquaredBitLength == 0 )
XSquaredBitLength = 1;
// if( ItIsTheAnswerAlready( XSquared )) It's too unlikely.
// QuadResCombinatorics could run in parallel to check for that,
// and it would be way ahead of this.
GetOneMainFactor();
if( OneMainFactor.IsEqual( XSquared ))
{
MakeFastExpNumber( ExpNumber );
}
else
{
if( OneMainFactor.IsZero())
throw( new Exception( "OneMainFactor.IsZero()." ));
IntMath.Divide( XSquared, OneMainFactor, Quotient, Remainder );
ExpNumber.SetFromTraditionalInteger( Quotient );
ExpNumber.Multiply( ExpOneMainFactor );
ExpNumber.GetTraditionalInteger( Test );
if( !Test.IsEqual( XSquared ))
throw( new Exception( "!Test.IsEqual( XSquared )." ));
}
if( ExpNumber.IsBSmooth())
{
BSmoothCount++;
string DelimS = IntMath.ToString10( Y ) + "\t" +
ExpNumber.ToDelimString();
Worker.ReportProgress( 1, DelimS );
if( (BSmoothCount & 0x3F) == 0 )
{
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "BitLength: " + BitLength.ToString());
Worker.ReportProgress( 0, "XSquaredBitLength: " + XSquaredBitLength.ToString());
Worker.ReportProgress( 0, ExpNumber.ToString() );
// What should BSmoothLimit be?
// (Since FactorDictionary.cs will reduce the final factor base.)
if( BSmoothCount > BSmoothLimit )
{
Worker.ReportProgress( 0, "Found enough to make the matrix." );
Worker.ReportProgress( 0, "BSmoothCount: " + BSmoothCount.ToString( "N0" ));
Worker.ReportProgress( 0, "BSmoothLimit: " + BSmoothLimit.ToString( "N0" ));
Worker.ReportProgress( 0, "Seconds: " + StartTime.GetSecondsToNow().ToString( "N1" ));
double Seconds = StartTime.GetSecondsToNow();
int Minutes = (int)Seconds / 60;
int Hours = Minutes / 60;
Minutes = Minutes % 60;
Seconds = Seconds % 60;
string ShowS = "Hours: " + Hours.ToString( "N0" ) +
" Minutes: " + Minutes.ToString( "N0" ) +
" Seconds: " + Seconds.ToString( "N0" );
Worker.ReportProgress( 0, ShowS );
return;
}
}
}
}
}
catch( Exception Except )
{
throw( new Exception( "Exception in MakeBaseNumbers():\r\n" + Except.Message ));
}
}
internal void Subtract( Integer Result, Integer ToSub )
{
// This checks that the sign is equal too.
if( Result.IsEqual( ToSub ))
{
Result.SetToZero();
return;
}
// ParamIsGreater() handles positive and negative values, so if the
// parameter is more toward the positive side then it's true. It's greater.
// The most common form. They are both positive.
if( !Result.IsNegative && !ToSub.IsNegative )
{
if( ToSub.ParamIsGreater( Result ))
{
SubtractPositive( Result, ToSub );
return;
}
// ToSub is bigger.
TempSub1.Copy( Result );
TempSub2.Copy( ToSub );
SubtractPositive( TempSub2, TempSub1 );
Result.Copy( TempSub2 );
Result.IsNegative = true;
return;
}
if( Result.IsNegative && !ToSub.IsNegative )
{
TempSub1.Copy( Result );
TempSub1.IsNegative = false;
TempSub1.Add( ToSub );
Result.Copy( TempSub1 );
Result.IsNegative = true;
return;
}
if( !Result.IsNegative && ToSub.IsNegative )
{
TempSub1.Copy( ToSub );
TempSub1.IsNegative = false;
Result.Add( TempSub1 );
return;
}
if( Result.IsNegative && ToSub.IsNegative )
{
TempSub1.Copy( Result );
TempSub1.IsNegative = false;
TempSub2.Copy( ToSub );
TempSub2.IsNegative = false;
// -12 - -7 = -12 + 7 = -5
// Comparing the positive numbers here.
if( TempSub2.ParamIsGreater( TempSub1 ))
{
SubtractPositive( TempSub1, TempSub2 );
Result.Copy( TempSub1 );
Result.IsNegative = true;
return;
}
// -7 - -12 = -7 + 12 = 5
SubtractPositive( TempSub2, TempSub1 );
Result.Copy( TempSub2 );
Result.IsNegative = false;
return;
}
}
// Finding the square root of a number is similar to division since
// it is a search algorithm. The TestSqrtBits method shown next is
// very much like TestDivideBits(). It works the same as
// FindULSqrRoot(), but on a bigger scale.
