# Extension for Galois Field of order 2

**URL:** <https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334>\
**Category:** New to Julia\
**Tags:** nemo\
**Created:** [April 11, 2023, 9:57am UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334 "2023-04-11T09:57:24Z")\
**Posts on this page:** 8\
**Page:** 1

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**Author:** ![SSF](https://avatars.discourse-cdn.com/v4/letter/s/8dc957/32.png) [@SSF](https://discourse.julialang.org/u/SSF)\
**Post date:** [April 11, 2023, 9:57am UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/1 "2023-04-11T09:57:24Z")

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Hello,  
New to Julia and to Galois Field, I would like to calculate in GF(2^8). Using Nemo, here is what I start with:

```julia
julia> using Nemo

Welcome to Nemo version 0.33.7

Nemo comes with absolutely no warranty whatsoever

julia> R, x = FiniteField(2, 8, "x")
(Finite field of degree 8 over F_2, x)

```

Then I need to use elements of GF(256). I tried to cast them from integers like this:

```julia
julia> R(3)
1

```

but it doesn’t seem to work as I expected (output 3). How can I input GF(256) values? Should I use polynomial expressions only? If so, is there a way to convert Int to polynomial expression?  
thanks for your help,  
B.R.

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<div class="post-metadata">

**Author:** ![SSF](https://avatars.discourse-cdn.com/v4/letter/s/8dc957/32.png) [@SSF](https://discourse.julialang.org/u/SSF)\
**Post date:** [April 11, 2023, 3:21pm UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/2 "2023-04-11T15:21:10Z")

</div>

I found this way to do it. I post it if anyone is interested (or to get expert’s point of view). You will find respectively the GF-\> Int conversion, the Int → GF conversion (restricted to GF(256) but change UInt8 to greater if needed), and a simple test.

```julia
function Int(a::fqPolyRepFieldElem)
    dum = 0
    if !iszero(a)
        for i=a.length-1:-1:0
            if coeff(a,i)>0
                dum+=2^i
            end
        end
    end
    return dum
end

function GF(a::Integer,F::fqPolyRepField)
    b = bitstring(UInt8(mod(a,2^8)))
    res = F(0);
    for i=1:8
        if b[i]=='1'

            res+=(gen(R))^(8-i)
        end
    end
    return res
end

R, x = FiniteField(2, 8, "x")
a = 253
agf = GF(a,R)
Int(agf)==a

```

thanks!

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<div class="post-metadata">

**Author:** ![thofma](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/thofma/32/1691_2.png) [@thofma](https://discourse.julialang.org/u/thofma)\
**Post date:** [April 11, 2023, 3:27pm UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/3 "2023-04-11T15:27:48Z")

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The element `3` is the same as `1` in `GF(256)` (or any `GF(2^n)`).

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<div class="post-metadata">

**Author:** ![SSF](https://avatars.discourse-cdn.com/v4/letter/s/8dc957/32.png) [@SSF](https://discourse.julialang.org/u/SSF)\
**Post date:** [April 11, 2023, 3:36pm UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/5 "2023-04-11T15:36:32Z")

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To me GF(2^p) has 2^p elements, not only 2 like in GF(2), no?

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**Author:** ![thofma](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/thofma/32/1691_2.png) [@thofma](https://discourse.julialang.org/u/thofma)\
**Post date:** [April 11, 2023, 4:17pm UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/6 "2023-04-11T16:17:25Z")

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Sure. I did not claim that there are only two elements. But one has 1 + 1 = 0 in \mathrm{GF}(2^n) for any n \geq 1 (all these fields contain \mathrm{GF}(2) or in other words, have characteristic 2). In particular 3 = 1 + 1 + 1 = 1 in \mathrm{GF}(2^n).

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<div class="post-metadata">

**Author:** ![SSF](https://avatars.discourse-cdn.com/v4/letter/s/8dc957/32.png) [@SSF](https://discourse.julialang.org/u/SSF)\
**Post date:** [April 12, 2023, 6:47am UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/7 "2023-04-12T06:47:31Z")

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I agree with your 1+1+1 = 1. But how do you manipulate other elements than 0 and 1 in GF(256)? can you write an element in GF(256) that is not 0 or 1? (it seems that there is no other way than the form using the root of the irreducible polynomial, am I correct?)  
I thought the integer representation could also be useful (and quite common), hence my 2 conversion functions, but I do not know if this was already provided by Nemos?

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<div class="post-metadata">

**Author:** ![thofma](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/thofma/32/1691_2.png) [@thofma](https://discourse.julialang.org/u/thofma)\
**Post date:** [April 12, 2023, 7:03am UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/8 "2023-04-12T07:03:53Z")

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Yes, one just writes an arbitrary element using powers of `x`, e.g., `1 + x + x^2 + x^7`. What I have seen in a more applied context is something like `11001101` to represent the element `a = 1 + x + 0*x^2 + 0*x^3 + x^4 + x^5 + 0*x^6 + 1*x^7` (or maybe reversed, I don’t remember). I have never seen the “integer representation” in a math paper before. The “integer representation” has the big disadvantage, that it does not respect addition nor multiplication. It is just a bijection from \{1,\dotsc,2^n\} \to \mathrm{GF}(2^n).

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<div class="post-metadata">

**Author:** ![SSF](https://avatars.discourse-cdn.com/v4/letter/s/8dc957/32.png) [@SSF](https://discourse.julialang.org/u/SSF)\
**Post date:** [April 12, 2023, 7:12am UTC](https://discourse.julialang.org/t/extension-for-galois-field-of-order-2/97334/9 "2023-04-12T07:12:09Z")

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Yes the integer representation might seem strange when looking at the multiplication and addition tables, but it is quite convenient to write C programs for example. It just a data format trick, nothing to be worth describing in an article 🙂 You can see it also in matlab, so I thought it might be the same.  
However thank you for your confirmation and your replies!