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Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields


Author: Greg Stein
Journal: Math. Comp. 70 (2001), 1237-1251
MSC (2000): Primary 11T06, 11T24, 11T22; Secondary 12Y05, 13P05
DOI: https://doi.org/10.1090/S0025-5718-00-01233-3
Published electronically: March 24, 2000
MathSciNet review: 1709159
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Abstract:

We examine the problem of factoring the $r$th cyclotomic polynomial, $\Phi _r(x),$ over $\mathbb{F}_p$, $r$ and $p$ distinct primes. Given the traces of the roots of $\Phi _r(x)$ we construct the coefficients of $\Phi _r(x)$ in time $O(r^4)$. We demonstrate a deterministic algorithm for factoring $\Phi _r(x)$ in time $O((r^{1/2+\epsilon}\log p)^9)$ when $\Phi _r(x)$ has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of $\Phi _r(x)$ in time $O(r^6)$.


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Additional Information

Greg Stein
Affiliation: The City University of New York, 300 Jay Street, Brooklyn, New York 11201
Email: gregstein@member.ams.org

DOI: https://doi.org/10.1090/S0025-5718-00-01233-3
Keywords: Finite field, factorization, cyclotomic, polynomial
Received by editor(s): December 1, 1998
Received by editor(s) in revised form: June 22, 1999
Published electronically: March 24, 2000
Additional Notes: Supported in part by PSC-CUNY Research Foundation Grant 69666-00 29
Article copyright: © Copyright 2000 American Mathematical Society

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