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| ///////////////////////////////////////////////////////////////////////////
// //
// Copyright (c) 2015 by Charta Software B.V. //
// All rights reserved //
// //
// Version: 1.7.0.83525 //
// Web site: https://pascal.chartasoftware.com/ //
// //
// This code and information are provided "as is" without warranty of //
// any kind. Dissemination of this information or reproduction of //
// this material is strictly forbidden unless prior written permission //
// is obtained from Charta Software B.V.. //
// //
///////////////////////////////////////////////////////////////////////////
unit Integer;
interface
uses
Boolean,
Byte,
Comparator,
Exception,
FloatingPoint._32,
FloatingPoint._64,
Hash.Function_,
Integer._16,
Integer._16.Positive,
Integer._32,
Integer._32.Positive,
Integer._64,
Integer._64.Positive,
Integer._8,
Integer._8.Positive,
Mathematics.Sign,
Stream.Input,
Stream.Output,
Text;
type
EInteger = class(EException);
TInteger = record
private
FSign: TSign;
FMagnitude: TArray<TByte>;
function AddMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TArray<TByte>;
function SubtractMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TInteger;
function MultiplyMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TArray<TByte>;
function DivideMagnitude(const Dividend, Divisor: TArray<TByte>): TArray<TByte>;
procedure DivideAndRemainderMagnitude(const Dividend, Divisor: TArray<TByte>; out Quotient, Remainder: TArray<TByte>);
function RemainderMagnitude(const Dividend, Divisor: TArray<TByte>): TArray<TByte>;
function ModularExponentMagnitude(const Base, Exponent, Modulus: TArray<TByte>): TArray<TByte>;
function ModularInverseMagnitude(const Value, Modulus: TArray<TByte>): TArray<TByte>;
function SquareMagnitude(const Value: TArray<TByte>): TArray<TByte>;
function GreatestCommonDivisorMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TArray<TByte>;
function CompareMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TSign;
procedure SetZeroMagnitude();
procedure SetPositive64Magnitude(Value: TPositiveInteger64; SignificantByteCount: TPositiveInteger32);
procedure SetValue64(Value: TInteger64; SignificantByteCount: TPositiveInteger32 = SizeOf(TInteger64));
procedure SetValuePositive64(Value: TPositiveInteger64; SignificantByteCount: TPositiveInteger32 = SizeOf(TPositiveInteger64));
function ToBoundedInteger(ByteCount: TInteger32): TInteger64;
function ToBoundedPositiveInteger(ByteCount: TInteger32): TPositiveInteger64;
private
class procedure InitializeConstants(); static;
class function StripLeadingZeroBytes(const Magnitude: TArray<TByte>): TArray<TByte>; static;
class function MakePositive(const TwosComplement: TArray<TByte>): TArray<TByte>; static;
public
class var
Zero: TInteger;
One: TInteger;
class function New(Sign: TSign; const Magnitude: TArray<TByte>): TInteger; static;
class function NewFromTwosComplement(const TwosComplement: TArray<TByte>): TInteger; static;
public
function ToText(): TText;
function HashCode(): TPositiveInteger32;
function Multiply(const Value: TInteger): TInteger;
function Add(const Value: TInteger): TInteger;
function Subtract(const Value: TInteger): TInteger;
function Divide(Divisor: TInteger): TInteger;
procedure DivideAndRemainder(const Divisor: TInteger; out Quotient, Remainder: TInteger);
function Modulo(const Divisor: TInteger): TInteger;
function Modulo2(Divisor: TInteger32): TInteger;
function Remainder(const Divisor: TInteger): TInteger;
function ModularExponent(Exponent: TInteger; const Modulus: TInteger): TInteger;
function ModularExponent2(Exponent: TInteger; ModulusExponent: TInteger32): TInteger;
function ModularInverse(const Modulus: TInteger): TInteger;
function GreatestCommonDivisor(Value: TInteger): TInteger;
function MultiplyWithPowerOfTen(Exponent: TInteger32): TInteger;
function CompareTo(const Value: TInteger): TSign;
function TestBit(Index: TInteger32): TBoolean;
function GetLowestSetBit(): TInteger32;
function BitLength(): TInteger32;
function OctetLength(): TInteger32;
function GetByteCount(): TInteger32;
function IsZero(): TBoolean;
function IsPositive(): TBoolean;
function IsNegative(): TBoolean;
function IsOne(): TBoolean;
function AbsoluteValue(): TInteger;
function SignByte(): TByte;
function GetByte(Index: TInteger32): TByte;
function GetFirstNonZeroByteIndex(): TInteger32;
function RoundUpToPrecision(const Precision: TInteger): TInteger;
function ToInteger8(): TInteger8;
function ToInteger16(): TInteger16;
function ToInteger32(): TInteger32;
function ToInteger64(): TInteger64;
function ToPositiveInteger8(): TPositiveInteger8;
function ToPositiveInteger16(): TPositiveInteger64;
function ToPositiveInteger32(): TPositiveInteger32;
function ToPositiveInteger64(): TPositiveInteger64;
function ToFloatingPoint32(): TFloatingPoint32;
function ToFloatingPoint64(): TFloatingPoint64;
function ToTwosComplementBigEndian(ByteCount: TInteger32): TArray<TByte>;
function ToTwosComplementLittleEndian(ByteCount: TInteger32): TArray<TByte>;
function Coerce(SignType: TSignType; ByteCount: TInteger32): TInteger;
property Sign: TSign read FSign;
property Magnitude: TArray<TByte> read FMagnitude;
public
class operator Implicit(Value: TInteger8): TInteger;
class operator Implicit(Value: TPositiveInteger8): TInteger;
class operator Implicit(Value: TInteger16): TInteger;
class operator Implicit(Value: TPositiveInteger16): TInteger;
class operator Implicit(Value: TInteger32): TInteger;
class operator Implicit(Value: TPositiveInteger32): TInteger;
class operator Implicit(Value: TInteger64): TInteger;
class operator Implicit(Value: TPositiveInteger64): TInteger;
class operator Implicit(const Value: TInteger): TPositiveInteger8;
class operator Implicit(const Value: TInteger): TPositiveInteger16;
class operator Implicit(const Value: TInteger): TPositiveInteger32;
class operator Implicit(const Value: TInteger): TPositiveInteger64;
class operator Implicit(const Value: TInteger): TInteger8;
class operator Implicit(const Value: TInteger): TInteger16;
class operator Implicit(const Value: TInteger): TInteger32;
class operator Implicit(const Value: TInteger): TInteger64;
class operator Implicit(const Value: TInteger): TFloatingPoint32;
class operator Implicit(const Value: TInteger): TFloatingPoint64;
