The explicit construction of irreducible polynomials over finite fields
Designs, Codes and Cryptography
Normal and Self-dual Normal Bases from Factorization of$c x^{q+1} + d x^{q} - ax - b$
SIAM Journal on Discrete Mathematics
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CRYPTO '95 Proceedings of the 15th Annual International Cryptology Conference on Advances in Cryptology
Irreducible compositions of polynomials over finite fields
Designs, Codes and Cryptography
On different families of invariant irreducible polynomials over F2
Finite Fields and Their Applications
Iterated constructions of irreducible polynomials over finite fields with linearly independent roots
Finite Fields and Their Applications
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Let g(x) = x n + a n-1 x n-1 + . . . + a 0 be an irreducible polynomial over $${\mathbb{F}_q}$$ . Varshamov proved that for a = 1 the composite polynomial g(x p 驴ax驴b) is irreducible over $${\mathbb{F}_q}$$ if and only if $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a = 1 and $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$$ and for the case a 驴 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form $${g(x^{r^kp}-x^{r^k})}$$ are also presented. Moreover, Cohen's method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q 3 驴 q) = 1.