Designs, Codes and Cryptography

, Volume 78, Issue 3, pp 629–654 | Cite as

Quadratic zero-difference balanced functions, APN functions and strongly regular graphs

Article

Abstract

Let \(F\) be a function from \(\mathbb {F}_{p^n}\) to itself and \(\delta \) a positive integer. \(F\) is called zero-difference \(\delta \)-balanced if the equation \(F(x+a)-F(x)=0\) has exactly \(\delta \) solutions for all nonzero \(a\in \mathbb {F}_{p^n}\). As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over \(\mathbb {F}_{2^n}\) are zero-difference 2-balanced. In this paper, we study the relationship between this notion and differential uniformity; we show that all quadratic zero-difference \(\delta \)-balanced functions are differentially \(\delta \)-uniform and we investigate in particular such functions with the form \(F=G(x^d)\), where \(\gcd (d,p^n-1)=\delta +1\) and where the restriction of \(G\) to the set of all nonzero \((\delta +1)\)th powers in \(\mathbb {F}_{p^n}\) is an injection. We introduce new families of zero-difference \(p^t\)-balanced functions. More interestingly, we show that the image set of such functions is a regular partial difference set, and hence yields strongly regular graphs; this generalizes the constructions of strongly regular graphs using planar functions by Weng et al. Using recently discovered quadratic APN functions on \(\mathbb {F}_{2^8}\), we obtain \(15\) new \((256, 85, 24, 30)\) negative Latin square type strongly regular graphs.

Keywords

Zero-difference balanced functions Almost perfect nonlinear functions Strongly regular graphs 

Mathematics Subject Classification

11T06 11T71 05E30 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.LAGA, Universities of Paris 8 and Paris 13, CNRSSaint-Denis Cedex 02France
  2. 2.Department of MathematicsUniversity of Paris 8Saint-Denis Cedex 02France
  3. 3.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada

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