Constructing New APN Functions Through Relative Trace Functions
Lijing Zheng, Haibin Kan, Yanjun Li, Jie Peng, Deng Tang · IEEE Transactions on Information Theory · 2022
Let$n=2m$. In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered a family of quadrinomials over$\mathbb {F}_{2^{n}}$of the form$x^{3}+a(x^{2^{s}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{s+m}+2^{m}})^{2^{k}}$. They showed that two infinite classes of almost perfect nonlinear (APN) functions belong to this family when$\gcd (6,m)=1$. We observe that these two infinite classes of APN quadrinomials and the infinite class of APN polynomials from the Budaghyan-Carlet family belong to a more general family of polynomials over$\mathbb {F}_{2^{n}} $with the form$f(x)=a{\mathrm{ Tr}}^{n}_{m}(F(x))+a^{2^{m}}{\mathrm{ Tr}}^{n}_{m}(G(x))$, where$a \in \mathbb {F}_{2^{n}}\backslash \mathbb {F}_{2^{m}} $, and both$F$and$G$are quadratic functions over$\mathbb {F}_{2^{n}}$. We characterize when$f(x) $is APN. With the help of our characterization, letting$F(x)=bx^{2^{i}+1} $and$G(x)=cx^{2^{s}+1}$with$b, c\in \mathbb {F}_{2^{n}} $, we obtain an infinite family of APN functions of the form$f(x) $when${\mathrm{ gcd}}(2,m)=1 $and verify that for$n=10 $two APN instances from this infinite family are CCZ-inequivalent to each other, and to any APN function over$\mathbb {F}_{2^{10}} $from the previously known infinite families.