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In this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form f (x) = x 4 q + 1 + λ 1 x 5 q + λ 2 x q + 4 over the finite field F 5 k , which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on λ 1 , λ 2 ∈ F 5 k so that f(x) is a permutation monomial, binomial, or trinomial of F 5 2 k . [ABSTRACT FROM AUTHOR]

We completely characterize all permutation trinomials of the form f (x) = x 3 (1 + a x q - 1 + b x 2 (q - 1)) over F q 2 , where a , b ∈ F q ∗ and all permutation trinomials of the form f (x) = x 3 (1 + b x 2 (q - 1) + c x 3 (q - 1)) over F q 2 , where b , c ∈ F q ∗ in both even and odd characteristic cases. [ABSTRACT FROM AUTHOR]

We classify all permutation polynomials of the form x 3 g (x q - 1) of F q 2 where g (x) = x 3 + b x + c and b , c ∈ F q ∗ . Moreover we find new examples of permutation polynomials and we correct some contradictory statements in the recent literature. We assume that gcd (3 , q - 1) = 1 and we use a well known criterion due to Wan and Lidl, Park and Lee, Akbary and Wang, Wang, and Zieve. [ABSTRACT FROM AUTHOR]

The subset sum problem is an important theoretical problem with many applications, such as in coding theory, cryptography, graph theory and other fields. One of the many aspects of this problem is to answer the solvability of the k -subset sum problem. It has been proven to be NP-hard in general. However, if the evaluation set has some special algebraic structure, it is possible to obtain some good conclusions. Zhu, Wan and Keti proposed partial results of this problem over two special kinds of evaluation sets. We generalize their conclusions in this paper, and propose asymptotical results of the solvability of the k -subset sum problem by using estimates of additive character sums over the evaluation set, together with the Brun sieve and the new sieve proposed by Li and Wan. We also apply the former two examples as application of our results. [ABSTRACT FROM AUTHOR]