# Create Morphism's between Finite Fields and VectorSpaces

Hallo,

I am interested in creating morphims between $GF(2^{n+1}) \stackrel{h_1}{\longrightarrow} H_u \stackrel{h_2}\longrightarrow GF(2^n)$ where $H_u={x^{2^i} + x : x \in GF(2^{n+1})}$ where $gcd(i,n+1) =1$. I am interested in constructing $h_1$ and $h_2$,

My attempt to create $h_1$ is:


n = 4
i = 1
f = lambda x : x^(2^i) + x
Kn_1 = GF(2^(n+1),'x')
H_u = map(f, Kn_1)
hs = Hom(Kn_1,H_u)
pi = SetMorphism(Hom(Kn_1,Kn), lambda y : y^(2^i) + y)
pi(Kn_1.random_element()) in Kn
 

now $hs$ is a set of morphism and $h_1 \in hs$. Can I get a (the) specific $h_1$ from $hs$? My attempt is $pi$ but 'pi(Kn_1.random_element()) in Kn' fails.

To construct $h_2$ I have more success.


n = 4
i = 2
f = lambda x: x^(2^i) + x

Kn_1 = GF(2^(n+1),'x1')
Vn_1 = Kn_1.vector_space()
Sn_1 = Vn_1.subspace([Vn_1(f(u)) for u in Kn_1])

Kn = GF(2^n,'x')
Vn = Kn.vector_space()

h_a = Sn_1.basis_matrix().transpose()
 

Now $h_a : Vn\rightarrow Sn_1$ where $Vn$ and $Sn_1$ is vector space representation of $GF(2^n)$ and $H_u$. To get from $GF(2^n)$ to $Vn$ and back I am good with But to create $h_2$ I have no luck. My attempts to use


MatrixMorphism(Hom(Vn,Sn_1), Sn_1.basis_matrix().transpose())
 

but get errors with regards to the dimensions of the matrix.

Regards

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It is a morphism for which structure? Vector space? If so, is that clear that $H_u$ is a vector space?

( 2013-07-23 19:41:21 +0200 )edit

The curly brackets got lost in the definition of H_u. The function $l(x)=x^(2^i)+x$ is linear over GF(2^(n+1)) and H_u is image of $l(x)$ giving that H_u is a subgroup of GF and can also be seen as a vector space.

( 2013-07-24 03:08:27 +0200 )edit

Your first code excerpt does not work for me: on Hom(Kn_1, H_u) it complains that H_u is a list. Set morphisms forget a lot of structure about you vector spaces. Have you considered using Kn_1.vector_space() and working with matrices?

( 2013-07-24 09:39:24 +0200 )edit

H_u was suppose to the a multiplicative subgroup of Kn_1 and now can't get away to generate a subgroup in sage. The reason I want to use a morphism is that I am only interested in the Range and Domain and the mapping. This will also help met to learn a functionality of sage.

( 2013-07-28 04:46:20 +0200 )edit

Did you mean "additive" instead of "multiplicative"? Vector spaces are additive groups, so you may be happy with them.

( 2013-07-30 06:04:00 +0200 )edit

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Here's my shot at it. Not very different from your code, but all your spaces are represented by a VectorSpace object, now.

n, i = 4, 1

Kn = GF(2^n,'x')
Vn = Kn.vector_space()
Kn_1.<x> = GF(2^(n+1))
Vn_1 = Kn_1.vector_space()

f = lambda x: Vn_1(Kn_1(x)^(2^i)) + x
pi = Vn_1.hom(map(f, Vn_1.basis()))
H_u = pi.image()

h1 = Hom(Vn_1, H_u)(random_matrix(GF(2), n+1, n))
h2 = Hom(H_u, Vn)(random_matrix(GF(2), n))


Be careful that vector spaces homomorphisms are represented by matrices multiplied on the left.

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