Planar functions and perfect nonlinear monomials over finite fields

Abstract

Abstract. The study of finite projective planes involves planar functions, namely, functions f : Fq → Fq such that, for each a ∈ F * q , the function c → f (c + a) − f (c) is a bijection on Fq. Planar functions are also used in the construction of DES-like cryptosystems, where they are called perfect nonlinear functions. We determine all planar functions on Fq of the form c → c t , under the assumption that q ≥ (t − 1) 4 . This implies two recent conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci from 1971 concerning monomial hyperovals in finite Desarguesian projective planes.

Exceptional planar polynomials

“…The polynomial (2) is planar on F p r for odd r/ gcd(k, r) [4], the polynomial (3) is planar on F 3 r for gcd(k, r) = 1 [4], and the polynomial (4) is planar on F 3 rn for odd r [7] (see also [4] for the case u = −1). As shown in two papers by Leducq [11] and Zieve [18], up to EA-equivalence, the polynomials (2) and (3) are the only exceptional planar monomials.…”

“…In this paper, we study polynomials f ∈ F q [X] that induce planar functions on F q r for infinitely many r; a polynomial f satisfying this property will be called an exceptional planar polynomial. Exceptional planar monomials have been completely classified by Leducq [11] and Zieve [18] in Date: 14 February 2014. 2010 Mathematics Subject Classification.…”

Low-degree planar polynomials over finite fields of characteristic two

“…This approach seems to be first used by Janwa and Wilson [15] for monomial APN functions and by Aubry, McGuire, and Rodier [1] for general APN functions. Besides the classification problem for APN functions, classification problems for other combinatorial objects have been attacked with this method, for example for planar functions in odd characteristic [6,17,26], hyperovals [5,13,26], and maximum scattered linear sets [2]. However a complete classification, as in Corollary 1.2, has been obtained so far only in one other case, namely in the classification problem for polynomials that induce hyperovals in finite Desarguesian planes [5].…”

Exceptional scattered polynomials

“…Later these ideas were developed by Jedlicka [26] and Hernando and McGuire [22]. The same approach has been applied in [18] to prove a conjecture on monomial hyperovals and in [32] to get partial results towards the classification of monomial planar functions for infinitely many n, which was later completely solved by Zieve [48] by using the classification of indecomposable exceptional (permutation) polynomials. Similar results and approaches can also be found in [6,7,8,44,45].…”