BigIntegers unit
An uptodate version of this and other files can be found in my DelphiBigNumbers project on GitHub.
BigIntegers
The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can’t even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions. — Ronald Graham
On many occasions, I missed a BigInteger
type in Delphi. I wanted to have a type with which you could do simple things like:
var
A, B, C, D: BigInteger;
begin
A := 3141592653589793238462643383279;
B := 1414213562373095 shl 10;
C := 2718281828459045235360287471352;
D := A + B * C;
Writeln(D.ToString);
But in Delphi, there is no proper BigInteger
type that
 is easy to use,
 can handle very long integers,
 is optimized for speed and precision,
 can handle different numeric bases (binary, hex, decimal, etc.) and
 can do more than basic arithmetic.
I have been thinking of using the GNU multiprecision library, but that has one big problem: it is GPLlicensed, which makes it unsuitable for my needs. So I decided to write my own, using assembler where appropriate, but also thus that pure Pascal routines would be optimal too. It works well and, as far as I know, faultless.
I initially modelled it after the .NET System.Numerics.BigInteger
type, but took up influences from other languages with builtin BigIntegers
, like Common Lisp, Python and Java. And yet I think I managed to make it as “Delphi”like as possible. It is described below.
In the course of writing this, I had to implement a few other things, which can be useful on their own too. They are included in the bigintegers.zip file.
But for the most uptodate version, rather download from GitHub.
Take a look at my BigDecimal
implementation too. It uses BigIntegers
for most of the hard work. It can also be found in the bigintegers.zip file and on GitHub.
Note that I wrote this all on my own. I did not use any code from other implementations, but I did implement algorithms described in papers from, for instance, Burnikel and Ziegler, Karatsuba, etc.
The BigInteger type
Unlike the integral types in Delphi, the BigInteger
type uses a signmagnitude format, i.e. the magnitude (or absolute value) is always unsigned, and the sign is an extra bit. A Delphi Integer
1000 is stored as 32bit value of $FFFFFC18
, which is called two’s complement. A BigInteger
−1000 is stored as an always positive magnitude (or absolute value) of 1000 (or hex $000003E8
) and a sign bit. This has some implications for bitwise operations, but more about that later.
The magnitude is stored as a dynamic array of UInt32
. I, and as it turned out, also others, call such a single large digit a “limb” (a term you will see being used later on too). This is because in his book series “The Art of Computer Programming” (a.k.a. TAOCP), Donald Knuth, who describes a few base case algorithms for multiple precision math in TAOCP Volume 2, calls them so. I also like another term I have seen lately, “bigit” (biginteger digit).
BigInteger
is a record type. This allows it to be used as a value type, like normal integers, i.e. there is no need to worry about the lifetime, and it also allows the use of operators like *
, div
, mod
, +
, 
, and
, or
, xor
(and some others), as well as a big number of methods.
The main thing about a BigInteger
is that there is a very low chance (virtually nil) that it will overflow, since it will simply grow if more bits are needed. It can hold enormously huge integer values, theoretically — given enough memory is present — in the order of approximately 2^{4,000,000,000} or 10^{1,290,000,000}, in other words, an integer consisting of more than 1 billion digits. The total number of particles in the known universe is estimated as 10^{80} or even 10^{85}, so a BigInteger
can theoretically hold a vast multitude of that number.
Initialization
Unfortunately, there is no way to define a large integer literal like:
B := 3141592653589793238462643383279;
But yet, there are several ways to get values into a BigInteger. You can use constructors, like
A := BigInteger.Create(17);
or (implicit or explicit) conversion operators like
B := '3141592653589793238462643383279'; // string
C := 17; // integral type
E := BigInteger(6e30); // floating point type
or functions and expressions like
D := 6 * BigInteger.Pow(10, 30); // 6 * 10^30 exactly, unlike the floating point
// value 6e30 which is only an approximation.
Unfortunately, you can’t enter very long values as integers directly, so the following does not work:
// Does not compile.
B := 3141592653589793238462643383279;
because Delphi does not know such big integer literals. The alternative is to use a string, like in the second example above.
Constructors
Although BigInteger
is a record type and record constructors are no real constructors, and the syntax used can deceive you into thinking a new instance is allocated somewhere, I think records should have them, for initialization. But note that it doesn’t matter if you do:
A := BigInteger.Create(17);
Or just:
A.Create(17);
Both will do exactly the same and initialize A with the value 17.
The constructors defined are:
constructor Create(const Limbs: array of TLimb; Negative: Boolean); overload;  Sets up a BigInteger using the Limbs array as the (unsigned) magnitude and Negative as the sign 
constructor Create(const Data: TMagnitude; Negative: Boolean); overload;  Sets up a BigInteger using the Magnitude array as the (unsigned) magnitude and Negative as the sign 
constructor Create(const Int: BigInteger ); overload; 
Sets up a BigInteger with the value of the given BigInteger 
constructor Create(const Int: Int32); overload;  Sets up a BigInteger with the value of the given signed integer 
constructor Create(const Int: UInt32); overload;  Sets up a BigInteger with the value of the given unsigned integer 
constructor Create(const Int: Int64); overload;  Sets up a BigInteger with the value of the given 64bit signed integer 
constructor Create(const Int: UInt64); overload;  Sets up a BigInteger with the value of the given 64bit unsigned integer 
constructor Create(const ADouble: Double); overload;  Sets up a BigInteger with the value of the given Double . How this is done depends on the setting of the property BigInteger.RoundingMode (see below) 
constructor Create( const Bytes: array of Byte); overload;  Sets up a BigInteger using the bytes in Bytes as a two’s complement value. The opposite of the function ToByteArray . 
constructor Create(NumBits: Integer; const Random: IRandom); overload;  Sets up a BigInteger with a random value, with a size of NumBits bits and a value determined by the given IRandom interface (see below) 
Byte array parameter
The bytes in the byte array are considered to be in littleendian two’s complement format. That means that if the top bit of the most significant byte is set, then the value in the bytes is interpreted as negative. Say you want to pass a value of 129. Then your array must contain two bytes: $81, $00
, because a single $81
is interpreted as 127. Likewise, if you want to pass 129, you must pass $7F, $FF
, because $7F
is interpreted as +127.
