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Fast construction of irreducible polynomials over finite fields. (English) Zbl 1287.11141
Summary: We present a randomized algorithm that on inputting a finite field $$\mathbf K$$ with $$q$$ elements and a positive integer $$d$$ outputs a degree $$d$$ irreducible polynomial in $$\mathbf K[x]$$. The running time is $$d^{1+\varepsilon(d)}\times(\log q)^{5+\varepsilon(q)}$$ elementary operations. The function $$\varepsilon$$ in this expression is a real positive function belonging to the class $$o(1)$$, especially, the complexity is quasi-linear in the degree $$d$$. Once given such an irreducible polynomial of degree $$d$$, we can compute random irreducible polynomials of degree $$d$$ at the expense of $$d^{1+\varepsilon(d)} \times (\log q)^{1+\varepsilon(q)}$$ elementary operations only.

##### MSC:
 11Y16 Number-theoretic algorithms; complexity 11T06 Polynomials over finite fields 12E05 Polynomials in general fields (irreducibility, etc.) 68Q25 Analysis of algorithms and problem complexity 68W20 Randomized algorithms 68W30 Symbolic computation and algebraic computation
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