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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Computing in $ {\rm GF}\,(q)$


Author: Jacob T. B. Beard
Journal: Math. Comp. 28 (1974), 1159-1166
MSC: Primary 12C05; Secondary 12-04
MathSciNet review: 0352058
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Abstract: This paper gives an elementary deterministic algorithm for completely factoring any polynomial over $ {\text{GF}}(q),q = {p^d}$, criteria for the identification of three types of primitive polynomials, an exponential representation for $ {\text{GF}}(q)$ which permits direct rational calculations in $ {\text{GF}}(q)$ as opposed to modular arithmetic over $ {\text{GF}}[p,x]$, and a matrix representation for $ \overline {{\text{GF}}} (p)$ which admits computer computations. The third type of primitive polynomial examined permits the given representation of $ {\text{GF}}(q)$ to display a primitive normal basis over $ {\text{GF}}(p)$. The techniques developed require only the usual addition and multiplication of square matrices over $ {\text{GF}}(p)$. Partial tables from computer programs based on certain of these results will appear in later papers.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1974-0352058-9
PII: S 0025-5718(1974)0352058-9
Keywords: Factorization, arithmetic in finite fields, irreducibility criterion, primitive polynomials, primitive normal bases, Euler function, exponent, linear polynomial, algebraic closure
Article copyright: © Copyright 1974 American Mathematical Society