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Inverse function is not component-wise uniform ¨ glu ˘ 1 Faruk Golo Received: 27 September 2019 / Accepted: 1 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract Recently, Carlet introduced (Carlet Finite. Fields Appl. 53, 226–253 (2018)) the concept of component-wise uniform (CWU) functions which is a stronger notion than being almost perfect nonlinear (APN). Carlet showed that crooked functions (in particular quadratic functions, including Gold functions) and the compositional inverse of a specific Gold function are CWU. Carlet also compiled a table on the component-wise uniformity of APN power functions, where for the extension degrees less than 12, the APN power functions which are always CWU are Gold, the compositional inverse of the specific Gold function and Kasami. In this paper, we show that the Inverse function is not CWU if n > 5. We also show that any additive shift of a derivative of the Inverse function is not a cyclic difference set. Keywords Boolean functions · APN functions · Cyclic difference sets Mathematics Subject Classification (2010) 94A60 · 11T71 · 06E30

1 Introduction A vectorial Boolean function F : Fq → Fq = F, q = 2n is said to be almost perfect nonlinear (APN) if the nonzero derivatives DF (α) = {F (x) + F (x + α) : x ∈ F} with respect to α ∈

F∗

have maximal cardinality q/2. This implies

F (c) + F (x + c) + F (y + c) + F (x + y + c) = 0

(1)

if and only if at least one of x = 0, y = 0, x = y holds. The Walsh transform of F then can be used to characterize APNness of F , which is defined by  (a, b) = F χ (bF (x) + ax), x∈F

 Faruk G¨olo˘glu

[email protected] 1

Department of Algebra, Charles University, Prague, Czech Republic

Cryptography and Communications

n−1 2i for all a, b ∈ F and χ (x) = (−1)tr(x) and tr(x) = i=0 x . The number of zeroes of (1) can be quantified by the fourth moment of the Walsh transform. We have, F is APN if and only if  (u, v))4 = 23n+1 (2n − 1). (F u∈F,v∈F∗

Carlet shows [4] that another more complex but also useful characterization is possible. He shows that F is APN if and only if  (u1 , v1 ))2 (F (u2 , v2 ))2 (F (u1 + u2 , v1 + v2 ))2 = 25n (2n − 1)(2n − 2). (F u1 ,u2 ,v1 ,v2 ∈F v1 =0,v2 =0,v1 =v2

Note that both numbers appearing on the right hand sides of the last two equalities are divisible by the number of possible v’s (i.e., q − 1 for the first one and (q − 1)(q − 2) for the second). The functions F which are uniform among those v’s is the topic of the paper [4]. The following two definitions are based on these observations. Definition 1 The vectorial Boolean function F : F → F is said to be component-wise APN (CAPN) if for any v ∈ F∗ ,  (u, v))4 = 23n+1 . (F u∈F

This notion was introduced by Berger et al. in [2]; naming is due to Carlet [4]. The second characterization introduced in [4] corresponds to the notion of component-wise uniformity. Definition 2 The vectorial Boolean function F : F → F is said to be component-wise uniform (CWU) if for any v1 , v2 ∈ F∗ with v1  = v2 ,  (u1 ,