## BigInt

### 1.13.0

Arbitrary-precision integer arithmetic in Swift

### What's New

###### 2023-05-22T08:12:58Z

New in release 1.13.0:

1. A new BFraction constructor from a string representation

public init?(_ x: String)

for example

``````BFraction("3.1415") // = 6283 / 200
BFraction("123E-3") // = 123 / 1000
BFraction("abc") // = nil
``````
1. A new BFraction 'mod' method to compute the value modulo an integer

public func mod(_ m: BInt) -> BInt?
public func mod(_ m: Int) -> Int?

for example

BFraction(13, 3).mod(5) // = Optional(1) because 3^(-1) mod 5 = 2 and (13 * 2).mod(5) = 1

Returns nil if the denominator and modulus are not coprime

2. The BFraction method 'asDecimalString' has a new API

public func asDecimalString(precision: Int, exponential: Bool = false) -> String

where precision is the number of significant digits
and exponential determines whether to use exponential or plain notation. For example

BFraction(712, 11001).asDecimalString(precision: 5, exponential: false) // = "0.064721"
BFraction(712, 11001).asDecimalString(precision: 5, exponential: true ) // = "6.4721E-2"

3. A new static BFraction method 'bernoulliSequence'

public static func bernoulliSequence(_ n: Int) -> [BFraction]

BFraction.bernoulliSequence(n) computes the n even indexed Bernoulli numbers B(0), B(2) ... B(2 * n - 2)
This is much faster than computing the same numbers individually.

## BigInt

### Contents:

The BigInt package provides arbitrary-precision integer arithmetic in Swift. Its functionality falls in the following categories:
• Arithmetic: add, subtract, multiply, divide, remainder and exponentiation
• Comparison: the six standard operations == != < <= > >=
• Shifting: logical left shift and rigth shift
• Logical: bitwise and, or, xor, and not
• Modulo: normal modulus, inverse modulus, and modular exponentiation
• Conversion: to double, to integer, to string, to magnitude byte array, and to 2's complement byte array
• Primes: prime number testing, probable prime number generation and primorial
• Miscellaneous: random number generation, greatest common divisor, least common multiple, n-th root, square root modulo an odd prime, Jacobi symbol, Kronecker symbol, Factorial function, Binomial function, Fibonacci numbers, Lucas numbers and Bernoulli numbers
• Fractions: Standard arithmetic on fractions whose numerators and denominators are of unbounded size

BigInt requires Swift 5.0. It also requires that the Int and UInt types be 64 bit types.

## Usage

``````  dependencies: [
.package(url: "https://github.com/leif-ibsen/BigInt", from: "1.13.0"),
]
``````

## Examples

### Creating BInt's

``````  // From an integer
let a = BInt(27)

// From a decimal value
let x = BInt(1.12345e30) // x = 1123450000000000064996914495488

// From string literals
let b = BInt("123456789012345678901234567890")!
let c = BInt("1234567890abcdef1234567890abcdef", radix: 16)!

// From magnitude and sign
let m: Limbs = [1, 2, 3]
let d = BInt(m) // d = 1020847100762815390427017310442723737601
let e = BInt(m, true) // e = -1020847100762815390427017310442723737601

// From a big-endian 2's complement byte array
let f = BInt(signed: [255, 255, 127]) // f = -129

// From a big-endian magnitude byte array
let g = BInt(magnitude: [255, 255, 127]) // g = 16777087

// Random value with specified bitwidth
let h = BInt(bitWidth: 43) // For example h = 3965245974702 (=0b111001101100111011000100111110100010101110)

// Random value less than a given value
let i = h.randomLessThan() // For example i = 583464003284
``````

### Converting BInt's

``````  let x = BInt(16777087)

// To double
let d = x.asDouble() // d = 16777087.0

// To strings
let s1 = x.asString() // s1 = "16777087"
let s2 = x.asString(radix: 16) // s2 = "ffff7f"

// To big-endian magnitude byte array
let b1 = x.asMagnitudeBytes() // b1 = [255, 255, 127]

// To big-endian 2's complement byte array
let b2 = x.asSignedBytes() // b2 = [0, 255, 255, 127]
``````

## Performance

To assess the performance of BigInt, the execution times for a number of common operations were measured on an iMac 2021, Apple M1 chip. The results are in the table below. It shows the operation being measured and the time it took (in microseconds or milliseconds).

Four large numbers 'a1000', 'b1000', 'c2000' and 'p1000' were used throughout the measurements. Their actual values are shown under the table.

