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The Lower Bounds on the Second Order Nonlinearity of Cubic Boolean Functions

Authors:
Xuelian Li
Yupu Hu
Juntao Gao
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URL: http://eprint.iacr.org/2010/009
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Abstract: 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 $f_\mu(x)$.
BibTeX
@misc{eprint-2010-22910,
  title={The Lower Bounds on the Second Order Nonlinearity of Cubic Boolean Functions},
  booktitle={IACR Eprint archive},
  keywords={secret-key cryptography / cryptography, derivative, the second nonlinearity, trace function, quadratic form},
  url={http://eprint.iacr.org/2010/009},
  note={This paper has not been published elsewhere. xuelian202@163.com 14619 received 10 Jan 2010},
  author={Xuelian Li and Yupu Hu and Juntao Gao},
  year=2010
}