Note that this is like in C#, which also accepts a byte array in littleendian order, but unlike in Java, which requires a bigendian byte array.
RoundingMode
The global RoundingMode property governs how a Double
is converted to a BigInteger
. It can have the following values.
rmTruncate 
Any fraction of the Double is discarded. This rounds toward 0 and is the default. 
rmSchool 
How I learned to round in school: any absolute fraction >= 0.5 rounds “up” (away from 0). So 1.5 rounds to 2 and 1.5 rounds to 2. But 1.49999 rounds to 1. 
rmRound 
Rounds the same way as the System.Round function. Any (absolute) fraction > 0.5 (this excludes 0.5 itself) is rounded away from 0. So 1.5 rounds to 1, but 1.50001 rounds to 2. 
Note that, due to the fact that floating point values can only have a limited precision, the results may not always be what you expect. A double value like 6e30
will not result in a BigInteger
with an exact value of 6000000000000000000000000000000
(a 6 with 30 zeroes), because the Double is not precise enough to hold such an exact value. The result is actually 5999999999999999556357795610624
, which is the conversion of the closest a Double
can get to 6e30
. If you want exact values, do not use floating point values as input. Use strings or functions like BigInteger.Pow
instead (see below). To obtain exactly 6e30
, you can use 6 * BigInteger.Pow(10, 30)
.
IRandom
I was not sure how to generate a random BigInteger
. Other languages seem to use separate Random
classes for this. I wanted to avoid any manual memory management for random numbers, so I declared an interface (called, you guessed it, IRandom
) with a few suitable methods and two implementations, the first being TRandom
, using the simple algorithm Knuth proposes and which Java seems to use (using an 48 bit seed) and the second, TDelphiRandom
, which uses the builtin System.Random
functions to generate random values. Both classes can be found in unit RandomNumbers.pas
, included in the download. If you want to create your own, inherit from TRandom
and override the virtual parameterless function Next()
.
Implicit conversions and class functions
To get a value into a BigInteger
, you don’t have to use constructors. There are implicit operators that allow you to assign several integer types directly, for instance:
var
A, B, C: BigInteger;
begin
A := 1;
B := 200*1000*1000; // My usual way to write a literal like 200,000,000
C := $8000000;
D := BigInteger(Pi * 1e20);
Another way is to use functions like Pow()
, e.g.
var
A: BigInteger;
begin
A := BigInteger.Pow(10, 85); // 10^85
The most convenient way, however, to get a well defined big number into a BigInteger
is to use its string parsing capabilities, which also allow you to assign a string, like so:
var
A, B: BigInteger;
begin
A := '671998030559713968361666935769';
B := '$2940C583C5C79C8A70261FEE080A73B5B23556A5CF802BAB81DB08546F3623D5';
Numeric base
Beside parsing simple decimal numbers like the ones in the code snippet above, the parser can do a bit more. But I will first have to explain the numeric base. This is the base used for text input as well as text output of a BigInteger
. I guess everyone knows decimal (base 10) numbers like 1000 or 17. But in Delphi, we know hexadecimal numbers (base 16) too, like $7F00. Some languages, like C or Java, also know octal (base 8), in the form 017 which is the same as 1*8 + 7, or decimal 15. The BigIntegers unit knows a bit more. It can have any default base from 2 to 36 (where base 36 has the “digits” 0..9 and A..Z, where A=10 and Z=35). You can set or query the default base by accessing the BigInteger.Base
class property. That it is a class property means that it is the same for all BigIntegers.
The default numeric base affects input and output. So if you do:
var
A, B: BigInteger;
begin
BigInteger.Base := 16;
A := '100';
BigInteger.Base := 10;
Writeln('A in decimal is: ', A.ToString);
The output is:
A in decimal is: 256
BigInteger
knows a few shortcut methods for the most usual base values, so instead of writing
BigInteger.Base := 16;
you can write
BigInteger.Hex;
The convenience methods are:
Name  Base value  

Binary 
2  Binary, e.g. '01011010' 
Octal 
8  Octal, e.g. '377' 
Decimal 
10  Decimal 
Hexadecimal 
16  Hexadecimal, e.g. '12BEEF34' 
Hex 
16  Short for Hexadecimal 
Parsing
The methods Parse
, TryParse
and the implicit conversion operator (which allows you to assign a string to a BigInteger
) allow for a few tricks to make it easy to enter numbers.
class function TryParse(const S: string; Base: TNumberBase; out Res: BigInteger): Boolean; overload; static;  Tries to parse the specified string into a valid BigInteger value in the specified numeric base. Returns False if this failed. 
class function TryParse(const S: string; out Res: BigInteger): Boolean; overload; static;  Tries to parse the specified string into a valid BigInteger value in the default BigInteger numeric base. Returns False if this failed. 
class function Parse(const S: string): BigInteger; static;  Parses the specified string into a BigInteger , using the default numeric base. Raises an exception if this is not possible. 
class operator Implicit( const Value: string): BigInteger;  Implicitly (i.e. without a cast) converts the specified string to a BigInteger . The BigInteger is the result of a call to Parse(Value) . 
