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# The lower bounds on the second-order nonlinearity of three classes of Boolean functions

• * Corresponding author: Qian Liu

This work was supported by Educational Research Projects of Young and Middle-aged Teachers in Fujian Province (No. JAT200033) and the Talent Fund project of Fuzhou University (No. 0030510858)

• In this paper, by calculating the lower bounds on the nonlinearity of the derivatives of the following three classes of Boolean functions, we provide the tight lower bounds on the second-order nonlinearity of these Boolean functions: (1) $f_1(x) = Tr_1^n(x^{2^{r+1}+2^r+1})$, where $n = 2r+2$ with even $r$; (2) $f_2(x) = Tr_1^n(\lambda x^{2^{2r}+2^{r+1}+1})$, where $\lambda \in \mathbb{F}_{2^r}^*$ and $n = 4r$ with even $r$; (3) $f_3(x,y) = yTr_1^n(x^{2^r+1})+Tr_1^n(x^{2^r+3})$, where $(x, y)\in \mathbb{F}_{2^n}\times \mathbb{F}_2$, $n = 2r$ with odd $r$. The results show that our bounds are better than previously known lower bounds in some cases.

Mathematics Subject Classification: Primary: 94A60, 11T71; Secondary: 06E30.

 Citation: • • Table 1.  The Walsh spectrum of $f$

 $W_f(\omega)$ Number of $\omega$ 0 $2^n-2^{n-k}$ $2^{\frac{n+k}{2}}$ $2^{n-k-1}+(-1)^{f(0)}2^{\frac{n-k-2}{2}}$ $-2^{\frac{n+k}{2}}$ $2^{n-k-1}-(-1)^{f(0)}2^{\frac{n-k-2}{2}}$

Table 2.  Comparison of the lower bound on second-order nonlinearity with the known results

 $n$ bound in  bound in  bound in  bound in  Our new bound in Theorem 3.6 8 62 63 64 38 80 16 28615 24561 28672 26974 29815 24 8125467 7339906 8126464 8017875 8202293 32 2130690153 2013264900 2130706432 2123757059 2135604974 40 548681810337 532575936520 548682072064 548237313548 548996316833

Table 3.  Comparison of the lower bound on second-order nonlinearity with the known results for odd $r$

 $r$ 1 3 5 7 9 11 13 bound in  0 19 662 13487 238971 4008935 65625942 Our new bound in Theorem 3.9 1 32 763 14309 245641 4062737 66058274
• Tables(3)

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