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October 2020, Vol. 63 209101:1–209101:3 https://doi.org/10.1007/s11432-018-9741-6

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Permutation polynomials x2 +3 + ax2 +2 + bx over F22k and their differential uniformity Jie PENG1 , Lijing ZHENG2 , Chunsheng WU3 & Haibin KAN4,5,6,7* 1

Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China; School of Mathematics and Physics, University of South China, Hengyang 421001, China; Department of Mathematics, Lianyungang Normal University, Lianyungang 222006, China; 4 Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, China; 5 Fudan-Zhongan Joint Laboratory of Blockchain and Information Security, Shanghai Blockchain Engineering Research Center, Shanghai 200433, China; 6 Shanghai Institute for Advanced Communication and Data Science, Shanghai 200433, China; 7 Shanghai Institute of Intelligent Electronics & Systems, Shanghai 200433, China 2

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Received 23 September 2018/Accepted 11 January 2019/Published online 25 August 2020 k+1

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+3 Citation Peng J, Zheng L J, Wu C S, et al. Permutation polynomials x2 + ax2 +2 + bx over F22k and their differential uniformity. Sci China Inf Sci, 2020, 63(10): 209101, https://doi.org/10.1007/s11432-018-9741-6

Dear editor, A polynomial f (x) ∈ Fq [x] is called a permutation polynomial (PP) over the finite field Fq if the associated mapping f : c 7→ f (c) from Fq to itself is bijective. A PP f is called a complete permutation polynomial if f (x) + x is a PP. PPs over finite fields of even characteristic have wide applications, including cryptography, coding theory, and communication theory. In many block ciphers with substitution-permutation network structure, the substitution box is usually a PP over F22k for some positive integer k. To resist the differential attacks, the differential uniformity of this polynomial should be as low as possible. However, finding such polynomials is difficult. Even for most known differentially low uniform permutations over F22k , their polynomial representations are difficult to obtain ([1–8]). Thus far, only a few classes of differentially low uniform PPs over F22k of few terms have been constructed. Monomial permutations with Gold exponents 2i + 1, Kasami exponents 22i − 2i + 1, Inverse exponents 2n − 2, and the Bracken-Leander exponents 22k + 2k + 1 are differentially 4-uniform. Moreover, when k > 5 is odd, monomials with exponents 2k+1 + 3 and k+1 2k + 2 2 + 1 were conjectured by Blondeau et al.

and finally verified by Xiong et al. [9] to be differentially 8-uniform permutations. A class of differentially 4-uniform permutation binomial, known as the Bracken-Tan-Tan function, was also determined. However, differentially low uniform permutation trinomials have not been found yet. k+1

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In this study, PPs of the form x2 +3 +ax2 +2 + bx over F22k are determined. A class of permutation monomials, a class of permutation binomials, and two classes of permutation trinomials are obtained. All those monomials and binomials are shown to