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 EAequivalence contains the first infinite family of functions, whose differential equivalence class is nontrivial.
Metadata
 Available format(s)
 Publication info
 Published elsewhere. Minor revision. Cryptography and communications
 DOI
 10.1007/s120950180329y
 Keywords
 Boolean functionAlmost perfect nonlinear functionAlmost bent functionCrooked functionDifferential equivalenceLinear spectrum
 Contact author(s)
 gorodilova @ math nsc ru
 History
 20180920: last of 2 revisions
 20170924: 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/s120950180329y}, note = {\url{https://eprint.iacr.org/2017/907}}, url = {https://eprint.iacr.org/2017/907} }