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In GF($2^m$) each element is a polynomial also it is a binary message. I wonder how to make maple help me convert vectors of maple Bits into GF elements and back?

Also it is not the question of mine but it is indeed quite interesting how to convert elements from GF with any characteristic having some alphabet in and out.

Is such thing possible with maple and how to do it?

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    Element of $GF(2^m)$ -> List: If you are using a monomial basis, then you simply want a list of coefficients of the polynomial. List -> element of $GF(2^m)$: you take the inner product of the list and a vector containing the powers of whatever you denote the generating element with. If you are using another basis, then use that instead.2011-12-15
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    Technically, the elements of $\mathrm{GF}(2^m)$ are *equivalence classes of polynomials* under the usual construction ($\mathrm{GF}(2)/(q(x))$, where $q(x)$ is an irreducible polynomial of degree $m$).2011-12-15

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For example:

G:= GF(2,30); r:= G:-random();

$r := \left({T}^{28}+{T}^{26}+{T}^{25}+{T}^{22}+{T}^{21}+{T}^{20}+{T}^{19}+{T}^{18 }+{T}^{16}+{T}^{12}+{T}^{11}+{T}^{8}+{T}^{7}+{T}^{6}+{T}^{5}+{T}^{4}+{ T}^{3}\right) \mod 2$

G:-output(r);

$377297400$

Bits:-Split(%);

$[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1]$

Bits:-Join(%);

$377297400$

G:-input(%);

$\left({T}^{28}+{T}^{26}+{T}^{25}+{T}^{22}+{T}^{21}+{T}^{20}+{T}^{19}+{T}^{18 }+{T}^{16}+{T}^{12}+{T}^{11}+{T}^{8}+{T}^{7}+{T}^{6}+{T}^{5}+{T}^{4}+{ T}^{3}\right) \mod 2$