Original scientific paper
https://doi.org/10.21857/ygjwrcdkgy
On known constructions of APN and AB functions and their relation to each other
Marco Calderini
; Department of Informatics, University of Bergen, Bergen 5005, Norway
Lilya Budaghyan
; Department of Informatics, University of Bergen, Bergen 5005, Norway
Claude Carlet
orcid.org/0000-0002-6118-7927
; Department of Informatics, University of Bergen, Bergen 5005, Norway LAGA, University of Paris 8, Saint-Denis 93526, France
Abstract
This work is dedicated to APN and AB functions which are optimal against differential and linear cryptanlysis when used as Sboxes in block ciphers. They also have numerous applications in other branches of mathematics and information theory such as coding theory, sequence design, combinatorics, algebra and projective geometry. In this paper we give an overview of known constructions of APN and AB functions, in particular, those leading to infinite classes of these functions. Among them, the bivariate construction method, the idea first introduced in 2011 by the third author of the present paper, turned out to be one of the most fruitful. It has been known since 2011 that one of the families derived from the bivariate construction contains the infinite families derived by Dillon’s hexanomial method. Whether the former family is larger than the ones it contains has stayed an open problem which we solve in this paper. Further we consider the general bivariate construction from 2013 by the third author and study its relation to the recently found infinite families of bivariate APN functions.
Keywords
Boolean functions; almost perfect nonlinear (APN) functions, almost bent (AB) functions; cryptography
Hrčak ID:
261468
URI
Publication date:
25.8.2021.
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