class operator Add(const Left: TInteger; const Right: TInteger): TInteger;
class operator Subtract(const Left: TInteger; const Right: TInteger): TInteger;
class operator Multiply(const Left: TInteger; const Right: TInteger): TInteger;
class operator IntDivide(const Left: TInteger; const Right: TInteger): TInteger;
class operator Divide(const Left: TInteger; const Right: TInteger): TFloatingPoint64;
class operator Modulus(const Left: TInteger; const Right: TInteger): TInteger;
class operator LeftShift(const Operand: TInteger; Count: TInteger32): TInteger;
class operator RightShift(const Operand: TInteger; Count: TInteger32): TInteger;
class operator Equal(const Left: TInteger; const Right: TInteger): TBoolean;
class operator NotEqual(const Left: TInteger; const Right: TInteger): TBoolean;
class operator GreaterThan(const Left: TInteger; const Right: TInteger): TBoolean;
class operator GreaterThanOrEqual(const Left: TInteger; const Right: TInteger): TBoolean;
class operator LessThan(const Left: TInteger; const Right: TInteger): TBoolean;
class operator LessThanOrEqual(const Left: TInteger; const Right: TInteger): TBoolean;
class operator Positive(const Value: TInteger): TInteger;
class operator Negative(const Value: TInteger): TInteger;
public
class constructor Create();
class destructor Destroy();
class function Minimum(const Left: TInteger; const Right: TInteger): TInteger; static;
class function Maximum(const Left: TInteger; const Right: TInteger): TInteger; static;
class function Compare(const Left: TInteger; const Right: TInteger): TSign; static;
class function ReadLittleEndian(Stream: TInputStream; ByteCount: TInteger32; Signed: TBoolean): TInteger; static;
class procedure WriteLittleEndian(Stream: TOutputStream; ByteCount: TInteger32; Signed: TBoolean; const Value: TInteger); static;
class function ReadBigEndian(Stream: TInputStream; ByteCount: TInteger32; Signed: TBoolean): TInteger; static;
class procedure WriteBigEndian(Stream: TOutputStream; ByteCount: TInteger32; Signed: TBoolean; const Value: TInteger); static;
public
type
TCompare = class(TComparator<TInteger>)
public
function Compare(const Left: TInteger; const Right: TInteger): TSign; override;
public
class var
Instance: TCompare;
end;
THash = class(THashFunction<TInteger>)
public
function Compute(const Value: TInteger): TPositiveInteger32; override;
public
class var
Instance: THash;
end;
end;
implementation
uses
Exception.Enumeration,
Integer.Positive,
Integer.Positive.Addition,
Integer.Positive.Division,
Integer.Positive.EuclideanAlgorithm,
Integer.Positive.ModularExponentiation,
Integer.Positive.ModularExponentiation.Montgomery,
Integer.Positive.ModularInversion,
Integer.Positive.Modulo,
Integer.Positive.Multiplication,
Integer.Positive.Subtraction,
Integer.TextFormat;
type
TBitStatistics8 = record
private
class function CountTrailingZeros(Value: TPositiveInteger8): TPositiveInteger8; static;
class function CountBitLength(Value: TPositiveInteger8): TPositiveInteger8; static;
class function CountOneBits(Value: TPositiveInteger8): TPositiveInteger8; static;
public
Value: TPositiveInteger8;
BitLength: TPositiveInteger8;
BitCount: TPositiveInteger8;
TrailingZeroCount: TPositiveInteger8;
class function New(Value: TPositiveInteger8): TBitStatistics8; static;
end;
var
BitStatistics: array [TPositiveInteger8] of TBitStatistics8;
procedure InitializeBitStatistics8();
var
Index: TInteger32;
begin
for Index := Low(BitStatistics) to High(BitStatistics) do
BitStatistics[Index] := TBitStatistics8.New(Index);
end;
{ TInteger }
class function TInteger.New(Sign: TSign; const Magnitude: TArray<TByte>): TInteger;
begin
Result.FMagnitude := StripLeadingZeroBytes(Magnitude);
if Length(Result.FMagnitude) = 0 then
Result.FSign := sZero
else
begin
if Sign = sZero then
raise EInteger.Create('Sign cannot be zero while the magnitude is different from zero');
Result.FSign := Sign;
end;
end;
function TInteger.AbsoluteValue(): TInteger;
begin
if Sign = sNegative then
Result := -Self
else
Result := Self;
end;
function TInteger.Add(const Value: TInteger): TInteger;
begin
if Value.IsZero then
Result := Self
else if Self.IsZero then
Result := Value
else if Value.Sign = Self.Sign then
Result := TInteger.New(Sign, AddMagnitude(Self.Magnitude, Value.Magnitude))
else if Self.Sign = sNegative then
Result := SubtractMagnitude(Value.Magnitude, Self.Magnitude)
else
Result := SubtractMagnitude(Self.Magnitude, Value.Magnitude);
end;
class operator TInteger.Add(const Left: TInteger; const Right: TInteger): TInteger;
begin
Result := Left.Add(Right);
end;
function TInteger.AddMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TArray<TByte>;
var
PositiveSource: TPositiveInteger;
PositiveTarget: TPositiveInteger;
PositiveResult: TPositiveInteger;
Addition: TPositiveIntegerAddition;
begin
PositiveTarget := TPositiveInteger.New(Left);
PositiveSource := TPositiveInteger.New(Right);
Addition := TPositiveIntegerAddition.Create(PositiveSource, PositiveTarget);
PositiveResult := Addition.Apply();
Addition.Free();
Result := PositiveResult.ToByteArray();
end;
function TInteger.ToBoundedInteger(ByteCount: TInteger32): TInteger64;
var
Index: TInteger32;
begin
if
(ByteCount > Length(Magnitude))
or (
(ByteCount = Length(Magnitude))
and (
(Magnitude[0] <= $7f)
or ((Sign = sNegative) and (Magnitude[0] = $80) and (GetFirstNonZeroByteIndex() = ByteCount - 1))
)
)
then
begin
Result := 0;
for Index := High(Magnitude) downto 0 do
Result := (Result shl 8) or Magnitude[High(Magnitude) - Index];
if Sign = sNegative then
Result := -Result;
end
else
raise EInteger.Create('Value does not not fit the target precision');
end;
function TInteger.ToBoundedPositiveInteger(ByteCount: TInteger32): TPositiveInteger64;
var
Index: TInteger32;
begin
if ByteCount >= Length(Magnitude) then
begin
Result := 0;
for Index := Minimum(ByteCount, Length(Magnitude)) - 1 downto 0 do
Result := (Result shl 8) or Magnitude[High(Magnitude) - Index];
end
else
raise EInteger.Create('Value does not not fit the target precision');
end;
function TInteger.ToFloatingPoint32(): TFloatingPoint32;
begin
// TODO: Currently we first convert to TInteger64, exceptions will be raised of the value does not fit
// that TInteger64. We might be able to prevent a number of exceptions by investigating how to
// convert any TInteger to a TFloatingPoint32, if possible.
