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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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