International Association for Cryptologic Research

International Association
for Cryptologic Research

CryptoDB

Xuelian Li

Publications

Year
Venue
Title
2010
EPRINT
The Lower Bounds on the Second Order Nonlinearity of Cubic Boolean Functions
Xuelian Li Yupu Hu Juntao Gao
It is a difficult task to compute the $r$-th order nonlinearity of a given function with algebraic degree strictly greater than $r>1$. Even the lower bounds on the second order nonlinearity is known only for a few particular functions. We investigate the lower bounds on the second order nonlinearity of cubic Boolean functions $F_u(x)=Tr(\sum_{l=1}^{m}\mu_{l}x^{d_{l}})$, where $u_{l} \in F_{2^n}^{*}$, $d_{l}=2^{i_{l}}+2^{j_{l}}+1$, $i_{l}$ and $j_{l}$ are positive integers, $n>i_{l}> j_{l}$. Especially, for a class of Boolean functions $G_u(x)=Tr(\sum_{l=1}^{m}\mu_{l}x^{d_{l}})$, we deduce a tighter lower bound on the second order nonlinearity of the functions, where $u_{l} \in F_{2^n}^{*}$, $d_{l}=2^{i_{l}\gamma}+2^{j_{l}\gamma}+1$, $i_{l}> j_{l}$ and $\gamma\neq 1$ is a positive integer such that $gcd(n,\gamma)=1$. \\The lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by $f_\mu(x)=Tr(\mu x^{2^i+2^j+1})$, $\mu\in F_{2^n}^*$, $i$ and $j$ are positive integers such that $i>j$, have recently (2009) been obtained by Gode and Gangopadhvay. Our results have the advantages over those of Gode and Gangopadhvay as follows. We first extend the results from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of $n$. Also, our bounds are better than those of Gode and Gangopadhvay for monomial functions $f_\mu(x)$.

Coauthors

Juntao Gao (1)
Yupu Hu (1)