There are a few things to know about the strings that can be valid BigIntegers:
 To make it easier to increase the legibility of large numbers, any
'_'
or' '
, anywhere in the numeric string, will completely be ignored, so'1_000_000_000'
,'1 000 000 000'
and'1000000000'
are exactly equivalent.  The string to be parsed is considered case insensitive, so
'$ABC'
and'$abc'
represent exactly the same value.  The number can be prefixed with a sign, either
'+'
or''
with the usual meaning, i.e. a prefix of''
will result in a negative number, the prefix'+'
in a positive one. If a sign prefix is omitted,'+'
is assumed.
Usually, the string that is parsed is assumed to be in the base that is set with BigInteger.Base
. But there are a few overrides that disregard the default base and give the number a specified base. Like in Delphi, this is done with a prefix:
Prefix  Base  

$ 
16  Hex, like in Delphi, e.g. '$8000' for decimal 32768 
0x 
16  Hex, like in C and C++, e.g. '0x8000' for decimal 32768 
0d 
10  Decimal, e.g. '0d1234' for decimal 1234 
0b 
2  Binary, e.g. '0b11001010' for decimal 202 
0o 
8  Octal, e.g. '0o377' for decimal 255 
0k 
8  (Better readable) alternative form for octal, e.g. '0k377' for decimal 255 
%nnR 
nn  nn stands for one or two decimal digits and gives the base (or radix) to be used. So '%36rRudyVelthuis' is a valid BigInteger number in base 36 
I made sure that simple numbers starting with a '0'
are not automatically regarded as octal, like they are in C. Any of the valid prefixes starting with '0'
has an alphabetic second character.
An example:
procedure Test;
var
A: BigInteger;
begin
BigInteger.Decimal;
A := '%36r Rudy Velthuis'; // Yes, that's my name
Writeln(A.ToString);
BigInteger.Base := 36;
Writeln(A.ToString);
BigInteger.Base := 35;
Writeln(A.ToString);
end;
The output is:
3664889415200015812
RUDYVELTHUIS
12XJI7QCQ4S0C
Note that the three lines above represent the exact same number, but in a different base.
Examples of valid BigInteger
strings:
String  Decimal value 

'0' 
0 
'1' 
1 
'01' 
1 
'0x123' 
291 
'$0123' 
291 
'$17' 
−23 
'12340' 
12340 
'%10r12345678901234567890' 
12345678901234567890 
'0d12345678901234567890' 
12345678901234567890 
'$12345678901234567890' 
85968058271978839505040 
'%16R12345678901234567890' 
85968058271978839505040 
'0x12345678901234567890' 
85968058271978839505040 
'17234' 
−17234 
'+17234' 
17234 
'0O7771234567' 
1071987063 
'0k7771234567' 
1071987063 
'%8R7771234567' 
1071987063 
'0b000100100011010001010110011110001001101010111100' 
−20015998343868 
'0b0001_0010_0011_0100_0101_0110_0111_1000_1001_0000' 
78187493520 
'$7fffffff9876543289abcdef01234567' 
170141183428425841568023956577411351911 
'$7fffffff_98765432_89abcdef_01234567' 
170141183428425841568023956577411351911 
'%16R 7FFFFFFF 98765432 89ABCDEF 01234567' 
170141183428425841568023956577411351911 
'$1234567898765432FFFFFF80' 
5634002667517048507802320768 
'%36rRudyVelthuis' 
3664889415200015812 
'%35rRudyVelthuis' 
2690686858144915658 
'$DEADBEEF' 
3735928559 
'%16r_DEADB_EE_F' 
3735928559 
'$ De ad Be ef' 
3735928559 
'%26rDead_Beef' 
108863310779 
'%36rDeadBeef' 
1049836114599 
'$cc ' 
−204 
'+0X0' 
0 
'0X00000000000000d' 
−13 
'%36rABCDEFGHIJKLMNOPQRSTUVWXYZ' 
8337503854730415241050377135811259267835 
'%36Rabcdefghijklmnopqrstuvwxyz' 
−8337503854730415241050377135811259267835 
Seeing 'RUDYVELTHUIS'
as a valid big integer still needs some getting used to. <g>
Output
There are two simple output routines.
function ToString: string; overload;  Returns a string representation of the BigInteger in the current default numeric base 
function ToString(Base: Byte): string; overload;  Returns a string representation of the BigInteger in the given numeric base 
The second version, ToString(Base)
accepts only Base
values in the range 2..36.
I also implemented a few convenience methods for this:
function ToBinaryString;  Shortcut for ToString(2) 
function ToOctalString;  Shortcut for ToString(8) 
function ToDecimalString;  Shortcut for ToString(10) 
function ToHexString;  Shortcut for ToString(16) 
UPDATE (7 Feb 2016)
The classic ToString(Base)
method (which is still available as ToStringClassic
) simply did a DivMod
by 10, turned the remainder into a digit and then repeated this with the quotient until the value of the BigInteger
was 0. That worked fine, but when I wanted to turn the currently largest known prime, 2^{74,207,281} − 1, into a string, I thought my computer had hung up. Well, the result is a 22 million digit string, and (what is now) ToStringClassic
simply repeatedly divided that huge number by 10, leaving a still huge quotient, with one digit less each time. This was, of course, even with Burnikel & Ziegler’s division algorithm, dead slow.