/*
private void TestSqrtBits( int TestIndex, Integer Square, Integer SqrRoot )
{
Integer Test1 = new Integer();
uint BitTest = 0x80000000;
for( int BitCount = 31; BitCount >= 0; BitCount-- )
{
Test1.Copy( SqrRoot );
Test1.D[TestIndex] |= BitTest;
Test1.Square();
if( !Square.ParamIsGreater( Test1 ) )
SqrRoot.D[TestIndex] |= BitTest; // Use the bit.
BitTest >>= 1;
}
}
*/
// In the SquareRoot() method SqrRoot.Index is half of Square.Index.
// Compare this to the Square() method where the Carry might or
// might not increment the index to an odd number. (So if the Index
// was 5 its square root would have an Index of 5 / 2 = 2.)
// The SquareRoot1() method uses FindULSqrRoot() either to find the
// whole answer, if it's a small number, or it uses it to find the
// top part. Then from there it goes on to a bit by bit search
// with TestSqrtBits().
public bool SquareRoot( Integer Square, Integer SqrRoot )
{
ulong ToMatch;
if( Square.IsULong() )
{
ToMatch = Square.GetAsULong();
SqrRoot.SetD( 0, FindULSqrRoot( ToMatch ));
SqrRoot.SetIndex( 0 );
if( (SqrRoot.GetD(0 ) * SqrRoot.GetD( 0 )) == ToMatch )
return true;
else
return false;
}
int TestIndex = Square.GetIndex() >> 1; // LgSquare.Index / 2;
SqrRoot.SetDigitAndClear( TestIndex, 1 );
// if( (TestIndex * 2) > (LgSquare.Index - 1) )
if( (TestIndex << 1) > (Square.GetIndex() - 1) )
{
ToMatch = Square.GetD( Square.GetIndex());
}
else
{
// LgSquare.Index is at least 2 here.
ToMatch = Square.GetD( Square.GetIndex()) << 32;
ToMatch |= Square.GetD( Square.GetIndex() - 1 );
}
SqrRoot.SetD( TestIndex, FindULSqrRoot( ToMatch ));
TestIndex--;
while( true )
{
// TestSqrtBits( TestIndex, LgSquare, LgSqrRoot );
SearchSqrtXPart( TestIndex, Square, SqrRoot );
if( TestIndex == 0 )
break;
TestIndex--;
}
// Avoid squaring the whole thing to see if it's an exact square root:
if( ((SqrRoot.GetD( 0 ) * SqrRoot.GetD( 0 )) & 0xFFFFFFFF) != Square.GetD( 0 ))
return false;
TestForSquareRoot.Copy( SqrRoot );
DoSquare( TestForSquareRoot );
if( Square.IsEqual( TestForSquareRoot ))
return true;
else
return false;
}
private void DoCRTTest( Integer StartingNumber )
{
CRTMath CRTMath1 = new CRTMath( Worker );
ECTime CRTTestTime = new ECTime();
ChineseRemainder CRTTest = new ChineseRemainder( IntMath );
ChineseRemainder CRTTest2 = new ChineseRemainder( IntMath );
ChineseRemainder CRTAccumulate = new ChineseRemainder( IntMath );
ChineseRemainder CRTToTest = new ChineseRemainder( IntMath );
ChineseRemainder CRTTempEqual = new ChineseRemainder( IntMath );
ChineseRemainder CRTTestEqual = new ChineseRemainder( IntMath );
Integer BigBase = new Integer();
Integer ToTest = new Integer();
Integer Accumulate = new Integer();
Integer Test1 = new Integer();
Integer Test2 = new Integer();
CRTTest.SetFromTraditionalInteger( StartingNumber );
// If the digit array size isn't set right in relation to
// Integer.DigitArraySize then it can cause an error here.
CRTMath1.GetTraditionalInteger( Accumulate, CRTTest );
if( !Accumulate.IsEqual( StartingNumber ))
throw( new Exception( " !Accumulate.IsEqual( Result )." ));
CRTTestEqual.SetFromTraditionalInteger( Accumulate );
if( !CRTMath1.IsEqualToInteger( CRTTestEqual, Accumulate ))
throw( new Exception( "IsEqualToInteger() didn't work." ));
// Make sure it works with even numbers too.