Result := ToInteger64();
end;
function TInteger.ToFloatingPoint64(): TFloatingPoint64;
begin
// TODO: Currently we first convert to TInteger64, exceptions will be raised of the value does not fit
// that TInteger64. We might be able to prevent a number of exceptions by investigating how to
// convert any TInteger to a TFloatingPoint64, if possible.
Result := ToInteger64();
end;
function TInteger.ToInteger16(): TInteger16;
begin
Result := ToBoundedInteger(SizeOf(Result));
end;
function TInteger.ToInteger32(): TInteger32;
begin
Result := ToBoundedInteger(SizeOf(Result));
end;
function TInteger.ToInteger64(): TInteger64;
begin
Result := ToBoundedInteger(SizeOf(Result));
end;
function TInteger.ToInteger8(): TInteger8;
begin
Result := ToBoundedInteger(SizeOf(Result));
end;
function TInteger.ToPositiveInteger16(): TPositiveInteger64;
begin
Result := ToBoundedPositiveInteger(SizeOf(Result));
end;
function TInteger.ToPositiveInteger32(): TPositiveInteger32;
begin
Result := ToBoundedPositiveInteger(SizeOf(Result));
end;
function TInteger.ToPositiveInteger64(): TPositiveInteger64;
begin
Result := ToBoundedPositiveInteger(SizeOf(Result));
end;
function TInteger.ToPositiveInteger8(): TPositiveInteger8;
begin
Result := ToBoundedPositiveInteger(SizeOf(Result));
end;
function TInteger.BitLength(): TInteger32;
var
MagnitudeBitLength: TInteger32;
IsPowerOfTwo: TBoolean;
Index: TInteger32;
begin
if IsZero then
Result := 0
else
begin
// Calculate the bit length of the magnitude
MagnitudeBitLength := 8 * (Length(Magnitude) - 1) + BitStatistics[Magnitude[0]].BitLength;
if IsNegative then
begin
// Check if magnitude is a power of two
IsPowerOfTwo := BitStatistics[Magnitude[0]].BitCount = 1;
Index := 1;
while IsPowerOfTwo and (Index < Length(Magnitude)) do
begin
IsPowerOfTwo := Magnitude[Index] = 0;
Inc(Index);
end;
if IsPowerOfTwo then
Result := MagnitudeBitLength - 1
else
Result := MagnitudeBitLength;
end
else
Result := MagnitudeBitLength;
end;
end;
function TInteger.Coerce(SignType: TSignType; ByteCount: TInteger32): TInteger;
begin
case SignType of
TSignType.Unsigned:
Result := TInteger.New(sPositive, ToTwosComplementBigEndian(ByteCount));
TSignType.Signed:
Result := TInteger.NewFromTwosComplement(ToTwosComplementBigEndian(ByteCount));
else
raise EUnknownEnumerationValue.Create('Unknown sign type');
end;
end;
class function TInteger.Compare(const Left: TInteger; const Right: TInteger): TSign;
begin
Result := Left.CompareTo(Right);
end;
function TInteger.CompareMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TSign;
var
Index: TInteger32;
ByteLeft: TByte;
ByteRight: TByte;
begin
if Length(Left) < Length(Right) then
Result := -1
else if Length(Left) > Length(Right) then
Result := 1
else
begin
Result := 0;
Index := 0;
// Argument lengths are equal; compare the values
while (Result = 0) and (Index < Length(Left)) do
begin
ByteLeft := Left[Index];
ByteRight := Right[Index];
if ByteLeft < ByteRight then
Result := -1
else if ByteLeft > ByteRight then
Result := 1;
Inc(Index);
end;
end;
end;
function TInteger.CompareTo(const Value: TInteger): TSign;
begin
if Self.Sign = Value.Sign then
Result := Sign * CompareMagnitude(Self.Magnitude, Value.Magnitude)
else if Self.Sign > Value.Sign then
Result := 1
else
Result := -1;
end;
class constructor TInteger.Create();
begin
InitializeConstants();
InitializeBitStatistics8();
TCompare.Instance := TCompare.Create();
TCompare.Instance.AddReference();
THash.Instance := THash.Create();
THash.Instance.AddReference();
end;
class destructor TInteger.Destroy();
begin
TCompare.Instance.ReleaseReference();
THash.Instance.ReleaseReference();
end;
function TInteger.Divide(Divisor: TInteger): TInteger;
begin
if Divisor.IsZero then
raise EInteger.Create('Division by zero')
else if Self.IsZero then
Result := TInteger.Zero
else
Result := TInteger.New(Self.Sign * Divisor.Sign, DivideMagnitude(Self.Magnitude, Divisor.Magnitude));
end;
class procedure TInteger.InitializeConstants();
begin
TInteger.Zero := TInteger.New(sZero, nil);
TInteger.One := 1;
end;
class operator TInteger.IntDivide(const Left: TInteger; const Right: TInteger): TInteger;
begin
Result := Left.Divide(Right);
end;
class operator TInteger.Divide(const Left: TInteger; const Right: TInteger): TFloatingPoint64;
begin
// TODO: This is a rough approximation, we should provide a much better implementation
Result := Left.ToFloatingPoint64() / Right.ToFloatingPoint64();
end;
procedure TInteger.DivideAndRemainder(const Divisor: TInteger; out Quotient, Remainder: TInteger);
var
QuotientMagnitude: TArray<TByte>;
RemainderMagnitude: TArray<TByte>;
begin
if Divisor.IsZero then
raise EInteger.Create('Division by zero')
else if IsZero then
begin
Quotient := Zero;
