By Jürgen Müller

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Q d −1 2 . = 1} ∪ {α ∈ F∗qd ; α Fq [X]/ Φi , i. e. Φi | gcd(Ψ, Θ = Hence for Θ ∈ Fq [X]

Note that by the above we have dimFq (ker(QΨ − En )) = r, hence this yields the so far unknown number r of prime divisors of Ψ, in particular Ψ is irreducible 46 if and only if rkFq (QΨ − En ) = n − 1. We finally have to pick Θ ∈ UΨ yielding non-trivial factors: For i = j ∈ {1, . . , r} there is Θ ∈ UΨ such that αi := Θπi = Θπj =: αj ∈ Fq , i. e. we have Φi | gcd(Ψ, Θ − αi ) and Φj | gcd(Ψ, Θ − αi ), as well as Φj | gcd(Ψ, Θ − αj ) and Φi | gcd(Ψ, Θ − αj ). Hence given an Fq -basis {Θ0 , . . , Θr−1 } ⊆ UΨ , where we may assume Θ0 = 1, by Fq -linearity there is an element Θk , for some k ∈ {1, .

Bn } ⊆ Rn be the associated Gram-Schmidt R-basis, and let µij := bi ,bj ||bj ||2 ∈ R, for 1 ≤ j < i ≤ n. Then B is called LLL reduced if the following holds, with respect to some fixed µ2ij α−1 α 1 4 < γ ≤ 1 and where we let α := 1 γ− 14 : i) ≤ for all 1 ≤ j < i − 1 ≤ n, and ii) |µi,i−1 | ≤ 21 for all i ∈ {2, . . , n}, as well as 2 2 iii) Lovasz condition: ||bi || ≥ (γ − µ2i,i−1 ) · ||bi−1 || for all i ∈ {2, . . , n}. 2 2 Note that the Lovasz condition is equivalent to ||bi + bi−1 µi,i−1 || = ||bi || + 2 2 µ2i,i−1 · ||bi−1 || ≥ γ · ||bi−1 || , for all i ∈ {2, .