Generalized isotopic shift construction for APN functions
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Generalized isotopic shift construction for APN functions Lilya Budaghyan1 · Marco Calderini1 Irene Villa1
· Claude Carlet1,2 · Robert Coulter3 ·
Received: 18 March 2020 / Revised: 24 June 2020 / Accepted: 8 September 2020 © The Author(s) 2020
Abstract In this work we give several generalizations of the isotopic shift construction, introduced recently by Budaghyan et al. (IEEE Trans Inform Theory 66:5299–5309, 2020), when the initial function is a Gold function. In particular, we derive a general construction of APN functions which covers several unclassified APN functions for n = 8 and produces fifteen new APN functions for n = 9. Keywords APN functions · Isotopic shift · Vectorial Boolean functions Mathematics Subject Classification 94A60 · 11T71 · 06E30
1 Introduction For n a positive integer, let F2n be the finite field with 2n elements. By F2n we denote the multiplicative group of F2n and, throughout the paper, ζ denotes one of its primitive elements,
Communicated by A. Pott. Parts of this work were presented at WCC 2019: The Eleventh International Workshop on Coding and Cryptography.
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Marco Calderini [email protected] Lilya Budaghyan [email protected] Claude Carlet [email protected] Robert Coulter [email protected] Irene Villa [email protected]
1
Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway
2
LAGA, University of Paris 8, Saint-Denis, France
3
Department of Mathematical Sciences, University of Delaware, Newark, DE, USA
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L. Budaghyan et al.
so that F2n = ζ = {1, ζ, ζ 2 , ζ 3 , . . . , ζ 2 −2 }. An (n, n)-function is a map from F2n to itself. Such function admits a unique representation as a univariate polynomial of degree at most 2n − 1, that is n
F(x) =
n −1 2
ajx j,
a j ∈ F2 n .
j=0
The kernel of F is defined as ker(F) = {u ∈ F2n s.t. F(u) = 0}. The function F is n−1 i ci x 2 ; – linear if F(x) = i=0 – affine if it is the sum of a linear function and a constant; i j – DO (Dembowski-Ostrom) polynomial if F(x) = 0≤i< j 2), i.e. differentially 1-uniform functions. Also here, given a planar function, it is possible to obtain an inequivalent planar function from its isotopic shifts. In the present paper we further study the isotopic shift construction over fields of even characteristic. Firstly, we verify that, over F26 , any quadratic APN map can be obtained as an isotopic shift of any other quadratic APN map. Then, we consider different generalizations of the isotopic shift construction when the initial function is a monomial with a Gold exponent. i In [6], we studied the APN property of the isotopic shift of Gi (x) = x 2 +1 over F2n , with n = km, given by i
i
Gi,L (x) = x L(x)2 + x 2 L(x),
(2)
k−1 im Ai x 2 for some Ai ∈ F2n . This conwhere L is a 2m -polynomial, that is L(x) = i=0 struction provides a new APN function over F29 . i i In the present work, we study the APN property of x L 1 (x)2 + x 2 L 2 (x) where both L 1 and L 2 are 2m -polynomials. From this construction, we obtain fifteen new APN functions for n = 9. M
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