On a generalization of planar functions
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On a generalization of planar functions Daniele Bartoli1
· Marco Timpanella2
Received: 6 January 2019 / Accepted: 10 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Planar functions have deep applications in different areas of mathematics. We give a generalization of planar functions, and we obtain construction and classification results concerning these new objects. More in detail, for a prime power q = pr and β ∈ Fq \{0, 1}, we call a function f : Fq → Fq β-planar function in Fq if for each γ ∈ Fq we have that f (x + γ ) − β f (x) permutes Fq . In this work, we study β-planar monomials: We provide some necessary conditions for general r , whereas for r ≤ 3 we construct examples of β-planar monomials. Connections with algebraic curves are used to prove nonexistence results. Keywords Planar functions · β-planar polynomials · Finite fields
1 Introduction Let q = pr , p a prime, r a positive integer, be a prime power. A planar function is a function f : Fq → Fq such that for every a ∈ Fq∗ , the function D f c → f (c + a) − f (c)
(1.1)
is a bijection on Fq . Planar functions have deep applications in different areas of mathematics: They can be used to construct finite projective planes [9], they are useful in DES-like cryptosystems [29], and they are connected with relative difference sets [15]. Carlet, Ding, and Yuan [4,10,33], among others, used planar functions to construct error-correcting codes, which are then employed to design secret sharing schemes.
B
Daniele Bartoli [email protected] Marco Timpanella [email protected]
1
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy
2
Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Potenza, Italy
123
Journal of Algebraic Combinatorics
Planar functions are also applied to the construction of authentication codes [11], constant composition codes [13], and signal sets [12]. Besides, planar functions induce many combinatorial objects such as skew Hadamard difference sets and Paley type partial difference sets [34]. Since for q even the function D f in (1.1) can never be a permutation, the concept of almost perfect nonlinear functions (APN for short) has been introduced. Namely, a function is APN if the corresponding D f is 2-to-1. Although APN functions have relevant connections with the construction of S-boxes in block ciphers [29], there is no apparent link between APN functions and projective planes. For a comprehensive survey of the state of the art of APN functions, the reader is referred to [3,8,30]. A new definition of planar functions in even characteristic has been recently proposed by Zhou [37]: As shown in [31,37], such planar functions have similar properties and applications as their odd characteristic counterparts. In the literature, the planarity of monomial functions f (x) = x t , with t > 0, has been mostly investigated; see [25, Theorem 4.6] for the case F p , [5, Theorem 2.1] for the case F p2 , an
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