Test1.Copy( StartingNumber );
Test1.SetD( 0, Test1.GetD( 0 ) & 0xFE );
CRTTest.SetFromTraditionalInteger( Test1 );
CRTMath1.GetTraditionalInteger( Accumulate, CRTTest );
if( !Accumulate.IsEqual( Test1 ))
throw( new Exception( "For even numbers. !Accumulate.IsEqual( Test )." ));
////////////
// Make sure the size of this works with the Integer size because
// an overflow is hard to find.
CRTTestTime.SetToNow();
Test1.SetToMaxValueForCRT();
CRTTest.SetFromTraditionalInteger( Test1 );
CRTMath1.GetTraditionalInteger( Accumulate, CRTTest );
if( !Accumulate.IsEqual( Test1 ))
throw( new Exception( "For the max value. !Accumulate.IsEqual( Test1 )." ));
// Worker.ReportProgress( 0, "CRT Max test seconds: " + CRTTestTime.GetSecondsToNow().ToString( "N1" ));
// Worker.ReportProgress( 0, "MaxValue: " + IntMath.ToString10( Accumulate ));
// Worker.ReportProgress( 0, "MaxValue.Index: " + Accumulate.GetIndex().ToString());
// Multiplicative Inverse test:
Integer TestDivideBy = new Integer();
Integer TestProduct = new Integer();
ChineseRemainder CRTTestDivideBy = new ChineseRemainder( IntMath );
ChineseRemainder CRTTestProduct = new ChineseRemainder( IntMath );
TestDivideBy.Copy( StartingNumber );
TestProduct.Copy( StartingNumber );
IntMath.Multiply( TestProduct, TestDivideBy );
CRTTestDivideBy.SetFromTraditionalInteger( TestDivideBy );
CRTTestProduct.SetFromTraditionalInteger( TestDivideBy );
CRTTestProduct.Multiply( CRTTestDivideBy );
CRTMath1.GetTraditionalInteger( Accumulate, CRTTestProduct );
if( !Accumulate.IsEqual( TestProduct ))
throw( new Exception( "Multiply test was bad." ));
IntMath.Divide( TestProduct, TestDivideBy, Quotient, Remainder );
if( !Remainder.IsZero())
throw( new Exception( "This test won't work unless it divides it exactly." ));
ChineseRemainder CRTTestQuotient = new ChineseRemainder( IntMath );
CRTMath1.MultiplicativeInverse( CRTTestProduct, CRTTestDivideBy, CRTTestQuotient );
// Yes, multiplicative inverse is the same number
// as with regular division.
Integer TestQuotient = new Integer();
CRTMath1.GetTraditionalInteger( TestQuotient, CRTTestQuotient );
if( !TestQuotient.IsEqual( Quotient ))
throw( new Exception( "Modular Inverse in DoCRTTest didn't work." ));
// Subtract
Test1.Copy( StartingNumber );
IntMath.SetFromString( Test2, "12345678901234567890123456789012345" );
CRTTest.SetFromTraditionalInteger( Test1 );
CRTTest2.SetFromTraditionalInteger( Test2 );
CRTTest.Subtract( CRTTest2 );
IntMath.Subtract( Test1, Test2 );
CRTMath1.GetTraditionalInteger( Accumulate, CRTTest );
if( !Accumulate.IsEqual( Test1 ))
throw( new Exception( "Subtract test was bad." ));
// Add
Test1.Copy( StartingNumber );
IntMath.SetFromString( Test2, "12345678901234567890123456789012345" );
CRTTest.SetFromTraditionalInteger( Test1 );
CRTTest2.SetFromTraditionalInteger( Test2 );
CRTTest.Add( CRTTest2 );
IntMath.Add( Test1, Test2 );
CRTMath1.GetTraditionalInteger( Accumulate, CRTTest );
if( !Accumulate.IsEqual( Test1 ))
throw( new Exception( "Add test was bad." ));
/////////
CRTBaseMath CBaseMath = new CRTBaseMath( Worker, CRTMath1 );
ChineseRemainder CRTInput = new ChineseRemainder( IntMath );
CRTInput.SetFromTraditionalInteger( StartingNumber );
Test1.Copy( StartingNumber );
IntMath.SetFromString( Test2, "12345678901234567890123456789012345" );
IntMath.Add( Test1, Test2 );
Integer TestModulus = new Integer();
TestModulus.Copy( Test1 );
ChineseRemainder CRTTestModulus = new ChineseRemainder( IntMath );
CRTTestModulus.SetFromTraditionalInteger( TestModulus );
Integer Exponent = new Integer();
Exponent.SetFromULong( PubKeyExponentUint );
CBaseMath.ModularPower( CRTInput, Exponent, CRTTestModulus, false );
IntMath.IntMathNew.ModularPower( StartingNumber, Exponent, TestModulus, false );
if( !CRTMath1.IsEqualToInteger( CRTInput, StartingNumber ))
