Constructions of complete permutations over π½qn
Abstract
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 Ξ¨(X)=M(X+Ο(AX)) over π½qn, where π½q is a finite field of q elements with q being a prime power, Mβ GL(n, π½q), GL(n, π½q) is the general linear group of order n over π½q, Am Γ n is a full-rank matrix over π½q, and Ο=(Ο1,Ο2,β¦,Οn) with each component function Οi:π½qmβπ½q. Furthermore, we establish criteria for the permutation and complete permutation properties of the mapping F(X)=T(X+Btf(AX)) over π½qn, f: π½qm β π½qnβm, T β GL(n, π½q), Am Γ n and B(nβm)Γ n are full-rank matrices over π½q, Bt 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 π½qm to π½q nβm can be extended to a bijection over π½qn through the mapping F(X)=T(X+Btf(AX)), under the condition ABt=0. Here we do not impose the restriction that ABt=0.
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BibTeX
@misc{hasan2026constructions,
title = {{Constructions of complete permutations over $\mathbbF_q^n$}},
author = {Sartaj Ul Hasan and Ramandeep Kaur and Hridesh Kumar and Daniel Panario and Qiang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05564},
eprint = {2609.05564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05564},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Constructions of complete permutations over π½_q^n
AU - Sartaj Ul Hasan
AU - Ramandeep Kaur
AU - Hridesh Kumar
AU - Daniel Panario
AU - Qiang Wang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.05564
UR - https://arxiv.org/abs/2609.05564
AB - 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 Ξ¨(X)=M(X+Ο(AX)) over π½_q^n, where π½_q is a finite field of q elements with q being a prime power, Mβ GL(n, π½_q), GL(n, π½_q) is the general linear group of order n over π½_q, A_m Γ n is a full-rank matrix over π½_q, and Ο=(Ο_1,Ο_2,β¦,Ο_n) with each component function Ο_i:π½_q^mβπ½_q. Furthermore, we establish criteria for the permutation and complete permutation properties of the mapping F(X)=T(X+B^tf(AX)) over π½_q^n, f: π½_q^m β π½_q^nβm, T β GL(n, π½_q), A_m Γ n and B_(nβm)Γ n are full-rank matrices over π½_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 π½_q^m to π½_q^ nβm can be extended to a bijection over π½_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.
ER -