To improve things, I came up with a divideandconquer algorithm which divides the huge number by a power of ten that more or less splits it in half, and then recursively calls itself for the remainder and the quotient, so on each recursion, only half of the number has to be converted. Very small values are then converted using an optimized version of ToStringClassic
. This cut the time for ToString(10)
to form a 22,338,618 digit string down to approx. 150 seconds.
I really came up with this algorithm myself, although it turned out I was not the first. :(
For what it’s worth, I also modified ToString(Base)
for bases 2, 4 and 16 to take advantage of the fact that these only have to shift limbwise, in a simple nested loop, to get the digits. The same huge number was converted using ToString(16)
in – hold on, hold on… – 133ms, in other words, more than 1,000 times as fast as ToString(10)
. This shows that base conversions can be quite slow, and that it makes sense to treat different bases differently.
I intend to do something similar for the parser, but note that this is not a priority, at the moment.
Arithmetic operators and functions
The following functions and operators are defined for arithmetic with BigIntegers:
class operator Add( const Left, Right: BigInteger): BigInteger;  Adds two BigIntegers, operator + 
class function Add( const Left, Right: BigInteger): BigInteger; static;  Adds two BigIntegers 
class operator Subtract( const Left, Right: BigInteger): BigInteger;  Subtracts two BigIntegers, operator  
class function Subtract( const Left, Right: BigInteger): BigInteger; overload; static;  Subtracts two BigIntegers 
class operator Multiply( const Left, Right: BigInteger): BigInteger; overload; static;  Multiplies two BigIntegers, operator * 
class function Multiply( const Left, Right: BigInteger): BigInteger; overload; static;  Multiplies two BigIntegers. 
class function MultiplyBaseCase( const Left, Right: BigInteger): BigInteger; overload; static;  Multiplies two BigIntegers using base case or “schoolbook” multiplication. This may be called internally by Multiply and MultiplyKaratsuba . 
class function MultiplyKaratsuba( const Left, Right: BigInteger): BigInteger; overload; static;  Multiplies two BigIntegers using the Karatsuba algorithm. This may be called internally by Multiply and MultiplyToomCook3 . 
class function MultiplyToomCook3( const Left, Right: BigInteger): BigInteger; overload; static;  Multiplies two BigIntegers using the ToomCook 3way algorithm. This may be called internally by Multiply . 
class operator Multiply( const Left: BigInteger; Right: Word): BigInteger;  Multiplies a BigInteger with a Word , operator * 
class operator Multiply( Left: Word; const Right: BigInteger): BigInteger;  Multiplies a Word with a BigInteger , operator * 
class operator IntDivide( const Left, Right: BigInteger): BigInteger;  Divides two BigIntegers , operator div 
class function Divide( const Left, Right: BigInteger): BigInteger; static;  Divides two BigIntegers . 
class operator Modulus( const Left, Right: BigInteger): BigInteger;  Returns the remainder after the division of the given BigIntegers , operator mod 
class function Remainder( const Left, Right: BigInteger): BigInteger; static;  Returns the remainder after the divison of two BigIntegers 
class procedure DivMod( const Dividend, Divisor: BigInteger; var Quotient, Remainder: BigInteger); static;  Returns quotient and remainder after the integer division of Dividend by Divisor . They are the result of the same internal operation. Is called internally by Divide and Modulus . 
class function DivModBaseCase( const Left, Right: BigInteger): BigInteger; static;  Divides two BigIntegers using simple “schoolbook” division. My be called internally by DivMod . 
class function DivModBurnikelZiegler( const Left, Right: BigInteger): BigInteger; static;  Divides two BigIntegers using the BurnikelZiegler algorithm. May be called internally by DivMod . 
class operator Negative( const Int: BigInteger): BigInteger;  Returns the negation of a BigInteger , unary operator  
Bitwise operators
The following bitwise operators are defined for BigIntegers
. Note that these have two’s complement semantics, so before any bitwise operations are performed, the internal format is
converted to two’s complement. Aftwerward, the result is converted to signmagnitude again.
class operator BitwiseAnd( const Left, Right: BigInteger): BigInteger;  Returns the result of the bitwise AND operation.Operator and . 
class operator BitwiseOr( const Left, Right: BigInteger): BigInteger;  Returns the result of the bitwise OR operation.Operator or . 
class operator BitwiseXor( const Left, Right: BigInteger): BigInteger;  Returns the result of the bitwise XOR operation.Operator xor . 
class operator LogicalNot( const Int: BigInteger): BigInteger;  Returns the result of the bitwise NOT operation.Operator not . 
The following shift operators perform, unlike usual in Delphi, arithmetic shifts. This means that the sign bit is preserved, so negative numbers remain negative.  
class operator LeftShift( const Value: BigInteger; Shift: Integer): BigInteger;  Shifts the specified BigInteger value the specified number of bits to the left (away from 0).
The size of the BigInteger is adjusted accordingly.Operator shl . 
class operator RightShift( const Value: BigInteger; Shift: Integer): BigInteger;  Shifts the specified BigInteger value the specified number of bits to the right (toward 0).
The size of the BigInteger is adjusted accordingly. The sign is preserved, so 128 shr 8 does not end up as 0, but as 1 instead.Operator shr . 