Remainder := Zero;
end
else if Length(Self.Magnitude) < Length(Divisor.Magnitude) then
begin
Quotient := Zero;
Remainder := Self;
end
else
begin
DivideAndRemainderMagnitude(Magnitude, Divisor.Magnitude, QuotientMagnitude, RemainderMagnitude);
Quotient := TInteger.New(Sign * Divisor.Sign, QuotientMagnitude);
Remainder := TInteger.New(Sign, RemainderMagnitude);
end;
end;
procedure TInteger.DivideAndRemainderMagnitude(const Dividend, Divisor: TArray<TByte>; out Quotient, Remainder: TArray<TByte>);
var
PositiveDividend: TPositiveInteger;
PositiveDivisor: TPositiveInteger;
PositiveQuotient: TPositiveInteger;
Division: TPositiveIntegerDivision;
begin
PositiveDividend := TPositiveInteger.New(Dividend);
PositiveDivisor := TPositiveInteger.New(Divisor);
Division := TPositiveIntegerDivision.Create(PositiveDividend, PositiveDivisor);
PositiveQuotient := Division.Apply();
Quotient := PositiveQuotient.ToByteArray();
Remainder := Division.Remainder.ToByteArray();
Division.Free();
end;
function TInteger.DivideMagnitude(const Dividend: TArray<TByte>; const Divisor: TArray<TByte>): TArray<TByte>;
var
PositiveDividend: TPositiveInteger;
PositiveDivisor: TPositiveInteger;
PositiveQuotient: TPositiveInteger;
Division: TPositiveIntegerDivision;
begin
PositiveDividend := TPositiveInteger.New(Dividend);
PositiveDivisor := TPositiveInteger.New(Divisor);
Division := TPositiveIntegerDivision.Create(PositiveDividend, PositiveDivisor);
PositiveQuotient := Division.Apply();
Result := PositiveQuotient.ToByteArray();
Division.Free();
end;
class operator TInteger.Equal(const Left: TInteger; const Right: TInteger): TBoolean;
begin
Result := Left.CompareTo(Right) = sZero;
end;
function TInteger.GetFirstNonZeroByteIndex(): TInteger32;
begin
Result := High(Magnitude);
while (Result >= 0) and (Magnitude[Result] = 0) do
Dec(Result);
Result := High(Magnitude) - Result; // Convert endian
end;
function TInteger.GetByte(Index: TInteger32): TByte;
var
PositiveByte: TByte;
begin
if Index >= Length(Magnitude) then
Result := SignByte
else
begin
PositiveByte := Magnitude[High(Magnitude) - Index];
if Sign >= sZero then
Result := PositiveByte
else if Index <= GetFirstNonZeroByteIndex() then
Result := -TInteger8(PositiveByte)
else
Result := not PositiveByte;
end;
end;
function TInteger.GetByteCount(): TInteger32;
begin
Result := BitLength() div 8 + 1;
end;
function TInteger.GetLowestSetBit(): TInteger32;
var
Index: TInteger32;
Digit: TByte;
begin
if IsZero then
Result := -1
else
begin
Index := 0;
repeat
Digit := GetByte(Index);
Inc(Index);
until Digit <> 0;
Result := 8 * Index + BitStatistics[Digit].TrailingZeroCount;
end;
end;
class operator TInteger.GreaterThan(const Left: TInteger; const Right: TInteger): TBoolean;
begin
Result := Left.CompareTo(Right) > 0;
end;
class operator TInteger.GreaterThanOrEqual(const Left: TInteger; const Right: TInteger): TBoolean;
begin
Result := Left.CompareTo(Right) >= 0;
end;
function TInteger.GreatestCommonDivisor(Value: TInteger): TInteger;
begin
if Value.IsZero then
Result := Self.AbsoluteValue()
else if Self.IsZero then
Result := Value.AbsoluteValue()
else
Result := TInteger.New(1, GreatestCommonDivisorMagnitude(Self.Magnitude, Value.Magnitude));
end;
function TInteger.GreatestCommonDivisorMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TArray<TByte>;
var
PositiveLeft: TPositiveInteger;
PositiveRight: TPositiveInteger;
EuclideanAlgorithm: TEuclideanAlgorithm;
begin
PositiveLeft := TPositiveInteger.New(Left);
PositiveRight := TPositiveInteger.New(Right);
EuclideanAlgorithm := TEuclideanAlgorithm.Create(PositiveLeft, PositiveRight);
EuclideanAlgorithm.Calculate();
Result := EuclideanAlgorithm.GreatestCommonDivisor.ToByteArray();
EuclideanAlgorithm.Free();
end;
function TInteger.HashCode(): TPositiveInteger32;
var
Index: TInteger32;
begin
Result := 0;
for Index := 0 to High(Magnitude) do
Result := 31 * Result + Magnitude[Index];
Result := Result * Sign;
end;
class operator TInteger.Implicit(Value: TInteger32): TInteger;
begin
Result.SetValue64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(Value: TInteger64): TInteger;
begin
Result.SetValue64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(const Value: TInteger): TPositiveInteger8;
begin
Result := Value.ToPositiveInteger8();
end;
class operator TInteger.Implicit(const Value: TInteger): TPositiveInteger16;
begin
Result := Value.ToPositiveInteger16();
end;
class operator TInteger.Implicit(const Value: TInteger): TPositiveInteger32;
begin
Result := Value.ToPositiveInteger32();
end;
class operator TInteger.Implicit(const Value: TInteger): TPositiveInteger64;
begin
Result := Value.ToPositiveInteger64();
end;
class operator TInteger.Implicit(const Value: TInteger): TInteger8;
