Paper 2022/1059
Classification of all DO planar polynomials with prime field coefficients over GF(3^n) for n up to 7
Abstract
We describe how any function over a finite field $\mathbb{F}_{p^n}$ can be represented in terms of the values of its derivatives. In particular, we observe that a function of algebraic degree $d$ can be represented uniquely through the values of its derivatives of order $(d1)$ up to the addition of terms of algebraic degree strictly less than $d$. We identify a set of elements of the finite field, which we call the degree $d$ extension of the basis, which has the property that for any choice of values for the elements in this set, there exists a function of algebraic degree $d$ whose values match the given ones. We discuss how to reconstruct a function from the values of its derivatives, and discuss the complexity involved in transitioning between the truth table of the function and its derivative representation. We then specialize to the case of quadratic functions, and show how to directly convert between the univariate and derivative representation without going through the truth table. We thus generalize the matrix representation of qaudratic vectorial Boolean functions due to Yu et al. to the case of arbitrary characteristic. We also show how to characterize quadratic planar functions using the derivative representation. Based on this, we adapt the method of Yu et al. for searching for quadratic APN functions with prime field coefficients to the case of planar DO functions. We use this method to find all such functions (up to CCZequivalence) over $\mathbb{F}_{3^n}$ for $n \le 7$. We conclude that the currently known planar DO polynomials cover all possible cases for $n \le 7$. We find representatives simpler than the known ones for the ZhouPott, Dickson, and LunardonMarinoPolverinoTrombettiBierbrauer families for $n = 6$. Finally, we discuss the computational resources that would be needed to push this search to higher dimensions.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint.
 Keywords
 planar function APN differential uniformity derivative representation classification
 Contact author(s)

davidovadiana210 @ gmail com
nikolay kaleyski @ gmail com  History
 20220817: approved
 20220815: received
 See all versions
 Short URL
 https://ia.cr/2022/1059
 License

CC BY
BibTeX
@misc{cryptoeprint:2022/1059, author = {Diana Davidova and Nikolay Kaleyski}, title = {Classification of all DO planar polynomials with prime field coefficients over GF(3^n) for n up to 7}, howpublished = {Cryptology ePrint Archive, Paper 2022/1059}, year = {2022}, note = {\url{https://eprint.iacr.org/2022/1059}}, url = {https://eprint.iacr.org/2022/1059} }