Relational operators and function
The following relational operators and functiond are defined for BigIntegers.
class operator Equal( const Left, Right: BigInteger): Boolean;  operator = 
class operator NotEqual( const Left, Right: BigInteger): Boolean;  operator <> 
class operator GreaterThan( const Left, Right: BigInteger): Boolean;  operator > 
class operator GreaterThanOrEqual( const Left, Right: BigInteger): Boolean;  operator >= 
class operator LessThan( const Left, Right: BigInteger): Boolean;  operator < 
class operator LessThanOrEqual( const Left, Right: BigInteger): Boolean;  operator <= 
class function Compare( const Left, Right: BigInteger): Integer; static;  Returns 1 if Left < Right ; 1 if Left > Right ; 0 if Left = Right 
Conversion operators and functions
The following conversion functions perform a conversion to builtin Delphi types. If the value of the BigInteger
is too large, an EConvertError
is raised.
function AsDouble: Double;  Converts the BigInteger to a Double if that is possible. Raises an exception if not. 
function AsInteger: Integer;  Converts the BigInteger to an Integer if that is possible. Raises an exception if not. 
function AsCardinal: Cardinal;  Converts the BigInteger to a Cardinal if that is possible. Raises an exception if not. 
function AsInt64: Int64;  Converts the BigInteger to an Int64 if that is possible. Raises an exception if not. 
function AsUInt64: UInt64;  Converts the BigInteger to a UInt64 if that is possible. Raises an exception if not. 
The following implicit conversion operators (the conversions generally do not require a cast) perform a conversion from builtin Delphi types to a BigInteger
.
class operator Implicit( const Int: Integer): BigInteger;  Implicitly converts the specified Integer to a BigInteger . 
class operator Implicit( const Int: Cardinal): BigInteger;  Implicitly converts the specified Cardinal to a BigInteger . 
class operator Implicit( const Int: Int64): BigInteger;  Implicitly converts the specified Int64 to a BigInteger . 
class operator Implicit( const Int: UInt64): BigInteger;  Implicitly converts the specified UInt64 to a BigInteger . 
The following explicit conversion operators, (conversions requiring a cast) from BigInteger
to builtin Delphi types do not raise exceptions, but truncate or sign extend the values to make them fit, just like Delphi does.
class operator Explicit( const Int: BigInteger): Integer;  Explicitly converts the specified BigInteger to an Integer . 
class operator Explicit( const Int: BigInteger): Cardinal;  Explicitly converts the specified BigInteger to a Cardinal . 
class operator Explicit( const Int: BigInteger): Int64;  Explicitly converts the specified BigInteger to an Int64 . 
class operator Explicit( const Int: BigInteger): UInt64;  Explicitly converts the specified BigInteger to an UInt64 . 
class operator Explicit( const Int: BigInteger): Double;  Explicitly converts the specified BigInteger to a Double . 
class operator Explicit( const ADouble: Double): BigInteger;  Explicitly converts the specified Double to a BigInteger , using the constructor for that. More about that can be found under constructors, above. 
Conversion to bytes
The individual bytes in the array returned by the following method appear in littleendian order.
function ToByteArray: TArray<Byte>;  Converts a BigInteger value to a byte array. The array is in littleendian order, i.e. the least significant byte comes first. 
Negative values are written to the array using two’s complement representation in the most compact form possible. For example, 1 is represented as a single byte whose value is $FF instead of as an array with multiple elements, such as $FF, $FF or $FF, $FF, $FF, $FF.
Because two’s complement representation always interprets the highestorder bit of the last byte in
the array (the byte at position High(Array))
as the sign bit, the method returns a byte array with
an extra element whose value is zero to disambiguate positive values that could otherwise be interpreted
as having their sign bits set. For example, the value 120 or $78 is represented as a singlebyte array:
$78. However, 129, or $81, is represented as a twobyte array: $81, $00. Something similar applies to
negative values: 179 (or $B3) must be represented as $4D, $FF.
You can, inversely, convert such a byte array to a BigInteger
again, using the already mentioned:
constructor Create( const Bytes: array of Byte); overload;  Sets up a BigInteger using the bytes in Bytes as a two’s complement value. The opposite of the function ToByteArray . 
Mathematical methods
The following mathematical methods are implemented:
class function Abs( const Int: BigInteger): BigInteger; overload; static;  Returns the absolute value of a BigInteger 
function Abs: BigInteger; overload;  Returns the absolute value of the current BigInteger 
class function GreatestCommonDivisor( const Left, Right: BigInteger): BigInteger; static;  Returns the (positive) greatest common divisor of the specified BigInteger values. 
class function Ln( const Int: BigInteger): Double; overload; static;  Returns the natural logarithm of the value in Int. 
function Ln: Double; overload;  Returns the natural logarithm of the current value. 
class function Log( const Int: BigInteger; Base: Double): Double; overload; static;  Returns the logarithm to the specified base of the value in Int. 
function Log(Base: Double): Double; overload;  Returns the logarithm to the specified base of the current value. 
class function Log2( const Int: BigInteger): Double; overload; static;  Returns the logarithm to base 2 of the value in Int. 
function Log2: Double; overload;  Returns the logarithm to base 2 of the current value. 
class function Log10( const Int: BigInteger): Double; overload; static;  Returns the logarithm to base 10 of the value in Int. 
function Log10( const Int: BigInteger): Double; overload;  Returns the logarithm to base 10 of the current value. 
class function Max( const Left, Right: BigInteger): BigInteger; static;  Returns the greater of two specified values. 
class function Min( const Left, Right: BigInteger): BigInteger; static;  Returns the lesser of two specified values. 
class function ModPow( const Value: BigInteger; Exponent: Integer; const Modulus: BigInteger): BigInteger; static;  Returns the specified modulus value of the specified value raised to the specified power. 
class function Pow( const Value: BigInteger; Exponent: Integer): BigInteger; static;  Returns the specified value raised to the specified power. 
class function ModInverse( const Value, Modulus: BigInteger): BigInteger; static;  Returns the modular inverse of Value mod Modulus. 
Returns an EInvalidArgument exception if there is no modular inverse. 

