Paper 2017/907
On the differential equivalence of APN functions
Anastasiya Gorodilova
Abstract
C.~Carlet, P.~Charpin, V.~Zinoviev in 1998 defined the associated Boolean function $\gamma_F(a,b)$ in $2n$ variables for a given vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to itself. It takes value~$1$ if $a\neq {\bf 0}$ and equation $F(x)+F(x+a)=b$ has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function $G$ in $n$ variables, $n\leq 6$, that is differentially equivalent to a given quadratic APN function $F$, can be represented as $G = F + A$, where $A$ is affine. For the APN Gold function $F$, we completely described all affine functions $A$ such that $F$ and $F+A$ are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial.
Metadata
- Available format(s)
-
PDF
- Publication info
- Published elsewhere. Minor revision. Cryptography and communications
- DOI
- 10.1007/s12095-018-0329-y
- Keywords
- Boolean functionAlmost perfect nonlinear functionAlmost bent functionCrooked functionDifferential equivalenceLinear spectrum
- Contact author(s)
- gorodilova @ math nsc ru
- History
- 2018-09-20: last of 2 revisions
- 2017-09-24: received
- See all versions
- Short URL
- https://ia.cr/2017/907
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2017/907,
author = {Anastasiya Gorodilova},
title = {On the differential equivalence of APN functions},
howpublished = {Cryptology ePrint Archive, Paper 2017/907},
year = {2017},
doi = {10.1007/s12095-018-0329-y},
note = {\url{https://eprint.iacr.org/2017/907}},
url = {https://eprint.iacr.org/2017/907}
}