OperationSwift codeTime
As stringc2000.asString()13 uSec
As signed bytesc2000.asSignedBytes()0.30 uSec
Bitwise anda1000 & b10000.083 uSec
Bitwise ora1000 | b10000.083 uSec
Bitwise xora1000 ^ b10000.082 uSec
Bitwise not~c20000.087 uSec
Test bitc2000.testBit(701)0.017 uSec
Flip bitc2000.flipBit(701)0.018 uSec
Set bitc2000.setBit(701)0.018 uSec
Clear bitc2000.clearBit(701)0.018 uSec
Subtractiona1000 - b10000.08 uSec
Multiplicationa1000 * b10000.32 uSec
Divisionc2000 / a10002.2 uSec
Modulusc2000.mod(a1000)2.2 uSec
Inverse modulusc2000.modInverse(p1000)83 uSec
Modular exponentiationa1000.expMod(b1000, c2000)3.5 mSec
Equalc2000 + 1 == c20000.017 uSec
Less thanb1000 + 1 < b10000.021 uSec
Shift 1 leftc2000 <<= 10.05 uSec
Shift 1 rightc2000 >>= 10.06 uSec
Shift 100 leftc2000 <<= 1000.14 uSec
Shift 100 rightc2000 >>= 1000.11 uSec
Is probably primep1000.isProbablyPrime()5.8 mSec
Make probable 1000 bit primeBInt.probablePrime(1000)60 mSec
Next probable primec2000.nextPrime()730 mSec
PrimorialBInt.primorial(100000)8.5 mSec
BinomialBInt.binomial(100000, 10000)22 mSec
FactorialBInt.factorial(100000)57 mSec
FibonacciBInt.fibonacci(100000)0.22 mSec
Greatest common divisora1000.gcd(b1000)29 uSec
Extended gcda1000.gcdExtended(b1000)81 uSec
Least common multiplea1000.lcm(b1000)32 uSec
Make random numberc2000.randomLessThan()1.2 uSec
Squarec2000 ** 20.68 uSec
Square rootc2000.sqrt()13 uSec
Square root and remainderc2000.sqrtRemainder()13 uSec
Is perfect square(c2000 * c2000).isPerfectSquare()16 uSec
Square root modulob1000.sqrtMod(p1000)1.6 mSec
Powerc2000 ** 1111.9 mSec
Rootc2000.root(111)15 uSec
Root and remainderc2000.rootRemainder(111)17 uSec
Is perfect rootc2000.isPerfectRoot()13 mSec
Jacobi symbolc2000.jacobiSymbol(p1000)0.15 mSec
Kronecker symbolc2000.kroneckerSymbol(p1000)0.15 mSec
Bernoulli numberBFraction.bernoulli(1000)83 mSec

a1000 = 3187705437890850041662973758105262878153514794996698172406519277876060364087986868049465132757493318066301987043192958841748826350731448419937544810921786918975580180410200630645469411588934094075222404396990984350815153163569041641732160380739556436955287671287935796642478260435292021117614349253825
b1000 = 9159373012373110951130589007821321098436345855865428979299172149373720601254669552044211236974571462005332583657082428026625366060511329189733296464187785766230514564038057370938741745651937465362625449921195096442684523511715110908407508139315000469851121118117438147266381183636498494901233452870695
c2000 = 1190583332681083129323588684910845359379915367459759242106618067261956856381281184752008892106576666833853411939711280970145570546868549934865719229538926506588929417873149597614787608112658086250354719939407543740242931571462165384138560315454455247539461818779966171917173966217706187439870264672508450210272481951994459523586160979759782950984370978171111340529321052541588344733968902238813379990628157732181128074253104347868153860527298911917508606081710893794973605227829729403843750412766366804402629686458092685235454222856584200220355212623917637542398554907364450159627359316156463617143173
p1000 (probably a prime) = 7662841304438384296568220077355872003841475576593385710590818274399706072141018649398767137142090308734613594718593893634649122767374115742644499040193270857876678047220373151142747088797516044505739487695946446362769947024029728822155570722524629197074319602110260674029276185098937139753025851896997

## Fractions

Fractions are represented as BFraction values consisting of a numerator BInt value and a denominator BInt value. The representation is normalized:
• The numerator and denominator have no common factors except 1
• The denominator is always positive
• Zero is represented as 0/1

### Creating BFraction's

Fractions are created by
• Specifying the numerator and denominator explicitly f.ex. BFraction(17, 4)
• Specifying the decimal value explictly f.ex. BFraction(4.25)
• Using a string representation f.ex. BFraction("4.25")! or equivalently BFraction("425E-2")!
Defining a fraction by giving its decimal value (like 4.25) might lead to surprises, because not all decimal values can be represented exactly as a floating point number. For example, one might think that BFraction(0.1) would equal 1/10, but in fact it equals 3602879701896397 / 36028797018963968 = 0.1000000000000000055511151231257827021181583404541015625

### Converting BFraction's

BFraction values can be converted to String values, to decimal String values and to Double values.
``````  let x = BFraction(1000, 7)

// To String
let s1 = x.asString() // s1 = "1000 / 7"

// To decimal String
let s1 = x.asDecimalString(precision: 8, exponential: false) // s1 = "142.85714"
let s2 = x.asDecimalString(precision: 8, exponential: true) // s2 = "1.4285714E+2"

// To Double
let d = x.asDouble() // d = 142.8571428571429
``````