begin
Result := Value.ToInteger8();
end;
class operator TInteger.Implicit(const Value: TInteger): TInteger16;
begin
Result := Value.ToInteger16();
end;
class operator TInteger.Implicit(const Value: TInteger): TInteger32;
begin
Result := Value.ToInteger32();
end;
class operator TInteger.Implicit(const Value: TInteger): TInteger64;
begin
Result := Value.ToInteger64();
end;
class operator TInteger.Implicit(const Value: TInteger): TFloatingPoint32;
begin
Result := Value.ToFloatingPoint32();
end;
class operator TInteger.Implicit(const Value: TInteger): TFloatingPoint64;
begin
Result := Value.ToFloatingPoint64();
end;
class operator TInteger.Implicit(Value: TPositiveInteger8): TInteger;
begin
Result.SetValuePositive64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(Value: TInteger8): TInteger;
begin
Result.SetValue64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(Value: TPositiveInteger16): TInteger;
begin
Result.SetValuePositive64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(Value: TInteger16): TInteger;
begin
Result.SetValue64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(Value: TPositiveInteger32): TInteger;
begin
Result.SetValuePositive64(Value, SizeOf(Value));
end;
class operator TInteger.Implicit(Value: TPositiveInteger64): TInteger;
begin
Result.SetValuePositive64(Value, SizeOf(Value));
end;
function TInteger.IsNegative(): TBoolean;
begin
Result := Sign = sNegative;
end;
function TInteger.IsOne(): TBoolean;
begin
Result := IsPositive and (Length(Magnitude) = 1) and (Magnitude[0] = 1);
end;
function TInteger.IsPositive(): TBoolean;
begin
Result := Sign = sPositive;
end;
function TInteger.IsZero(): TBoolean;
begin
Result := Sign = sZero;
end;
class operator TInteger.LessThan(const Left: TInteger; const Right: TInteger): TBoolean;
begin
Result := Left.CompareTo(Right) < 0;
end;
class operator TInteger.LessThanOrEqual(const Left: TInteger; const Right: TInteger): TBoolean;
begin
Result := Left.CompareTo(Right) <= 0;
end;
class function TInteger.MakePositive(const TwosComplement: TArray<TByte>): TArray<TByte>;
var
SignByteCount: TInteger32;
AllZero: TBoolean;
ExtraByte: TInteger32;
Index: TInteger32;
begin
SignByteCount := 0;
while (SignByteCount < Length(TwosComplement)) and (TwosComplement[SignByteCount] = $FF) do
Inc(SignByteCount);
AllZero := True;
Index := SignByteCount;
while AllZero and (Index < Length(TwosComplement)) do
begin
AllZero := TwosComplement[Index] = 0;
Inc(Index);
end;
if AllZero then
ExtraByte := 1
else
ExtraByte := 0;
SetLength(Result, Length(TwosComplement) - SignByteCount + ExtraByte);
if AllZero then
Result[0] := 0;
for Index := ExtraByte to High(Result) do
Result[Index] := not TwosComplement[Index + SignByteCount];
Index := Length(Result);
repeat
Dec(Index);
Inc(Result[Index]);
until (Result[Index] <> 0) or (Index = 0);
end;
class function TInteger.Maximum(const Left: TInteger; const Right: TInteger): TInteger;
begin
if Left > Right then
Result := Left
else
Result := Right;
end;
class function TInteger.Minimum(const Left: TInteger; const Right: TInteger): TInteger;
begin
if Left < Right then
Result := Left
else
Result := Right;
end;
function TInteger.ModularExponent(Exponent: TInteger; const Modulus: TInteger): TInteger;
var
InvertResult: TBoolean;
Base: TInteger;
p: TInteger32;
m1, m2: TInteger;
a1, a2: TInteger;
y1, y2: TInteger;
Intermediate: TInteger;
begin
if not Modulus.IsPositive then
raise EInteger.Create('Modulus should be positive');
if Exponent.IsZero then
Result := TInteger.One
else
begin
InvertResult := Exponent.IsNegative();
if InvertResult then
Exponent := -Exponent;
if Self.IsNegative or (Self.CompareTo(Modulus) >= 0) then
Base := Self.Modulo(Modulus)
else
Base := Self;
if Modulus.TestBit(0) then
Result := TInteger.New(sPositive, ModularExponentMagnitude(Base.Magnitude, Exponent.Magnitude, Modulus.Magnitude))
else
begin
p := Modulus.GetLowestSetBit();
m1 := Modulus shr p;
m2 := TInteger.One shl p;
a1 := TInteger.New(1, ModularExponentMagnitude(Base.Magnitude, Exponent.Magnitude, m1.Magnitude));
a2 := Base.ModularExponent2(Exponent, p);
// Combine using Chinese Remainder Theorem
y1 := m2.ModularInverse(m1);
y2 := m1.ModularInverse(m2);
Intermediate := a2.Multiply(m1);
Intermediate := Intermediate.Multiply(y2);
Result := a1.Multiply(m2);
Result := Result.Multiply(y1);
Result := Result.Add(Intermediate);
Result := Intermediate.Modulo(Modulus);
end;
if InvertResult then
Result := Result.ModularInverse(Modulus);
end;
end;
function TInteger.ModularExponent2(Exponent: TInteger; ModulusExponent: TInteger32): TInteger;
var
BaseToPower2: TInteger;
begin
// Perform exponentiation using repeated squaring, truncate
// high order bits as indicated by modulus.
