# On APN exponents, characterizations of differentially uniform functions by the Walsh transform, and related cyclic-difference-set-like structures

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## Abstract

In this paper, we summarize the results obtained recently in three papers on differentially uniform functions in characteristic 2, and presented at the workshop WCC 2017 in Saint-Petersburg, and we give new results on these functions. Firstly, we recall the recent connection between almost perfect nonlinear (APN) power functions and the two notions in additive combinatorics of Sidon sets and sum-free sets; we also recall a characterization of APN exponents which leads to a property of Dickson polynomials in characteristic 2 previously unobserved, which is generalizable to all finite fields. We also give a new characterization of APN exponents in odd dimension by Singer sets. Secondly, after recalling the recent multiple generalization to differentially \(\delta \)-uniform functions of the Chabaud–Vaudenay characterization of APN functions by their Walsh transforms, we generalize the method to all criteria on vectorial functions dealing with the numbers of solutions of equations of the form \(\sum _{i\in I}F(x+u_{i,a})+L_a(x)+u_a=0\), with \(L_a\) linear; we give the examples of injective functions and of o-polynomials; we also deduce a generalization to differentially \(\delta \)-uniform functions of the Nyberg characterization of APN functions by means of the Walsh transforms of their derivatives. Thirdly, we recall the two notions of componentwise APNness (CAPNness) and componentwise Walsh uniformity (CWU). We recall why CAPN functions can exist only if *n* is odd and why crooked functions (in particular, quadratic APN functions) are CWU. We also recall that the inverse of one of the Gold permutations is CWU and not the others. Another potential class of CWU functions is that of Kasami functions. We consider the difference sets with Singer parameters equal to the complement of \(\varDelta _F=\{F(x)+F(x+1)+1; x\in \mathbb {F}_{2^n}\}\) where *F* is a Kasami function. These sets have another potential property, called the cyclic-additive difference set property, which is related to the CWU property in the case of power permutations (*n* odd). We study cyclic-additive difference sets among Singer sets. We recall the main properties of Kasami functions and of the related set \(\varDelta _F\) shown by Dillon and Dobbertin and we observe and prove new expressions for \(\varDelta _F\).

## Keywords

Boolean function Vectorial function Walsh–Hadamard transform APN function Kasami function Cyclic difference set## Notes

### Acknowledgements

The author is grateful to Stjepan Picek for his kind help with computer investigations.

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