Abstract
In this paper, according to the known results of some normalized permutation polynomials with degree 5 over F 2 n , we determine sufficient and necessary conditions on the coefficients b 1 , b 2 ∈ F 2 n 2 such that f x = x 3 x ¯ 2 + b 1 x 2 x ¯ + b 2 x permutes F 2 n . Meanwhile, we obtain a class of complete permutation binomials over F 2 n .
Highlights
Denote Fq as a finite field with q elements, where q is a prime power;,∗ q is its multiplicative group.e bijectivity of the associated polynomial mapping(s) f: x ⟼ f(x) from Fq into itself (and f(x) + x) makes a polynomial f(x) ∈ F q[x] as a permutation polynomial [1]
Let m and n be two positive integers with m|n and F 2n be a finite field with 2n elements; the trace function from F 2n to F2m is denoted by Trnm(·), where
Let b, c ∈ F q and a ∈ F ∗q ; for any permutation polynomial f(x) ∈ Fq[x], there exists a unique normalized form provided by g(x) af(x + b) + c
Summary
E bijectivity of the associated polynomial mapping(s) f: x ⟼ f(x) from Fq into itself (and f(x) + x) makes a polynomial f(x) ∈ F q[x] as a (complete) permutation polynomial [1]. Readers could refer to [1, 8] for a comprehensive survey about (complete) permutation polynomials. E following form over F2n from Niho exponents has drawn much attention: f(x) xr1 + b1xs(2m− 1) + b2xt(2m− 1),. Is motivates us to explore new permutation polynomials with general coefficients (b1, b2) from non-Niho exponents with even characteristics. Journal of Mathematics degree 5 over even characteristics, we determine the coefficients (b1, b2) for f(x) x3x2 + b1x2x + b2x being a permutation over F 2n. In. Section 3, the sufficient and necessary conditions are shown to determine the coefficients of a class of permutation polynomials over F2n.
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