class function NthRoot(const Radicand: BigInteger; Nth: Integer): BigInteger; static;  Returns the nth root of the radicand. This means that, for instance, with Nth = 3 , it returns the cube root. With Nth = 2 , it returns the same as Sqrt . 
class procedure NthRootRemainder( const Radicand: BigInteger; Nth: Integer; var Root, Remainder: BigInteger); static;  Puts the nth root of the radicand in Root and the remainder in Remainder . 
class function Sqrt( const Radicand: BigInteger): BigInteger; static;  Returns the square root of the radicand. 
class procedure SqrtRemainder( const Radicand: BigInteger; var Root, Remainder: BigInteger); static;  Puts the square root of the radicand in Root and the remainder in Remainder . 
class function Sqr( const Value: BigInteger): BigInteger; static;  Returns the square of the given BigInteger , i.e. Value * Value . 
Bit fiddling
The following methods to do bit fiddling are implemented. They follow two’s complement semantics, so a value of, say, $1234000
is regarded as $EDCC0000
and bits are accessed accordingly. You can access bits past the current size of the BigInteger
. The BigInteger
is expanded if that is necessary to contain the new value.
For what it’s worth: for negative values, these methods can be slow (because of the need to turn the internal signmagnitude format into two’s complement) and thus are not fit for the implementation of bit sets and the like.
function TestBit(Index: Integer): Boolean;  Returns True if the bit at the given position is set. 
function SetBit(Index: Integer): BigInteger;  Returns a new BigInteger with the given bit set. 
function ClearBit(Index: Integer): BigInteger;  Returns a new BigInteger with the given bit cleared. 
function FlipBit(Index: Integer): BigInteger;  Returns a new BigInteger with the given bit flipped. 
Note that the above does not mean that the size of a BigInteger
is always expanded if you access a bit past its size. Say, you have a BigInteger
like this:
var
Q1, Q2: BigInteger;
begin
Q1 := '$12';
The two’s complement value of that is $FFFFFFEE
, or $FFFFFFFFFFFFFFEE
or even more $F
‘s in front, because the value is negative. In other words, in two’s complement, any bit beyond the most significant bit of a negative value is regarded as being set (and likewise, any bit beyond the most significant bit of a positive value or 0 is regarded as being clear). So doing:
Q2 := Q1.SetBit(100000);
does not return a different value, because, in two’s complement, bit 100000 is regarded as being set anyway. The result is still $12
, so in this case, SetBit
simply returns Self
.
Note that FlipBit
will always return a different value.
The following functions also provide information about the bits of a BigInteger
:
function BitLength: Integer;  Returns the bit length, the minimum number of bits needed to represent the value, excluding the sign bit. At the same time, this is the index of the highest set bit. 
function BitCount: Integer;  Returns the number of set bits (cardinality) in the value. 
function LowestSetBit: Integer;  Returns the bit index of the lowest bit that is set, which is, at the same time, the number of trailing zero bits. 
Properties
The following properties are defined by a BigInteger
. Apart from the last one, Base
, they are all instance properties describing the BigInteger
.
property Size: Integer read;  The number of limbs used for the magnitude. 
property Allocated: Integer read;  The number of limbs actually allocated. If the RESETSIZE conditional is defined, the internal Compact procedure will occasionally shrink an allocated array that is too large. That can affect performance, so by default, allocations are never shrunk. Of course, if a BigInteger goes out of scope, the allocated array will be released completely. 
property Negative: Boolean read;  True@ if the BigInteger is negative. 
property Sign: Integer read write;  Either $80000000 if the BigInteger is negative, or $00000000 if it is positive. 
property Magnitude: TMagnitude read;  The array of unsigned 32bit integers containing the magnitude of the BigInteger . 
class property Base: TNumberBase read write;  The default numeric base to be used for input and output. 
Conditional defines
The following conditional defines can be used. Note that some are off by default. This is reflected in the source file, e.g. the conditional PUREPASCAL
is easily turned off by turning
{$DEFINE PUREPASCAL}
into a mere comment:
{ $DEFINE PUREPASCAL}
PUREPASCAL 
If this conditional is set, no assembler is used. 
RESETSIZE 
If this conditional is set, the Compact internal routine shrinks the internal buffer if the required size of the BigInteger gets smaller than half of the allocated size. This can reduce memory usage, bit is can also make code a little slower. By default, this is off, and buffers are never shrunk. 
Experimental 
This conditional is can be used to shield off working code to do experiments. See text box below. 
During development, I use the Experimental
conditional to shield off existing and working (but perhaps too slow) code from experimental code, by doing something like this:
{$IFDEF Experimental} // ... the new, experimental code {$ELSE !Experimental} // ... the working original code {$ENDIF !Experimental}
With this setup, it is easy to switch between working, original code and new code, e.g. to debug the values the new code should produce (i.e. the values the original working code already produces), just by turning Experimental off or on.
Internal format
The internal format of a BigInteger
is quite simple. It consists of two private data members (fields). They should only be manipulated indirectly, using the various methods and operators defined for the BigInteger
. Of course you can access them using a trick, but it is not recommended.
Field  Type  Function  

FData 
TArray<TLimb> 
Magnitude  The magnitude of the BigInteger 
FSize 
Integer 
Size and sign bit  The lower 31 bits contain the number of limbs the magnitude has, the top bit contains the sign bit (if the bit is set, the BigInteger is negative) 
Note that FSize
is not two’s complement either. The lower bits are unsigned, and the sign bit has no relation to them. So negating a BigInteger
is as simple as flipping the top bit of FSize
.
