Bracken, Carl and Byrne, Eimear and McGuire, Gary and Nebe, Gabriele (2011) On the Equivalence of Quadratic APN Functions. Designs, Codes and Cryptography, 61 (3). pp. 261-272. ISSN 0925-1022

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Abstract

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function x2r+1 if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2n where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.

Item Type:	Article
Additional Information:	Preprint of published article. The original publication is available at www.springerlink.com. Research supported by the Claude Shannon Institute, Science Foundation Ireland Grant 06/MI/006 and the Irish Research Council for Science, Engineering and Technology
Keywords:	almost perfect nonlinear; APN; automorphism group; CCZ-equivalence; EA-equivalence; Gold function;
Academic Unit:	Faculty of Science and Engineering > Mathematics and Statistics
Item ID:	2690
Identification Number:	https://doi.org/10.1007/s10623-010-9475-8
Depositing User:	Hamilton Editor
Date Deposited:	01 Sep 2011 11:27
Journal or Publication Title:	Designs, Codes and Cryptography
Publisher:	Springer
Refereed:	No
Funders:	Science Foundation Ireland, Irish Research Council for Science, Engineering and Technology
URI:	http://www.springer.com
Use Licence:	This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

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