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

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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.

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Acknowledgments

We would like to thank Eric Chen for kindly sending us the generator matrices of the binary \([85,8]\) codes in the database of two-weight codes he maintains and thank Alexander Pott for careful reading of this paper. We also thank the anonymous reviewers for their valuable suggestions which improve this paper. We particularly thank one reviewer for pointing out the relationship between zero-difference balanced functions and optimal constant composition codes, which leads to the results in Remark 5.

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Authors and Affiliations

  1. LAGA, Universities of Paris 8 and Paris 13, CNRS, Saint-Denis Cedex 02, France

    Claude Carlet

  2. Department of Mathematics, University of Paris 8, 2 rue de la liberté, 93526 , Saint-Denis Cedex 02, France

    Claude Carlet

  3. Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada

    Guang Gong & Yin Tan

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  1. Claude Carlet
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  2. Guang Gong
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  3. Yin Tan
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Correspondence to Yin Tan.

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Communicated by G. McGuire.

Appendix

Appendix

The generating polynomial of the field \(\mathbb {F}_{2^8}\) over \(\mathbb {F}_2\) is \(x^8+x^4+x^3+x^2+1\) (see Table 2).

Table 2 Quadratic APN functions on \(\mathbb {F}_{2^8}\) with the form \(G(x^3)\) and \(G|_{C_3}\) is an injection
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Carlet, C., Gong, G. & Tan, Y. Quadratic zero-difference balanced functions, APN functions and strongly regular graphs. Des. Codes Cryptogr. 78, 629–654 (2016). https://doi.org/10.1007/s10623-014-0022-x

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