Improving an algorithm for factoring polynomials over a finite field and constructing large irreducible polynomials

Abstract
Letf(x)in {bf F}_{q}[X]be a polynomial with simple roots to be factored. The so-called Berlekamp subalgebraBspanned over{bf F}_{q}by the idempotents ofA={bf F}_{q}[X]/(f(X))is considered. An exponential technique introduced earlier is based upon taking elements from B at random and enables us to obtain idempotents and, from that, the factors of f(X). This algorithm is speeded up in three ways. The concept of a separating subset of B is introduced and the McEliece operator mapping A onto B is used to construct a small separating set. {em Factoring} subsets of{bf F}_{q}were defined and investigated previously. The algorithm and these subsets are used together with a process introduced by F. J. McWilliams for the rapid construction of primitive idempotents. Finally, an algorithm is introduced for constructing irreducible polynomials of{bf F}_{q}[X]of degreed, for large values ofd, in which the most expensive operation is the Euclidian algorithm applied to two polynomials of degree2d.
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