Result := TInteger.One;
BaseToPower2 := Self.Modulo2(ModulusExponent);
while not Exponent.IsZero do
begin
if Exponent.TestBit(0) then
begin
Result := Result.Multiply(BaseToPower2);
Result := Result.Modulo2(ModulusExponent);
end;
Exponent := Exponent shr 1;
if not Exponent.IsZero then
begin
BaseToPower2 := TInteger.New(1, SquareMagnitude(BaseToPower2.Magnitude));
BaseToPower2 := BaseToPower2.Modulo2(ModulusExponent);
end;
end;
end;
function TInteger.ModularExponentMagnitude(const Base, Exponent, Modulus: TArray<TByte>): TArray<TByte>;
var
PositiveBase: TPositiveInteger;
PositiveExponent: TPositiveInteger;
PositiveModulus: TPositiveInteger;
PositiveResult: TPositiveInteger;
ModularExponentiation: TModularExponentiation;
begin
PositiveBase := TPositiveInteger.New(Base);
PositiveExponent := TPositiveInteger.New(Exponent);
PositiveModulus := TPositiveInteger.New(Modulus);
ModularExponentiation := TMontgomeryModularExponentiation.Create(PositiveExponent, PositiveModulus);
PositiveResult := ModularExponentiation.Apply(PositiveBase);
Result := PositiveResult.ToByteArray();
ModularExponentiation.Free();
end;
function TInteger.ModularInverse(const Modulus: TInteger): TInteger;
var
Remainder: TInteger;
GreatestCommonDivisor: TInteger;
begin
Remainder := Self.Modulo(Modulus);
GreatestCommonDivisor := Remainder.GreatestCommonDivisor(Modulus);
if not GreatestCommonDivisor.IsOne then
raise EInteger.Create('Integer not invertible');
Result := TInteger.New(1, ModularInverseMagnitude(Remainder.Magnitude, Modulus.Magnitude));
end;
function TInteger.ModularInverseMagnitude(const Value, Modulus: TArray<TByte>): TArray<TByte>;
var
PositiveValue: TPositiveInteger;
PositiveModulus: TPositiveInteger;
PositiveResult: TPositiveInteger;
Inversion: TPositiveIntegerModularInversion;
begin
PositiveValue := TPositiveInteger.New(Value);
PositiveModulus := TPositiveInteger.New(Modulus);
Inversion := TPositiveIntegerModularInversion.Create(PositiveValue, PositiveModulus);
PositiveResult := Inversion.Apply();
Result := PositiveResult.ToByteArray();
Inversion.Free();
end;
function TInteger.Modulo(const Divisor: TInteger): TInteger;
begin
if not Divisor.IsPositive then
raise EInteger.Create('Modulus must be positive');
Result := Self.Remainder(Divisor);
if Result.IsNegative then
Result := Result.Add(Divisor);
end;
function TInteger.Modulo2(Divisor: TInteger32): TInteger;
var
ByteCount: TInteger32;
NewMagnitude: TArray<TByte>;
Index: TInteger32;
ExcessBitCount: TInteger32;
begin
if BitLength() <= Divisor then
Result := Self
else
begin
// Copy remaining bytes of magnitude
ByteCount := (Divisor + 7) div 8;
SetLength(NewMagnitude, ByteCount);
for Index := 0 to ByteCount - 1 do
NewMagnitude[Index] := Magnitude[Index + (Length(Magnitude) - ByteCount)];
// Mask out any excess bits
ExcessBitCount := 8 * ByteCount - Divisor;
NewMagnitude[0] := NewMagnitude[0] and (1 shl (8 - ExcessBitCount)) - 1;
Result := TInteger.New(1, NewMagnitude);
end;
end;
class operator TInteger.Modulus(const Left: TInteger; const Right: TInteger): TInteger;
begin
Result := Left.Modulo(Right);
end;
function TInteger.Multiply(const Value: TInteger): TInteger;
begin
if (Value.Sign = sZero) or (Self.Sign = sZero) then
Result := TInteger.Zero
else
Result := TInteger.New(Sign * Value.Sign, MultiplyMagnitude(Magnitude, Value.Magnitude));
end;
class operator TInteger.Multiply(const Left: TInteger; const Right: TInteger): TInteger;
begin
Result := Left.Multiply(Right);
end;
function TInteger.MultiplyMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TArray<TByte>;
var
PositiveLeft: TPositiveInteger;
PositiveRight: TPositiveInteger;
PositiveResult: TPositiveInteger;
Multiplication: TPositiveIntegerMultiplication;
begin
PositiveLeft := TPositiveInteger.New(Left);
PositiveRight := TPositiveInteger.New(Right);
Multiplication := TPositiveIntegerMultiplication.Create(PositiveLeft, PositiveRight);
PositiveResult := Multiplication.Apply();
Multiplication.Free();
Result := PositiveResult.ToByteArray();
end;
function TInteger.MultiplyWithPowerOfTen(Exponent: TInteger32): TInteger;
var
Index: TInteger32;
begin
// TODO: This can be implemented more efficiently
Result := Self;
for Index := 1 to Exponent do
Result := Result * 10;
end;
class operator TInteger.Negative(const Value: TInteger): TInteger;
begin
Result := TInteger.New(-Value.Sign, Value.Magnitude);
end;
class function TInteger.NewFromTwosComplement(const TwosComplement: TArray<TByte>): TInteger;
begin
if Length(TwosComplement) = 0 then
raise EInteger.Create('2''s complement representation cannot be of zero length');
if TInteger8(TwosComplement[0]) < 0 then
begin
Result.FSign := sNegative;
Result.FMagnitude := MakePositive(TwosComplement);
end
else
begin
Result.FMagnitude := StripLeadingZeroBytes(TwosComplement);
if Length(Result.FMagnitude) = 0 then
Result.FSign := sZero
else
Result.FSign := sPositive;
end;
end;
class operator TInteger.NotEqual(const Left: TInteger; const Right: TInteger): TBoolean;
begin
