Paper 2019/994
A new family of APN quadrinomials
Lilya Budaghyan, Tor Helleseth, and Nikolay Kaleyski
Abstract
The binomial $B(x) = x^3 + \beta x^{36}$ (where $\beta$ is primitive in $\mathbb{F}_{2^4}$) over $\mathbb{F}_{2^{10}}$ is the first known example of an Almost Perfect Nonlinear (APN) function that is not CCZequivalent to a power function, and has remained unclassified into any infinite family of APN functions since its discovery in 2006. We generalize this binomial to an infinite family of APN quadrinomials of the form $x^3 + a (x^{2^i+1})^{2^k} + b x^{3 \cdot 2^m} + c (x^{2^{i+m}+2^m})^{2^k}$ from which $B(x)$ can be obtained by setting $a = \beta$, $b = c = 0$, $i = 3$, $k = 2$. We show that for any dimension $n = 2m$ with $m$ odd and $3 \nmid m$, setting $(a,b,c) = (\beta, \beta^2, 1)$ and $i = m2$ or $i = (m2)^{1} \mod n$ yields an APN function, and verify that for $n = 10$ the quadrinomials obtained in this way for $i = m2$ and $i = (m2)^{1} \mod n$ are CCZinequivalent to each other, to $B(x)$, and to any other known APN function over $\mathbb{F}_{2^{10}}$.
Note: Minor update to Table 2 and one paragraph added on F13 before Corollary 1.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint. MINOR revision.
 Keywords
 Boolean functionAPNdifferential uniformity
 Contact author(s)
 nikolay kaleyski @ uib no
 History
 20190905: received
 Short URL
 https://ia.cr/2019/994
 License

CC BY
BibTeX
@misc{cryptoeprint:2019/994, author = {Lilya Budaghyan and Tor Helleseth and Nikolay Kaleyski}, title = {A new family of APN quadrinomials}, howpublished = {Cryptology ePrint Archive, Paper 2019/994}, year = {2019}, note = {\url{https://eprint.iacr.org/2019/994}}, url = {https://eprint.iacr.org/2019/994} }