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leif-ibsen/BigInt 1.2.8
Arbitrary-precision integer arithmetic in Swift
⭐️ 1
🕓 1 week ago
.package(url: "https://github.com/leif-ibsen/BigInt.git", from: "1.2.8")

Description

The BigInt package provides arbitrary-precision integer arithmetic in Swift. Its functionality is comparable to that of the Java BigInteger class. It falls in the following categories:

  • Arithmetic: add, subtract, multiply, divide, remainder and exponentiation
  • Comparison: the six standard operators == != < <= > >=
  • 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 and probable prime number generation
  • Miscellaneous: random number generation, greatest common divisor, n-th root, square root modulo an odd prime, and Jacobi symbol

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: "1.2.8"),
  ]

Examples

Creating BInt's

  // From an integer
  let a = BInt(27)
  
  // 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. Each execution time was then compared to the execution time for the same operation in Java using the BigInteger class. The results are in the table below. It shows the operation being measured and the time it took in Swift and in Java (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 codeSwift timeJava time
As stringc2000.asString()23 uSec31 uSec
As signed bytesc2000.asSignedBytes()0.23 uSec1.0 uSec
Bitwise anda1000 & b10000.16 uSec0.076 uSec
Bitwise ora1000 | b10000.17 uSec0.051 uSec
Bitwise xora1000 ^ b10000.17 uSec0.048 uSec
Bitwise not~c20000.10 uSec0.13 uSec
Test bitc2000.testBit(701)0.0038 uSec0.0058 uSec
Flip bitc2000.flipBit(701)0.0078 uSec0.069 uSec
Set bitc2000.setBit(701)0.0023 uSec0.088 uSec
Clear bitc2000.clearBit(701)0.0046 uSec0.072 uSec
Additiona1000 + b10000.096 uSec0.12 uSec
Subtractiona1000 - b10000.11 uSec0.25 uSec
Multiplicationa1000 * b10000.32 uSec0.73 uSec
Divisionc2000 / a10002.8 uSec3.4 uSec
Modulusc2000.mod(a1000)2.8 uSec3.5 uSec
Inverse modulusc2000.modInverse(p1000)0.60 mSec0.17 mSec
Modular exponentiationa1000.expMod(b1000, c2000)3.8 mSec2.3 mSec
Equalc2000 + 1 == c20000.00095 uSec0.019 uSec
Less thanb1000 + 1 < b10000.012 uSec0.011 uSec
Shift 1 leftc2000 <<= 10.094 uSec0.060 uSec
Shift 1 rightc2000 >>= 10.11 uSec0.058 uSec
Shift 100 leftc2000 <<= 1000.17 uSec0.045 uSec
Shift 100 rightc2000 >>= 1000.14 uSec0.060 uSec
Is probably primep1000.isProbablyPrime()6.2 mSec12 mSec
Make probable 1000 bit primeBInt.probablePrime(1000)55 mSec34 mSec
Next probable primec2000.nextPrime()780 mSec510 mSec
Greatest common divisora1000.gcd(b1000)37 uSec31 uSec
Make random numberc2000.randomLessThan()0.69 uSec0.49 uSec
Squarec2000 ** 20.88 uSec0.99 uSec
Square rootc2000.sqrt()22 uSec139 uSec
Square root modulob1000.sqrtMod(p1000)2.3 mSecn.a.

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

References

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

  • [BRENT] - Brent and Zimmermann: Modern Computer Arithmetic, 2010
  • [CRANDALL] - Crandall and Pomerance: Prime Numbers - A Computational Perspective. Second Edition, Springer 2005
  • [GRANLUND] - Moller and Granlund: Improved Division by Invariant Integers, 2011
  • [HANDBOOK] - Menezes, Oorschot, Vanstone: Handbook of Applied Cryptography. CRC Press 1996
  • [KNUTH] - Donald E. Knuth: Seminumerical Algorithms. Addison-Wesley 1971

Acknowledgement

The BitSieve class used in the implementation is a translation to Swift of the corresponding class from Java BigInteger. The Karatsuba and ToomCook multiplication algorithms are modelled after the corresponding algorithms in Java BigInteger.

GitHub

link
Stars: 1
Last commit: 1 week ago

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Release Notes

Release 1.2.8
1 week ago

New instance method 'nextPrime' that returns the next prime number greater than 'self' e.g. BInt(34).nextPrime() returns 37

Performance improvements:

  1. The 'asString' method is about 300% faster

  2. The 'gcd' method now uses Lehmer's algorithm and is about 200% faster

  3. The 'modInverse' method is about 60% faster

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