Paper 2009/094
On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions
Sugata Gangopadhyay, Sumanta Sarkar, and Ruchi Telang
Abstract
The $r$th order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the ReedMuller code $\mathcal{R}(r, n)$. In this paper we deduce the lower bounds of the second order nonlinearity of the two classes of Boolean functions of the form \begin{enumerate} \item $f_{\lambda}(x) = Tr_1^n(\lambda x^{d})$ with $d=2^{2r}+2^{r}+1$ and $\lambda \in \mathbb{F}_{2^{n}}$ where $n = 6r$. \item $f(x,y)=Tr_1^t(xy^{2^{i}+1})$ where $x,y \in \mathbb{F}_{2^{t}}, n = 2t, n \ge 6$ and $i$ is an integer such that $1\le i < t$, $\gcd(2^t1, 2^i+1) = 1$. \end{enumerate} For some $\lambda$, the first class gives bent functions whereas Boolean functions of the second class are all bent, i.e., they achieve optimum first order nonlinearity.
Metadata
 Available format(s)
 Category
 Secretkey cryptography
 Publication info
 Published elsewhere. Unknown where it was published
 Keywords
 Boolean functionssecond order nonlinearity
 Contact author(s)
 gsugata @ gmail com
 History
 20090317: last of 4 revisions
 20090302: received
 See all versions
 Short URL
 https://ia.cr/2009/094
 License

CC BY
BibTeX
@misc{cryptoeprint:2009/094, author = {Sugata Gangopadhyay and Sumanta Sarkar and Ruchi Telang}, title = {On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions}, howpublished = {Cryptology ePrint Archive, Paper 2009/094}, year = {2009}, note = {\url{https://eprint.iacr.org/2009/094}}, url = {https://eprint.iacr.org/2009/094} }