Result := Left.CompareTo(Right) <> sZero;
end;
function TInteger.OctetLength(): TInteger32;
begin
// TODO: How does this compare to GetByteCount();
Result := Length(Magnitude);
end;
class operator TInteger.Positive(const Value: TInteger): TInteger;
begin
Result := TInteger.New(Value.Sign, Value.Magnitude);
end;
procedure TInteger.SetValue64(Value: TInteger64; SignificantByteCount: TPositiveInteger32 = SizeOf(TInteger64));
begin
if Value < 0 then
begin
FSign := sNegative;
SetPositive64Magnitude(-Value, SignificantByteCount);
end
else if Value > 0 then
begin
FSign := sPositive;
SetPositive64Magnitude(Value, SignificantByteCount);
end
else
begin
FSign := sZero;
SetZeroMagnitude();
end;
end;
function TInteger.SignByte(): TByte;
begin
if Sign = sNegative then
Result := $FF
else
Result := 0;
end;
function TInteger.SquareMagnitude(const Value: TArray<TByte>): TArray<TByte>;
var
PositiveValue: TPositiveInteger;
PositiveResult: TPositiveInteger;
Squaring: TPositiveIntegerSquaring;
begin
PositiveValue := TPositiveInteger.New(Value);
Squaring := TPositiveIntegerSquaring.Create(PositiveValue);
PositiveResult := Squaring.Apply();
Result := PositiveResult.ToByteArray();
Squaring.Free();
end;
class function TInteger.StripLeadingZeroBytes(const Magnitude: TArray<TByte>): TArray<TByte>;
var
ZeroByteCount: TInteger32;
begin
ZeroByteCount := 0;
while (ZeroByteCount < Length(Magnitude)) and (Magnitude[ZeroByteCount] = 0) do
Inc(ZeroByteCount);
if ZeroByteCount > 0 then
Result := Copy(Magnitude, ZeroByteCount, Length(Magnitude) - ZeroByteCount)
else
Result := Magnitude;
end;
function TInteger.Subtract(const Value: TInteger): TInteger;
begin
// TODO: This function might not need to be expressed as an Add operation
// if we carefully consider the different possibilities
Result := Add(TInteger.New(-Value.Sign, Value.Magnitude));
end;
class operator TInteger.Subtract(const Left: TInteger; const Right: TInteger): TInteger;
begin
Result := Left.Subtract(Right);
end;
function TInteger.SubtractMagnitude(const Left: TArray<TByte>; const Right: TArray<TByte>): TInteger;
var
PositiveSource: TPositiveInteger;
PositiveTarget: TPositiveInteger;
PositiveResult: TPositiveInteger;
Subtraction: TPositiveIntegerSubtraction;
Sign: TSign;
begin
PositiveTarget := TPositiveInteger.New(Left);
PositiveSource := TPositiveInteger.New(Right);
Subtraction := TPositiveIntegerSubtraction.Create(PositiveTarget, PositiveSource);
PositiveResult := Subtraction.Apply();
if PositiveResult.IsZero() then
Result := TInteger.Zero
else
begin
if Subtraction.Negative then
Sign := sNegative
else
Sign := sPositive;
Result := TInteger.New(Sign, PositiveResult.ToByteArray());
end;
Subtraction.Free();
end;
function TInteger.TestBit(Index: TInteger32): TBoolean;
begin
if Index < 0 then
raise EInteger.Create('Index should be posirive');
Result := (GetByte(Index div 8) and (1 shl (Index mod 8))) <> 0;
end;
function TInteger.ToText(): TText;
begin
Result := TIntegerTextFormat.Default.ToText(Self);
end;
function TInteger.ToTwosComplementBigEndian(ByteCount: TInteger32): TArray<TByte>;
var
SourceIndex: TInteger32;
TargetIndex: TInteger32;
begin
SetLength(Result, ByteCount);
TargetIndex := ByteCount - 1;
for SourceIndex := 0 to ByteCount - 1 do
begin
Result[TargetIndex] := GetByte(SourceIndex);
Dec(TargetIndex);
end;
end;
function TInteger.ToTwosComplementLittleEndian(ByteCount: TInteger32): TArray<TByte>;
var
Index: TInteger32;
begin
SetLength(Result, ByteCount);
for Index := 0 to ByteCount - 1 do
Result[Index] := GetByte(Index);
end;
class procedure TInteger.WriteBigEndian(Stream: TOutputStream; ByteCount: TInteger32; Signed: TBoolean; const Value: TInteger);
var
Bytes: TArray<TByte>;
begin
if Signed then
Bytes := Value.ToTwosComplementBigEndian(ByteCount)
else
begin
Bytes := Value.Magnitude;
Stream.WriteZeros(ByteCount - Length(Bytes));
end;
Stream.Write(Bytes[0], Length(Bytes));
end;
class procedure TInteger.WriteLittleEndian(Stream: TOutputStream; ByteCount: TInteger32; Signed: TBoolean; const Value: TInteger);
var
Bytes: TArray<TByte>;
Index: TInteger32;
begin
if Signed then
Bytes := Value.ToTwosComplementBigEndian(ByteCount)
else
Bytes := Value.Magnitude;
for Index := High(Bytes) downto Low(Bytes) do
Stream.WriteLittleEndian8(Bytes[Index]);
Stream.WriteZeros(ByteCount - Length(Bytes));
end;
class function TInteger.ReadBigEndian(Stream: TInputStream; ByteCount: TInteger32; Signed: TBoolean): TInteger;
var
Bytes: TArray<TByte>;
begin
System.SetLength(Bytes, ByteCount);
Stream.Read(Bytes[0], ByteCount);
if Signed then
Result := TInteger.NewFromTwosComplement(Bytes)
else
Result := TInteger.New(sPositive, Bytes);
end;
class function TInteger.ReadLittleEndian(Stream: TInputStream; ByteCount: TInteger32; Signed: TBoolean): TInteger;
var
Bytes: TArray<TByte>;
Index: TInteger32;
begin
System.SetLength(Bytes, ByteCount);
for Index := ByteCount - 1 downto 0 do
Stream.Read(Bytes[Index], 1);
if Signed then
Result := TInteger.NewFromTwosComplement(Bytes)
else