throw( new Exception( "CRTBase ModularPower() didn't work." ));
CRTBase ExpTest = new CRTBase( IntMath );
CBaseMath.SetFromCRTNumber( ExpTest, CRTInput );
CBaseMath.GetExponentForm( ExpTest, 37 );
// Worker.ReportProgress( 0, "CRT was good." );
}
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 ); // Give up the time slice. Let other things on the server 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;
IntegerBase TestP = new IntegerBase();
IntegerBaseMath IntBaseMath = new IntegerBaseMath( IntMath );
string TestS = IntMath.ToString10( PrimeP );
IntBaseMath.SetFromString( TestP, TestS );
string TestS2 = IntBaseMath.ToString10( TestP );
if( TestS != TestS2 )
throw( new Exception( "TestS != TestS2 for IntegerBase." ));
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;
// This would never happen since the public key exponent used here
// is one of the small primes in the array in IntegerMath that it
// was checked against. But it does show here in the code that
// they have to be co-prime to each other. And in the future it
// might be found that the public key exponent has to be much larger
// than the one used here.
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.
IntMathNewForP.SetupGeneralBaseArray( PrimeP );
IntMathNewForQ.SetupGeneralBaseArray( PrimeQ );
PrimePMinus1.Copy( PrimeP );
IntMath.SubtractULong( PrimePMinus1, 1 );
PrimeQMinus1.Copy( PrimeQ );
IntMath.SubtractULong( PrimeQMinus1, 1 );
// These checks should be more thorough.
if( Worker.CancellationPending )
return;
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, " " );
// Euler's Theorem:
// https://en.wikipedia.org/wiki/Euler's_theorem
// if x ≡ y (mod φ(n)),
// then a^x ≡ a^y (mod n).
// Euler's Phi function (aka Euler's Totient function) is calculated
// next.
// PhiN is made from the two factors: (P - 1)(Q - 1)
// PhiN is: (P - 1)(Q - 1) = PQ - P - Q + 1
// If I add (P - 1) to PhiN I get:
// PQ - P - Q + 1 + (P - 1) = PQ - Q.
// If I add (Q - 1) to that I get:
// PQ - Q + (Q - 1) = PQ - 1.
// (P - 1)(Q - 1) + (P - 1) + (Q - 1) = PQ - 1
// If (P - 1) and (Q - 1) had a larger GCD then PQ - 1 would have
// that same factor too.
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.IntMathNew.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.IntMathNew.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.Divide( Test1, PrimePMinus1, Quotient, Remainder );
Test1.Copy( Remainder );
if( !Test1.IsEqual( PrivKInverseExponentDP ))
throw( new Exception( "Bug. 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.IntMathNew.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.Divide( Test1, PrimeQMinus1, Quotient, Remainder );
Test1.Copy( Remainder );
if( !Test1.IsEqual( PrivKInverseExponentDQ ))
throw( new Exception( "Bug. 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.IntMathNew.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.IntMathNew.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.IntMathNew.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( "This is a bug. 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 ));
/*
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, " " );
DoCRTTest( PrivKInverseExponent );
Worker.ReportProgress( 0, "Finished CRT test." );
Worker.ReportProgress( 0, " " );
*/
return; // Comment this out to just leave it while( true ) for testing.
}
}
internal bool DecryptWithQInverse( Integer EncryptedNumber,
Integer DecryptedNumber,
Integer TestDecryptedNumber,
Integer PubKeyN,
Integer PrivKInverseExponentDP,
Integer PrivKInverseExponentDQ,
Integer PrimeP,
Integer PrimeQ,
BackgroundWorker Worker )
{
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "Top of DecryptWithQInverse()." );
// QInv and the dP and dQ numbers are normally already set up before
// you start your listening socket.
