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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing in $\textrm {GF} (q)$

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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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