### Operations

The operations available to fractions are:
• Arithmetic
• subtraction
• multiplication
• division
• modulo an integer
• exponentiation
• Rounding to an integer
• round - to nearest integer
• truncate - towards 0
• ceil - towards +infinity
• floor - towards -infinity
• Comparison - the six standard operations == != < <= > >=

### Bernoulli Numbers

The static function
``````let bn = BFraction.bernoulli(n)
``````

computes the n'th (n >= 0) Bernoulli number, which is a rational number.
For example

``````print(BFraction.bernoulli(60))
print(BFraction.bernoulli(60).asDecimalString(precision: 20, exponential: true))
``````

would print

``````-1215233140483755572040304994079820246041491 / 56786730
-2.1399949257225333665E+34
``````

The static function

``````let x = BFraction.bernoulliSequence(n)
``````

computes the n even numbered Bernoulli numbers B(0), B(2) ... B(2 * n - 2).

## Chinese Remainder Theorem

The CRT structure implements the Chinese Remainder Theorem. Construct a CRT instance from a given set of moduli, and then use the *compute* method to compute the CRT value for a given set of residues. The same instance can be reused for any set of input data, as long as the moduli are the same. This is relevant because it takes longer time to create the CRT instance than to compute the CRT value.
For example
``````let moduli = [3, 5, 7]
let residues = [2, 2, 6]
let crt = CRT(moduli)!
print("CRT value:", crt.compute(residues))
``````

would print

``````CRT value: 62
``````

## References

Algorithms from the following books and papers have been used in the implementation. There are references in the source code where appropriate.

• [BRENT] - Brent and Zimmermann: Modern Computer Arithmetic, 2010
• [BURNIKEL] - Burnikel and Ziegler: Fast Recursive Division, October 1998
• [CRANDALL] - Crandall and Pomerance: Prime Numbers - A Computational Perspective. Second Edition, Springer 2005
• [GRANLUND] - Moller and Granlund: Improved Division by Invariant Integers, 2011
• [HACKER] - Henry S. Warren, Jr.: Hacker's Delight. Second Edition, Addison-Wesley
• [HANDBOOK] - Menezes, Oorschot, Vanstone: Handbook of Applied Cryptography. CRC Press 1996
• [JEBELEAN] - Tudor Jebelean: An Algorithm for Exact Division. Journal of Symbolic Computation, volume 15, 1993
• [KNUTH] - Donald E. Knuth: Seminumerical Algorithms, Third Edition
• [KOC] - Cetin Kaya Koc: A New Algorithm for Inversion mod p^k

## Algorithms

Some of the algorithms used in BigInt are described below.

### Multiplication

• Schonhage-Strassen FFT based algorithm for numbers above 384000 bits
• ToomCook-3 algorithm for numbers above 12800 bits
• Karatsuba algorithm for numbers above 6400 bits
• Basecase - Knuth algorithm M

### Division and Remainder

• Burnikel-Ziegler algorithm for divisors above 3840 bits provided the dividend has at least 3840 bits more than the divisor
• Basecase - Knuth algorithm D
• Exact Division - Jebelean's exact division algorithm

### Greatest Common Divisor and Extended Greatest Common Divisor

Lehmer's algorithm [KNUTH] chapter 4.5.2, with binary GCD basecase.

### Modular Exponentiation

Sliding window algorithm 14.85 from [HANDBOOK] using Barrett reduction for exponents with fewer than 2048 bits and Montgomery reduction for larger exponents.

### Inverse Modulus

If the modulus is a (not too large) power of 2, the algorithm from [KOC] section 7. Else, it is computed via the extended GCD algorithm.

### Square Root

Algorithm 1.12 (SqrtRem) from [BRENT] with algorithm 9.2.11 from [CRANDALL] as basecase.

### Square Root Modulo a Prime Number

Algorithm 2.3.8 from [CRANDALL].

### Prime Number Test

Miller-Rabin test.

### Prime Number Generation

The algorithm from Java BigInteger translated to Swift.

### Factorial

The 'SplitRecursive' algorithm from Peter Luschny: https://www.luschny.de

### Fibonacci

The 'fastDoubling' algorithm from Project Nayuki: https://www.nayuki.io

### Jacobi and Kronecker symbols

Algorithm 2.3.5 from [CRANDALL].

### Bernoulli Numbers

Computed via Tangent numbers which is fast because it only involves integer arithmetic but no fractional arithmetic.

### Chinese Remainder Theorem

The Garner algorithm 2.1.7 from [CRANDALL].

### Description

• Swift Tools 5.7.0

### Dependencies

• None
Last updated: Wed May 24 2023 05:13:30 GMT-0500 (GMT-05:00)