ECTime DecryptTime = new ECTime();
DecryptTime.SetToNow();
// See section 5.1.2 of RFC 2437 for these steps:
// http://tools.ietf.org/html/rfc2437
// 2.2 Let m_1 = c^dP mod p.
// 2.3 Let m_2 = c^dQ mod q.
// 2.4 Let h = qInv ( m_1 - m_2 ) mod p.
// 2.5 Let m = m_2 + hq.
Worker.ReportProgress( 0, "EncryptedNumber: " + IntMath.ToString10( EncryptedNumber ));
// 2.2 Let m_1 = c^dP mod p.
TestForDecrypt.Copy( EncryptedNumber );
IntMathNewForP.ModularPower( TestForDecrypt, PrivKInverseExponentDP, PrimeP, true );
if( Worker.CancellationPending )
return false;
M1ForInverse.Copy( TestForDecrypt );
// 2.3 Let m_2 = c^dQ mod q.
TestForDecrypt.Copy( EncryptedNumber );
IntMathNewForQ.ModularPower( TestForDecrypt, PrivKInverseExponentDQ, PrimeQ, true );
if( Worker.CancellationPending )
return false;
M2ForInverse.Copy( TestForDecrypt );
// 2.4 Let h = qInv ( m_1 - m_2 ) mod p.
// How many is optimal to avoid the division?
int HowManyIsOptimal = (PrimeP.GetIndex() * 3);
for( int Count = 0; Count < HowManyIsOptimal; Count++ )
{
if( M1ForInverse.ParamIsGreater( M2ForInverse ))
M1ForInverse.Add( PrimeP );
else
break;
}
if( M1ForInverse.ParamIsGreater( M2ForInverse ))
{
M1M2SizeDiff.Copy( M2ForInverse );
IntMath.Subtract( M1M2SizeDiff, M1ForInverse );
// Unfortunately this long Divide() has to be done.
IntMath.Divide( M1M2SizeDiff, PrimeP, Quotient, Remainder );
Quotient.AddULong( 1 );
Worker.ReportProgress( 0, "The Quotient for M1M2SizeDiff is: " + IntMath.ToString10( Quotient ));
IntMath.Multiply( Quotient, PrimeP );
M1ForInverse.Add( Quotient );
}
M1MinusM2.Copy( M1ForInverse );
IntMath.Subtract( M1MinusM2, M2ForInverse );
if( M1MinusM2.IsNegative )
throw( new Exception( "This is a bug. M1MinusM2.IsNegative is true." ));
if( QInv.IsNegative )
throw( new Exception( "This is a bug. QInv.IsNegative is true." ));
HForQInv.Copy( M1MinusM2 );
IntMath.Multiply( HForQInv, QInv );
if( HForQInv.IsNegative )
throw( new Exception( "This is a bug. HForQInv.IsNegative is true." ));
if( PrimeP.ParamIsGreater( HForQInv ))
{
IntMath.Divide( HForQInv, PrimeP, Quotient, Remainder );
HForQInv.Copy( Remainder );
}
// 2.5 Let m = m_2 + hq.
