BigInt

The BigInt package provides arbitrary-precision integer arithmetic in Swift. Its functionality falls in the following categories:

• Performance:
  • Division 2x AttaSwift
  • Multiplication 7x AttaSwift
  • Logic functions 2x AttaSwift
  • Convert to String 3x AttaSwift
  • Shifts up to 16x AttaSwift
• 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

• Added SignedInteger, BinaryInteger, and Numeric protocol compliance.
• Optional support for StaticBigInt to allow BigInt number initialization from very large integer literals. Uncomment the BigInt-Extensions StaticBigInt support.

Why support protocols? By supporting them you have the ability to formulate generic algorithms and make use of algorithms from others that use the protocol type(s) you support. For example, Strideable compliance is free (with BinaryInteger) and lets you do things like

for i in BInt(1)...10 {
   print(i.words)
}

In addition, if a third party defines extensions for the supported protocols, these can be easily adapted by just conforming to that protocol.

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

In your projects Package.swift file add a dependency like

dependencies: [
    .package(url: "https://github.com/mgriebling/BigInt.git", from: "2.0.0"),
]

  // 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

  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]

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.

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

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

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

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

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 == != < <= > >=

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).

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 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

Some of the algorithms used in BigInt are described below.

• 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

• 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

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

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

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.

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

Algorithm 2.3.8 from [CRANDALL].

Miller-Rabin test.

The algorithm from Java BigInteger translated to Swift.

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

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

Algorithm 2.3.5 from [CRANDALL].

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

The Garner algorithm 2.1.7 from [CRANDALL].

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

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