In the above, the term “limb” is used a few times. A TLimb
is one of the elements of the array that forms the magnitude of the BigInteger
. It is declared as:
type
PLimb = ^TLimb;
TLimb = type UInt32;
Even in 64bit Windows, I decided to use the same 32bit limbs. This makes some 64bit code a lot easier, and probably not slower.
Speed
For Windows, most of the time critical routines are implemented in 32bit or 64bit builtin assembler. For other platforms, or if the PUREPASCAL
conditional is defined, they are completely implemented in plain Object Pascal. I optimized the routines as much as I could, sometimes with the help of the good people on StackOverflow.com (see sources).
Optimizations
Many of the routines use unrolled loops and other smallscale optimizations, not only in assembler, but also in PUREPASCAL
. Most of the 64 bit assembler processes 64 bit (i.e. 2 limbs) at once. If this made sense, 64 bit variables were used. Just read the source code and see for yourself.
After a lot of reading and testing I was able to implement higher level, divideandconquer algorithms for faster operations (see, for example, this pdf and this website as well). As the Wikipedia article says: A divide and conquer algorithm works by recursively breaking down a problem into two or more subproblems of the same (or related) type (divide), until these become simple enough to be solved directly (conquer).
Karatsuba and ToomCook 3way
For faster multiplication, recursive algorithms described by Anatolii Alexeevitch Karatsuba and by Andrei Toom and Stephen Cook (the 3way algorithm) were implemented.
These are much more complicated than simple base case multiplication, but also asymptotically faster. However, due to the overhead in these functions, they are only faster if the sizes of the BigIntegers
are above certain thresholds. That is why multiplication only uses Karatsuba when both sizes are above BigInteger.KaratsubaThreshold
and it uses ToomCook 3way when the size of one of both operands is above BigInteger.ToomCook3Threshold
. These thresholds differ for 32bit and 64bit, as well as for PUREPASCAL
or assemblerbased code.
Usually, the Multiply()
function will decide which algorithm is used, but I also made functions MultiplyBaseCase
(normal “schoolbook” multiplication), MultiplyKaratsuba
and MultiplyToomCook3
public, if you have a special reason to use one of these. Note that MultiplyToomCook
falls back to Karatsuba, and MultiplyKaratsuba
falls back to base case if the operands do not meet the threshold criteria.
Because of the recursion, both algorithms are much faster than normal “schoolbook” multiplication. I measure a speed increase of almost 10 for BigIntegers
with more than 6,400 limbs (
There are faster, even more complicated algorithms, e.g. the FFTbased SchönhageStrassen algorithm for very large BigIntegers
. I did not implement any of those yet.
BurnikelZiegler
For faster division (and modulus), I implemented the divideandconquer algorithm described by Christoph Burnikel and Joachim Ziegler. The speed of this is more or less limited by the speed of multiplication. It mainly consists of two routines, DivTwoDigitsByOne
and DivThreeHalvesByTwo
, which call each other recursively until sizes get below a certain threshold. Then normal base case (“Knuth”) division is used. The paper linked to explains the algorithm quite nicely, without much ado. It is quite a bit faster than normal “Knuth” division, especially for very large values.
There are faster algorithms, e.g. the Barrett algorithm or Montgomery reduction for extremely large BigIntegers
, but I did not implement any of those yet.
It took me a while to wrap my head around stuff like Karatsuba (OK, that one was easy), ToomCook, BurnikelZiegler, MillerRabin, NewtonRaphson, SchönhageStrassen, Barrett, Montgomery, etc. I have done a lot of reading, and while doing that, I learned a lot of other things as well.
Allocation
As described above, if the RESETSIZE
conditional is defined, the internal Compact
procedure will shrink the size of the allocated array if Size
becomes less than half the allocated size (Length
) of the FData
array. Such a reallocation can negatively affect performance, so by default, the conditional is not defined, and FData
is never shrunk. But this does not mean that eventually, you will get an enormous amount of memory leaks. FData
is a normal dynamic array, so if the last reference to it is lost, it will be freed entirely.
Copyonwrite (COW)
The methods and operators of BigInteger
follow the COW principle. This means that if you just assign one BigInteger
to another, only a shallow copy is made. The FData
pointer is copied and its reference count is updated, but the entire contents of the FData
are not copied. Because of this, several BigIntegers
with the same value can share one FData
array. Only if one of these BigIntegers
is about to be modified, a new copy of the data for that BigInteger
is made, while the others can still reference the original FData
.
An example:
var
A, B, C, D: BigInteger;
begin
A := 17; // A gets new FData array with contents 17
B := A; // B.FData points to same array as A.FData
C := B; // C.FData points to same array as B.FData and A.FData
B := A + 1; // B gets new array, but A and C still reference array with value 17.
...
Note that, despite of the above, most of the time, BigIntegers
should and can be treated as immutable, i.e. most methods or operators operating on BigIntegers
return a new one, instead of changing the value of one of the operands. Even a function like SetBit
does not modify the BigInteger
on which it is called. It returns a new one with the given bit set. Only some selfreferential instance methods modify the value of Self
and Inc
and Dec
modify the value of the operand.
Partialflags stall problems
This is a problem that caused me some headaches before I could solve it. I noticed that on some CPUs, against all expectations, the PUREPASCAL
routines were faster than the assembler routines. This turned out to be due to a partialflags stall.
In assembler, on some “older” CPUs, certain loop code combining the opcodes INC
and/or DEC
with ADC
and/or SBB
or other instructions reading the carry flag, can cause a considerate slowdown (up to 3 times as slow), because of a so called partialflags stall, which happens when instructions like INC
or DEC
write only some of the flags, while other instructions (e.g. ADC
and SBB
) read other flags.
