Swiftpack.co - leif-ibsen/BigInt as Swift Package

BigInt

Contents:

• Usage
• Examples
• Performance
• Fractions
• Chinese Remainder Theorem
• References
• Algorithms

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

In your projects Package.swift file add a dependency like

  dependencies: [
  .package(url: "https://github.com/leif-ibsen/BigInt", from: "1.14.0"),
  ]

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.

Operation	Swift code	Time
As string	c2000.asString()	13 uSec
As signed bytes	c2000.asSignedBytes()	0.30 uSec
Bitwise and	a1000 & b1000	0.083 uSec
Bitwise or	a1000 | b1000	0.083 uSec
Bitwise xor	a1000 ^ b1000	0.082 uSec
Bitwise not	~c2000	0.087 uSec
Test bit	c2000.testBit(701)	0.017 uSec
Flip bit	c2000.flipBit(701)	0.018 uSec
Set bit	c2000.setBit(701)	0.018 uSec
Clear bit	c2000.clearBit(701)	0.018 uSec
Addition	a1000 + b1000	0.07 uSec
Subtraction	a1000 - b1000	0.08 uSec
Multiplication	a1000 * b1000	0.32 uSec
Division	c2000 / a1000	2.2 uSec
Modulus	c2000.mod(a1000)	2.2 uSec
Inverse modulus	c2000.modInverse(p1000)	83 uSec
Modular exponentiation	a1000.expMod(b1000, c2000)	3.5 mSec
Equal	c2000 + 1 == c2000	0.017 uSec
Less than	b1000 + 1 < b1000	0.021 uSec
Shift 1 left	c2000 <<= 1	0.05 uSec
Shift 1 right	c2000 >>= 1	0.06 uSec
Shift 100 left	c2000 <<= 100	0.14 uSec
Shift 100 right	c2000 >>= 100	0.11 uSec
Is probably prime	p1000.isProbablyPrime()	5.8 mSec
Make probable 1000 bit prime	BInt.probablePrime(1000)	60 mSec
Next probable prime	c2000.nextPrime()	730 mSec
Primorial	BInt.primorial(100000)	8.5 mSec
Binomial	BInt.binomial(100000, 10000)	22 mSec
Factorial	BInt.factorial(100000)	57 mSec
Fibonacci	BInt.fibonacci(100000)	0.22 mSec
Greatest common divisor	a1000.gcd(b1000)	29 uSec
Extended gcd	a1000.gcdExtended(b1000)	81 uSec
Least common multiple	a1000.lcm(b1000)	32 uSec
Make random number	c2000.randomLessThan()	1.2 uSec
Square	c2000 ** 2	0.68 uSec
Square root	c2000.sqrt()	13 uSec
Square root and remainder	c2000.sqrtRemainder()	13 uSec
Is perfect square	(c2000 * c2000).isPerfectSquare()	16 uSec
Square root modulo	b1000.sqrtMod(p1000)	1.6 mSec
Power	c2000 ** 111	1.9 mSec
Root	c2000.root(111)	15 uSec
Root and remainder	c2000.rootRemainder(111)	17 uSec
Is perfect root	c2000.isPerfectRoot()	13 mSec
Jacobi symbol	c2000.jacobiSymbol(p1000)	0.15 mSec
Kronecker symbol	c2000.kroneckerSymbol(p1000)	0.15 mSec
Bernoulli number	BFraction.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")!
• Specifying a continued fraction like BFraction([3, 4, 12, 4]) equivalent to 649 / 200

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, to Double values and to Continued Fraction 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
  
  // To Continued Fraction
  let f = x.asContinuedFraction() // f = [BInt(142), BInt(1), BInt(6)]

Operations

The operations available to fractions are:

• Arithmetic
  • addition
  • 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).

Harmonic Numbers

The static function

let Hn = BFraction.harmonic(n)

computes the n'th harmonic number, that is, 1 + 1/2 + ... + 1/n
The static function

let x = BFraction.harmonicSequence(n)

returns an array containing the first n harmonic numbers.

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].