Characterizations and constructions of triple-cycle permutations of the form $$x^rh(x^s)$$ x r h ( x s )
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Characterizations and constructions of triple-cycle permutations of the form x r h(x s ) Mengna Wu1,2 · Chengju Li1,2 · Zilong Wang2,3 Received: 22 October 2019 / Revised: 25 May 2020 / Accepted: 27 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let Fq be the finite field with q elements and let f be a permutation polynomial over Fq . Let Sq denote the symmetric group on Fq . In this paper, we mainly investigate some characterizations on the elements f ∈ Sq of order 3, i.e., f ◦ f ◦ f = I , where f is also called a triple-cycle permutation in the literature. Some explicit triple-cycle permutations are constructed. Keywords Permutation polynomial · Triple-cycle permutation · Finite field · Block cipher Mathematics Subject Classification 06E30 · 14G50 · 94A60
1 Introduction Let Fq be the finite field with q elements, where q is a power of a prime. A polynomial f (x) ∈ Fq [x] is called a permutation polynomial if its induced mapping f : c −→ f (c) from Fq to itself is a bijection. Permutation polynomials over finite fields have been extensively studied due to their wide applications in cryptography, coding theory and combinatorial design theory. Therefore, finding new classes of permutations with desired properties is of great interest in both theoretical and applied aspects, and also a challenging task. It is clear that permutation polynomials form a group under composition and reduction modulo x q − x that is isomorphic to the symmetric group Sq on q letters. Therefore, there is a unique
Communicated by O. Ahmadi.
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Chengju Li [email protected] Mengna Wu [email protected] Zilong Wang [email protected]
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Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China
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State Key Laboratory of Integrated Services and Networks, Xi’an 710071, China
3
School of Cyber Engineering, Xidian University, Xi’an 710071, China
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M. Wu et al.
compositional inverse f −1 such that f ◦ f −1 = f −1 ◦ f = I , where I is the identity map. Mullen proposed the problem of computing the coefficients of the inverse polynomial of a permutation polynomial efficiently ([13], Problem 10). The inverse is also useful to investigate the boomerang uniformity of the permutation polynomial [7,11]. It is then an interesting problem to investigate the inverse of a permutation polynomial. In block ciphers, permutation polynomials are often used to design substitution boxes (Sboxes) which build the confusion layer during the encryption process. Further, it is desired that the permutation and its compositional inverse used in the decryption process are efficient in terms of implementation. Let Sq be the symmetric group on Fq . The element f ∈ Sq is called an involution if the order of f is equal to 2, i.e., f = f −1 . Recently, the involutions were extensively investigated (see [5,6,8,22]) due to their applications in block cipher designs [1,3,4]. Inspired by the work of [22], we mainly focus on the elements f ∈ Sq of order 3, i.e., f ◦ f ◦ f =
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