Result := TInteger.New(sPositive, Bytes);
end;
function TInteger.Remainder(const Divisor: TInteger): TInteger;
begin
if Divisor.IsZero then
raise EInteger.Create('Division by zero')
else if Self.IsZero then
Result := TInteger.Zero
else if Length(Self.Magnitude) < Length(Divisor.Magnitude) then
Result := Self
else
Result := TInteger.New(Sign, RemainderMagnitude(Magnitude, Divisor.Magnitude));
end;
function TInteger.RemainderMagnitude(const Dividend, Divisor: TArray<TByte>): TArray<TByte>;
var
PositiveDividend: TPositiveInteger;
PositiveDivisor: TPositiveInteger;
Modulo: TPositiveIntegerModulo;
begin
PositiveDividend := TPositiveInteger.New(Dividend);
PositiveDivisor := TPositiveInteger.New(Divisor);
Modulo := TPositiveIntegerModulo.Create(PositiveDividend, PositiveDivisor);
Result := Modulo.Apply().ToByteArray();
Modulo.Free();
end;
function TInteger.RoundUpToPrecision(const Precision: TInteger): TInteger;
var
Remainder: TInteger;
begin
Remainder := Self mod Precision;
if Remainder > 0 then
Result := Self - Remainder + Precision
else
Result := Self;
end;
procedure TInteger.SetPositive64Magnitude(Value: TPositiveInteger64; SignificantByteCount: TPositiveInteger32);
var
Index: TInteger32;
begin
SetLength(FMagnitude, SignificantByteCount);
for Index := SignificantByteCount - 1 downto 0 do
begin
FMagnitude[Index] := Value;
Value := Value shr 8;
end;
FMagnitude := StripLeadingZeroBytes(Magnitude);
end;
procedure TInteger.SetValuePositive64(Value: TPositiveInteger64; SignificantByteCount: TPositiveInteger32 = SizeOf(TPositiveInteger64));
begin
if Value = 0 then
begin
FSign := sZero;
SetZeroMagnitude();
end
else
begin
FSign := sPositive;
SetPositive64Magnitude(Value, SignificantByteCount);
end;
end;
procedure TInteger.SetZeroMagnitude();
begin
SetLength(FMagnitude, 0);
end;
class operator TInteger.LeftShift(const Operand: TInteger; Count: TInteger32): TInteger;
var
ResultBytes: TArray<TByte>;
ResultByte: TByte;
ByteCount: TInteger32;
BitCount: TInteger32;
Index: TInteger32;
begin
if Count = 0 then
Result := Operand
else if Count < 0 then
Result := Operand shr -Count
else
begin
ByteCount := Count div 8;
BitCount := Count mod 8;
SetLength(ResultBytes, (Operand.BitLength() + Count) div 8 + 1);
if BitCount = 0 then
begin
for Index := ByteCount to High(ResultBytes) do
ResultBytes[High(ResultBytes) - Index] := Operand.GetByte(Index - ByteCount);
end
else
begin
for Index := ByteCount to High(ResultBytes) do
begin
ResultByte := Operand.GetByte(Index - ByteCount) shl BitCount;
if Index <> ByteCount then
ResultByte := ResultByte or (Operand.GetByte(Index - ByteCount - 1) shr (8 - BitCount));
ResultBytes[High(ResultBytes) - Index] := ResultByte;
end;
end;
Result := TInteger.NewFromTwosComplement(ResultBytes);
end;
end;
class operator TInteger.RightShift(const Operand: TInteger; Count: TInteger32): TInteger;
var
ByteCount: TInteger32;
BitCount: TInteger32;
ResultBytes: TArray<TByte>;
Index: TInteger32;
begin
if Count = 0 then
Result := Operand
else if Count < 0 then
Result := Operand shl -Count
else if Count >= Operand.BitLength() then
begin
if Operand.IsNegative then
Result := -1
else
Result := TInteger.Zero;
end
else
begin
ByteCount := Count div 8;
BitCount := Count mod 8;
SetLength(ResultBytes, (Operand.BitLength() - Count) div 8 + 1);
if BitCount = 0 then
begin
for Index:= 0 to High(ResultBytes) do
ResultBytes[High(ResultBytes) - Index] := Operand.GetByte(ByteCount + Index);
end
else
begin
for Index := 0 to High(ResultBytes) do
ResultBytes[High(ResultBytes) - Index] := ((Operand.GetByte(ByteCount + Index + 1) shl 8 or Operand.GetByte(ByteCount + Index)) shr BitCount);
end;
Result := TInteger.NewFromTwosComplement(ResultBytes);
end;
end;
{ TBitStatistics8 }
class function TBitStatistics8.CountBitLength(Value: TPositiveInteger8): TPositiveInteger8;
begin
Result := 8;
while (Result > 0) and ((Value and $80) = 0) do
begin
Dec(Result);
Value := Value shl 1;
end;
end;
class function TBitStatistics8.CountOneBits(Value: TPositiveInteger8): TPositiveInteger8;
var
Index: TInteger32;
begin
Result := 0;
for Index := 1 to 8 do
begin
if Value and $1 = $1 then
Inc(Result);
Value := Value shr 1;
end;
end;
class function TBitStatistics8.CountTrailingZeros(Value: TPositiveInteger8): TPositiveInteger8;
begin
Result := 0;
while (Result < 8) and ((Value and $1) = 0) do
begin
Inc(Result);
Value := Value shr 1;
end;
end;
class function TBitStatistics8.New(Value: TPositiveInteger8): TBitStatistics8;
begin
Result.Value := Value;
Result.TrailingZeroCount := CountTrailingZeros(Value);
Result.BitLength := CountBitLength(Value);
Result.BitCount := CountOneBits(Value);
end;
{ TInteger.TCompare }
function TInteger.TCompare.Compare(const Left: TInteger; const Right: TInteger): TSign;
begin
Result := TInteger.Compare(Left, Right);
end;
{ TInteger.THash }
function TInteger.THash.Compute(const Value: TInteger): TPositiveInteger32;
begin
Result := Value.HashCode();
end;
end.
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