Two New Families of Quadratic APN Functions
Kangquan Li, Yue Zhou, Chunlei Li, Longjiang Qu · IEEE Transactions on Information Theory · 2022
In this paper, we present two new families of APN functions. The first family is in bivariate form$\big (x^{3}+xy^{2}+ y^{3}+xy, x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2} \big)\,\,\vphantom {_{\int _{\int }}}$over${\mathbb F}_{2^{m}}^{2}$. It is obtained by adding certain terms of the form$\sum _{i}(a_{i}x^{2^{i}}y^{2^{i}},b_{i}x^{2^{i}}y^{2^{i}})$to a family of APN functions recently proposed by Gölo&gcaron;lu. The$\vphantom {_{\int _{\int }}}$second family has the form$L(z)^{2^{m}+1}+vz^{2^{m}+1}$over${\mathbb F}_{{2^{3m}}}$, which generalizes a family of APN functions by Bracken et al. from 2011. By calculating the$\Gamma $-rank of the constructed APN functions over${\mathbb F}_{2^{8}}$and${\mathbb F}_{2^{9}}$, we demonstrate that the two families are CCZ-inequivalent to all known families. In addition, the two new families cover two known sporadic APN instances over${\mathbb F}_{2^{8}}$and${\mathbb F}_{2^{9}}$, which were found by Edel and Pott in 2009 and by Beierle and Leander in 2021, respectively.