[ANN] Released GaloisFields v1.0

Package Announcements
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I just released v1.0 of GaloisFields.jl. For those people not familiar – a Galois field (or ‘finite field’) is a finite set that supports addition and multiplication, both with their inverses subtraction and division. The most well-known examples are the integers modulo a prime number p. This package allows using these sets wherever a Number is expected.

Commonly, a bottleneck when using Galois fields is the modulo (mod) operation. An interesting feature of this package is the use of broadcasting to reduce the number of these: whenever there’s a sufficient number of bits in Int, we postpone computing this reduction until the end. Here’s the performance effect:

julia> using GaloisFields
julia> const F = GaloisField(29);
julia> x = rand(F, 1000); y = rand(F, 1000); z = rand(F);
julia> @btime x .+ z .* y; # fused modulo optimization
  2.547 μs (7 allocations: 1.22 KiB)
julia> @btime broadcast((x,z,y) -> x + z * y, x, z, y); # same operation, no special optimization
  4.998 μs (1 allocation: 1.06 KiB)

@b-reinke contributed bug fixes and @Keno contributed support for BitIntegers.jl and code to compute primitive roots of unity. For fields of prime power order, we use Frank Lübeck’s database of Conway polynomials.

10 Likes
2

How do cast 15 in the GaloisField(7)? It’s should be the identity.

Is there a function to check if an element is a generator in the field?

3

Thanks for asking!

julia> using GaloisFields
julia> const F = GaloisField(7);
julia> F(15)
1
julia> convert(F, 15)
1

There is, thanks to @Keno:

julia> GaloisFields.is_primitive_root.(F, F.(1:6), char(F) - 1)
6-element BitArray{1}:
 0
 0
 1
 0
 1
 0
1 Like
4

Hi everyone, Hi @tkluck !
I don’t know if it makes sense to post 2 years after package announcement, but I am trying to use GaloisField with 2^p extension. It seems I have issues with it. Could you provide some help / information? (new to Julia, sorry if my post is not in the appropriate place).

julia> using GaloisFields
julia> const F = GaloisField(2^8)
(𝔽₂₅₆, ##312)

I can see the tuple output which is not the same as without extension. Then when I want to convert, the following issue is output:

julia> F(256)
ERROR: MethodError: objects of type Tuple{DataType, 𝔽₂₅₆} are not callable
Stacktrace:
 [1] top-level scope
   @ REPL[8]:1

And if I use the first tuple element, it acts as GF(2):

julia> a = F[1](15)
1
julia> typeof(a)
𝔽₂₅₆

Thanks for your help!
B.R.

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I am sure you found the answer. But to have the answer also in writing here: 15 is equal to 1 in GF(2^8).

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Well yes I think I did. In fact I was expecting a behavior like e.g. in matlab where you can name all the elements of GF(2^p) by an integer between 0 and 2^p-1, instead of using a generator element \alpha. I understood that it was not implemented like this.
Thanks for your message however!

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@SSF thanks for using GaloisFields.jl . The same question came up on Github before and here’s my answer there: Mapping to/from extension fields · Issue #17 · tkluck/GaloisFields.jl · GitHub

I should probably add methods for reinterpret to do what you’re asking here.

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thanks for pointing this Github question and your answer, that’s fine for me!