Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-01-13 , DOI: 10.1016/j.disc.2019.111698
Haode Yan; Deng Tang

Boolean functions used in symmetric-key cryptosystems must have high second-order nonlinearity to withstand several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since the maximal second-order nonlinearity of all Boolean functions in $n$ variables equals the covering radius of the Reed–Muller code with length ${2}^{n}$ and order $r$. It is well-known that providing a tight lower bound on the second-order nonlinearity of a general Boolean function with high algebraic degree is a hard task, excepting a few special classes of Boolean functions. In this paper, we improve the lower bounds on the second-order nonlinearity of three classes of Boolean functions of the form ${f}_{i}\left(x\right)={\mathrm{Tr}}_{1}^{n}\left({x}^{{d}_{i}}\right)$ in $n$ variables for $i=1,2$ and 3, where ${\mathrm{Tr}}_{1}^{n}$ denotes the absolute trace mapping from ${\mathbb{F}}_{{2}^{n}}$ to ${\mathbb{F}}_{2}$ and ${d}_{i}$’s are of the form (1) ${d}_{1}={2}^{m+1}+3$ and $n=2m$, (2) ${d}_{2}={2}^{m}+{2}^{\frac{m+1}{2}}+1$, $n=2m$ with odd $m$, and (3) ${d}_{3}={2}^{2r}+{2}^{r+1}+1$ and $n=4r$ with even $r$.

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