Constructions of complete permutations over F q n

Complete permutation polynomials play an important role in cryptography, particularly in the design of cryptographic primitives such as the Lai--Massey scheme and S-boxes. We generalize a result of Sun, Li, Guo, and Qu (2021) by characterizing the complete permutation behavior of the mapping Ψ(X)=M(X+ψ(AX)) over F q n, where F q is a finite field of q elements with q being a prime power, M in GL(n, F q), GL(n, F q) is the general linear group of order n over F q, A m times n is a full-rank matrix over F q, and ψ=(ψ 1,ψ 2,,ψ n) with each component function ψ i: F q m to F q. Furthermore, we establish criteria for the permutation and complete permutation properties of the mapping F(X)=T(X+B tf(AX)) over F q n, f: F q m to F q n-m, T in GL(n, F q), A m times n and B(n-m) times n are full-rank matrices over F q, B t represents the transpose of the matrix B, and 0<m<n are integers. These results also generalize an earlier result of Gravel and Panario (2023), who showed that any arbitrary function f from F q m to F q,n-m can be extended to a bijection over F q n through the mapping F(X)=T(X+B tf(AX)), under the condition AB t=0. Here we do not impose the restriction that AB t=0.