I coded routines (using LEA
and JECXZ/JRCXZ
instead of INC/DEC/JNE
) that avoid that problem, but on CPUs that do not have the problem, these modified routines are a bit slower than the “plain” routines. So now, at startup, the unit determines, with a few timing loops, which of the routines it should use: the plain ones or the modified ones. So, on CPUs that are not affected, plain code using INC/DEC/JNE
is used. That is faster than the modified procedures. But on older, affected CPUs, modified code using LEA/JECXZ
is used. On these CPUs, the modified code is much faster than the plain code. At the moment, only code for addition and subtration is different. The problem does not occur in other routines, probably because in these routines another instruction (e.g. ADD
or CMP
) will set the full flags register before it is read.
But every now and then, due to unexpected events happening that mess up the timing, the decision taken by the unit can be wrong, or you may have other reasons to choose one implementation over the other. For that, you can use:
BigInteger.AvoidPartialFlagsStall(True); // True to make BigInteger use modified routines,
// False to make it use plain ones.
// Omit routine to keep the setting the unit
// measured.
Whether plain or modified routines are used can be queried from the property:
property StallAvoided: Boolean;  Returns True if modified routines are used, or False for plain routines. 
If you know a better way to determine the affected CPUs than the simple timing loops I am using, please send me a note.
Bitwise operators and negative values
As I explained already, bitwise operators like and
, or
and xor
use two’s complement semantics. This means that before the operations on the values are performed, they are converted from the usual signmagnitude to two’s complement. For positive values, that means that nothing has to be done, but negative values must be negated completely. I found ways to get around that, partially. People on StackOverflow helped me, as well as the many bitwise tricks mentioned in Hacker’s Delight.
Visualizer
All the time when working, to find the numeric value of a BigInteger
, let’s call it X
, in the debugger, I had to either add a watch entry with X.ToString
, or I had to run a little piece of code. So I decided to write a debugger visualizer for it. Here you can see it in action:
Here value A
, which was initialized with the string above it, is displayed in the tooltip.
I had hoped it would be easy, but it turned out that the IDE only passes you a string like
'(FData:(4294967295 {$FFFFFFFF}, 4294967295 {$FFFFFFFF}, 4294967295 {$FFFFFFFF}, 0, 0, 0, 0, 0); FSize:3)'
or, in the Local Variables pane, a simplified version (for the same BigInteger
) like:
'((4294967295, 4294967295, 4294967295, 0, 0, 0, 0, 0), 3)'
or a mix of these styles, which you then have to parse to get the data, turn them into an array and a sign, use these to create a BigInteger
and then apply ToString
to get the replacement string. It took me some debugging and quite a few IDE restarts to get the parsing straight for the different formats. Perhaps there is a better way, i.e. one that gives you access to the real data, but I haven’t found it yet.
The visualizer package currently comes in source format, which you only have to install in the IDE (click Install in the context menu in the Project Manager). Make sure that the BigNumbers.dpk
package was built first and that the visualizer package references its .dcp
file. I intend to write an installer for this, but please don’t hurry me, as this is not something I have done before.
Currently, the code on GitHub represents a DLL expert, which must be added to the known experts in the registry. It is very easy to make this a packagebased expert instead, though.
FreePascal
In {$mode delphi}
, it is possible to compile the units. FreePascal doesn’t seem to use segmented unit names (yet), so you’ll have to rename some of the used units and qualifiers like System.Math
or System.SysUtils
to simply Math
or SysUtils
. At first, you’ll have to {$DEFINE PUREPASCAL}
, because making the assembler work satisfactorily could be quite a job. It seems to work alright, if you set {$asmmode intel}
, but it is a little finicky. Stuff like LEA EAX,[EDX + EDI*ClimbSize]
seems to cause problems, but [EDX + CLimbSize*EDI]
works. That is a bit of a problem for me, because I most of the time use the former variety. But I guess that, with enough time on your hands, you can make it work.
Some of the tricky routines, e.g. ToString
, which uses RecursiveToString
, which in its turn needs an exact alignment of bytes to get a usable string, don’t work, but in the meantime — until you solved the problem — you can e.g. use (or wrap or rename) .ToStringClassic(10)
instead. I guess the tricky routines just need some work and some debugging to make them work under FreePascal. I only tried a quick and dirty test, did not do any extensive tests, but it is obviously possible to make it work. Just don’t give up immediately.
Types like TArray<Bla>
don’t seem to work. And there is no way you can define an array of BigInteger
before BigInteger
is defined, so you will have to define a type BigIntegerArray = array of BigInteger
inside the BigInteger
record, as a nested type. You will meet other, similar problems. Another was the fact that there is no DivMod()
function for UInt64
types, so you will have to either use the one for Longint
or use div
and mod
separately. It could be that the change I made there is one of the causes for the failure of RecursiveToString
. This needs some investigation.
There will be other, simple problems and it is well possible that there are a few notsosimple problems. But it looks as if BigIntegers
can be used in FreePascal.
Good luck!
Conclusion
I hope this code is useful to you. If you use some of it, please credit me. If you modify or improve the unit, please send me the modifications at this email address.
I may improve or enhance the unit myself, and I will try to post changes here. But this is not a promise. Please don’t request features.
Rudy Velthuis
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Disclaimer and Copyright
The coding examples presented here are for illustration purposes only. The author takes no responsibility for enduser use. All content herein is copyrighted by Rudy Velthuis, and may not be reproduced in any form without the author's permission. Source code written by Rudy Velthuis presented as download is subject to the license in the files.