Title: Cryptographic analysis and construction of complete permutations using a recursive approach

Authors: Shuang Xiang; Yingqi Tang; Yan Tong; Jinzhou Huang

Addresses: Data and Information Technology Engineering Center, ChangJiang Water Resources and Hydropower Development Group Co., Ltd., Wuhan, 430062, China; College of Informatics, Huazhong Agricultural University, Wuhan, 430070, China; School of Computer and Information Engineering, Hubei Normal University, Huangshi, 435002, China ' Data and Information Technology Engineering Center, ChangJiang Water Resources and Hydropower Development Group Co., Ltd., Wuhan, 430062, China ' College of Informatics, Huazhong Agricultural University, Wuhan, 430070, China; Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, 430062, China ' School of Computer Engineering, Hubei University of Arts and Science, Xiangyang, 441053, China

Abstract: This paper proposes a novel recursive approach for constructing generalised complete permutations over finite fields and analyses their cryptographic properties. The method can generate complete, strong complete, K-strong complete, and S-complete permutations by recursively combining component permutations and arbitrary mappings. Compared to prior recursive techniques, the approach provides larger classes of permutations with superior cryptographic strengths like higher algebraic degree, etc. Algebraic degree, nonlinearity, differential/boomerang uniformity, and other properties are investigated. For instance, constructed complete permutations can achieve optimal algebraic degrees to resist structural attacks. The analysis also derives tight lower and upper bounds on (c-)differential uniformity, (c-)boomerang uniformity and nonlinearity. Results demonstrate improved algebraic degree, differential and boomerang uniformities over previous recursive methods. Overall, this work makes significant contributions around complete permutation generation and analysis in cryptography.

Keywords: cryptography; complete permutation; algebraic degree; (c-)differential uniformity; (c-)boomerang uniformity; nonlinearity.

DOI: 10.1504/IJAHUC.2024.136833

International Journal of Ad Hoc and Ubiquitous Computing, 2024 Vol.45 No.2, pp.100 - 122

Received: 10 Aug 2023
Accepted: 16 Oct 2023
Published online: 22 Feb 2024 *

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