Compositional inverses and complete mappings over finite fields

We study compositional inverses of permutation polynomials and complete mappings over finite fields. Recently the compositional inverses of linearized permutation binomials were obtained in Wu (2013). In this paper we obtain compositional inverses of a class of linearized binomials permuting the kernel of the trace map. It was also shown in Tuxanidy and Wang (2014) that computing inverses of bijections of subspaces has an application in determining the compositional inverses of certain permutation classes related to linearized polynomials. Consequently, we give the compositional inverse of a new class of complete mappings. This complete mapping class extends several recent constructions given in Laigle-Chapuy (2007), Samardjiska and Gligoroski (2014), Wu and Lin (2013), Wu and Lin (2015), Wu et al. (2013). We also construct recursively a class of complete mappings involving multi-trace functions

On the inverses of some classes of permutations of finite fields

We study the compositional inverses of some general classes of permutation polynomials over finite fields. We show that we can write these inverses in terms of the inverses of two other polynomials bijecting subspaces of the finite field, where one of these is a linearized polynomial. In some cases we are able to explicitly obtain these inverses, thus obtaining the compositional inverse of the permutation in question. In addition we show how to compute a linearized polynomial inducing the inverse map over subspaces on which a prescribed linearized polynomial induces a bijection. We also obtain the explicit compositional inverses of two classes of permutation polynomials generalizing those whose compositional inverses were recently obtained in [22] and [24], respectively

Composed products and factors of cyclotomic polynomials over finite fields

Let q = p s be a power of a prime number p and let Fq be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over Fq. In particular we obtain the explicit factorization of the cyclotomic polynomial Φ2n r over Fq where both r ≥ 3 and q are odd, gcd(q, r) = 1, and n ε ℕ. Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of Φr over Fq is given to us as a known. Let n = p1 e1p2 e2...ps es be the factorization of n ε ℕ into powers of distinct primes p i, 1 ≤ i ≤ s. In the case that the multiplicative orders of q modulo all these prime powers pi ei are pairwise coprime, we show how to obtain the explicit factors of Φn from the factors of each p i ei. We also demonstrate how to obtain the factorization of Φmn from the factorization of Φn when q is a primitive root modulo m and gcd(m, n) = gcdφ(m)ordn(q)) = 1. Here φ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over Fq and generalize a result due to Varshamov (Soviet Math Dokl 29:334-336, 1984)

Characteristic digit-sum sequences

We introduce a new type of sequences using the sum of coefficients of characteristic polynomials for elements (in particular, primitive elements) in a finite field. These sequences are nonlinear filtering sequences of the well-known m-sequences. We show that they have large linear complexity and large period. We also provide some examples of such binary sequences with good autocorrelation values

On the number of N-free elements with prescribed trace

In this paper we derive a formula for the number of N-free elements over a finite field Fq with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive several explicit formulae in special cases. In addition we show that if all the prime factors of q-1 divide m, then the number of primitive elements in Fqm, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, Pq,m,N(c), of elements in Fqm with multiplicative order N and having trace c∈ Fq