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By Jürgen Müller

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1), while the transition function . τ : (X ∪ Y) × (S \ {s∞ }) −→ Pot((X ∪ Y) × {←, ↑, →} × S) allows for choices and thus branching. Let the non-determinateness be . defined as dT := max{|τ (x, s)|; x ∈ X ∪ Y, s ∈ S \ {s∞ }} ∈ N. The machine III Integer arithmetic 44 T halts if no further transition in either branch is possible. We assume that for all inputs T on halting either accepts or rejects, or outputs; for acceptance, rejection or output one of the branches is chosen randomly. A language L ⊆ X ∗ is called non-deterministically decidable, if there is a non-deterministic Turing machine T deciding L, i.

1     . . 1 . 1 . . 1 1 1 . 1    M =   1 1 . . . . 1 . 1 1 . 1   . 1 1 . . 1 1 1 . 1 . 1 1 1     . . . . . 1 1 . . 1  . 1 . 1 . 1 1 1 1 . 1 1 It turns out that M is given as follows:  1 1  1 1   .   . 1   .   . 1   1 . 1 . has maximal rank rkF2 (M ) = 8, and an F2 -basis of ker(M tr ) 1 . 1 . 1 . 1 . . 1 1 . 1 . 1 1 . . 1 1 . 1 . . 1 . . . 1 1 . 1 . 1 . . 1 1 . . 1 . . . . . 1 . . . . . 1 . . 1 . . . 1 . 1 . . . . .

II Public key cryptography 39 q 2m m · v = u′ · v q is If 0 = v ∈ Flq is fixed, the set of 0 = u′ ∈ Flq fulfilling u′ m l given as follows: Let u ∈ Fq be the unique element such that uq +1 = v, thus vq m −1 = uq 2m ′q m −1 2m . If u′ q 2m −1 · v = u′ · v q m for some u′ = 0, then we also have m (q m −1)(q m +1) m v q −1 = u . Thus we have u(q −1)(q +1) = u′ , and since q m −1 q m −1 m n ′ ′ gcd(q + 1, q − 1) = 1 we conclude u =u , and hence u · u−1 ∈ F is m a (q − 1)-st root of unity. Conversely, if ζ ∈ F is a (q m − 1)-st root of unity, 2m m m 2m then from uq · v = u · v q we get (u · ζ)q · v = (u · ζ) · v q .

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