On the Equivalence of Quadratic APN Functions

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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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: DOI: 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

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