Big Complex

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Big complex number calculation library for Go (with math/big).

Currently, the library supports:

  1. Gaussian integer, complex numbers whose real and imaginary parts are both integers:


  2. Hurwitz quaternion, quaternions whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded):



go get -u


The usage is quite similar to Golang math/big package.

package main

import (

   bc ""

func main() {
   // Gaussian integer calculation
   g1 := bc.NewGaussianInt(big.NewInt(5), big.NewInt(6)) // 5 + 6i
   g2 := bc.NewGaussianInt(big.NewInt(1), big.NewInt(2)) // 1 + 2i
   div := new(bc.GaussianInt).Div(g2, g1)
   fmt.Println(div) // 3 - i
   gcd := new(bc.GaussianInt).GCD(g1, g2)
   fmt.Println(gcd) // i

   // Hurwitz integer calculation
   // 1 + i + j + k
   h1 := bc.NewHurwitzInt(big.NewInt(1), big.NewInt(1), big.NewInt(1), big.NewInt(1), false)
   // 3/2 + i + j + 3k/2
   h2 := bc.NewHurwitzInt(big.NewInt(3), big.NewInt(2), big.NewInt(2), big.NewInt(3), true)
   prod := new(bc.HurwitzInt).Prod(h1, h2)
   fmt.Println(prod) // 2 + 3i + 2j + 3k

Why This Library?

Fan fact: Golang has native complex number types: complex64 and complex128.

c1 := complex(10, 11) // constructor init
c2 := 10 + 11i        // complex number init syntax

realPart := real(c1)    // gets real part
imagPart := imag(c1)    // gets imaginary part

complex64 represents float64 real and imaginary data, and complex128 represents float128 real and imaginary data. They are easy to use, but unfortunately they are incapable for handling very large complex numbers.

For instance, in finding the Lagrange Four Square Sum of a very large integer (1792 bits in size) for cryptographic range proof, we need to compute the Greatest Common Divisor (GCD) of Gaussian integers and the Greatest Common Right Divisor of Hurwitz integers. And the built-in complex number types can not handle such large numbers.

So I came up with the idea of building a library for large complex number calculation with Golang math/big package.


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