DecryptedNumber.Copy( HForQInv );
IntMath.Multiply( DecryptedNumber, PrimeQ );
DecryptedNumber.Add( M2ForInverse );
if( !TestDecryptedNumber.IsEqual( DecryptedNumber ))
throw( new Exception( "!TestDecryptedNumber.IsEqual( DecryptedNumber )." ));
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "DecryptedNumber: " + IntMath.ToString10( DecryptedNumber ));
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "TestDecryptedNumber: " + IntMath.ToString10( TestDecryptedNumber ));
Worker.ReportProgress( 0, " " );
Worker.ReportProgress( 0, "Decrypt with QInv time seconds: " + DecryptTime.GetSecondsToNow().ToString( "N2" ));
Worker.ReportProgress( 0, " " );
return true;
}
private bool GetFactors( Integer Y, ExponentVectorNumber XExp )
{
Integer XRoot = new Integer();
Integer X = new Integer();
Integer XwithY = new Integer();
Integer Gcd = new Integer();
XExp.GetTraditionalInteger( X );
if( !IntMath.SquareRoot( X, XRoot ))
throw( new Exception( "Bug. X should have an exact square root." ));
XwithY.Copy( Y );
XwithY.Add( XRoot );
IntMath.GreatestCommonDivisor( Product, XwithY, Gcd );
if( !Gcd.IsOne())
{
if( !Gcd.IsEqual( Product ))
{
SolutionP.Copy( Gcd );
IntMath.Divide( Product, SolutionP, Quotient, Remainder );
if( !Remainder.IsZero())
throw( new Exception( "The Remainder with SolutionP can't be zero." ));
SolutionQ.Copy( Quotient );
MForm.ShowStatus( "SolutionP: " + IntMath.ToString10( SolutionP ));
MForm.ShowStatus( "SolutionQ: " + IntMath.ToString10( SolutionQ ));
return true;
}
else
{
MForm.ShowStatus( "GCD was Product." );
}
}
else
{
MForm.ShowStatus( "GCD was one." );
}
MForm.ShowStatus( "XRoot: " + IntMath.ToString10( XRoot ));
MForm.ShowStatus( "Y: " + IntMath.ToString10( Y ));
XwithY.Copy( Y );
if( Y.ParamIsGreater( XRoot ))
throw( new Exception( "This can't be right. XRoot is bigger than Y." ));
IntMath.Subtract( Y, XRoot );
IntMath.GreatestCommonDivisor( Product, XwithY, Gcd );
if( !Gcd.IsOne())
{
if( !Gcd.IsEqual( Product ))
{
SolutionP.Copy( Gcd );
IntMath.Divide( Product, SolutionP, Quotient, Remainder );
if( !Remainder.IsZero())
throw( new Exception( "The Remainder with SolutionP can't be zero." ));
SolutionQ.Copy( Quotient );
MForm.ShowStatus( "SolutionP: " + IntMath.ToString10( SolutionP ));
MForm.ShowStatus( "SolutionQ: " + IntMath.ToString10( SolutionQ ));
return true;
}
else
{
MForm.ShowStatus( "GCD was Product." );
}
}
else
{
MForm.ShowStatus( "GCD was one." );
}
return false;
}
internal void ModularPower( Integer Result, Integer Exponent, Integer GeneralBase )
{
// The square and multiply method is in Wikipedia:
// https://en.wikipedia.org/wiki/Exponentiation_by_squaring
// x^n = (x^2)^((n - 1)/2) if n is odd.
// x^n = (x^2)^(n/2) if n is even.
if( Result.IsZero())
return; // With Result still zero.
if( Result.IsEqual( GeneralBase ))
{
// It is congruent to zero % ModN.
Result.SetToZero();
return;
}
// Result is not zero at this point.
if( Exponent.IsZero() )
{
Result.SetFromULong( 1 );
return;
}
if( GeneralBase.ParamIsGreater( Result ))
{
// throw( new Exception( "This is not supposed to be input for RSA plain text." ));
IntMath.Divide( Result, GeneralBase, Quotient, Remainder );
Result.Copy( Remainder );
}
if( Exponent.IsEqualToULong( 1 ))
{
// Result stays the same.
return;
}
// This could also be called ahead of time if the base (the modulus)
// doesn't change. Like when your public key doesn't change.
SetupGeneralBaseArray( GeneralBase );
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.Multiply( Result, XForModPower );
// Modular Reduction:
AddByGeneralBaseArrays( TempForModPower, Result );
Result.Copy( TempForModPower );
}
ExponentCopy.ShiftRight( 1 ); // Divide by 2.
if( ExponentCopy.IsZero())
break;
// Square it.
IntMath.Multiply( XForModPower, XForModPower );
// Modular Reduction:
AddByGeneralBaseArrays( TempForModPower, XForModPower );
XForModPower.Copy( TempForModPower );
}
// When AddByGeneralBaseArrays() gets called it multiplies a 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.
// If by chance you got a carry bit on _every_ addition that was done
// in AddByGeneralBaseArrays() then this number could increase in size
// by 1 bit for each addition that was done. It would take 32 bits of
// carrying for HowBig to increase by 1.
// See HowManyToAdd in AddByGeneralBaseArrays() for why this check is done.
int HowBig = Result.GetIndex() - GeneralBase.GetIndex();
if( HowBig > 2 ) // I have not seen this happen yet.
throw( new Exception( "The difference in index size was more than 2. Diff: " + HowBig.ToString() ));
// So this Quotient has only one or two 32-bit digits in it.
// And this Divide() is only called once at the end. Not in the loop.
IntMath.Divide( Result, GeneralBase, Quotient, Remainder );
Result.Copy( Remainder );
}