Paper 2017/1179
On the exponents of APN power functions and Sidon sets, sumfree sets, and Dickson polynomials
Claude Carlet and Stjepan Picek
Abstract
We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sumfree sets, on those exponents $d\in {\mathbb Z}/(2^n1){\mathbb Z}$ which are such that $F(x)=x^d$ is an APN function over ${\mathbb F}_{2^n}$ (which is an important cryptographic property). We study to which extent these new conditions may speed up the search for new APN exponents $d$. We also show a new connection between APN exponents and Dickson polynomials: $F(x)=x^d$ is APN if and only if the reciprocal polynomial of the Dickson polynomial of index $d$ is an injective function from $\{y\in {\Bbb F}_{2^n}^*; tr_n(y)=0\}$ to ${\Bbb F}_{2^n}\setminus \{1\}$. This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint. MINOR revision.
 Contact author(s)
 stjepan @ computer org
 History
 20171208: received
 Short URL
 https://ia.cr/2017/1179
 License

CC BY
BibTeX
@misc{cryptoeprint:2017/1179, author = {Claude Carlet and Stjepan Picek}, title = {On the exponents of APN power functions and Sidon sets, sumfree sets, and Dickson polynomials}, howpublished = {Cryptology ePrint Archive, Paper 2017/1179}, year = {2017}, note = {\url{https://eprint.iacr.org/2017/1179}}, url = {https://eprint.iacr.org/2017/1179} }