On CCZ-equivalence between a family of brivariate APN polynomials and power functions

抄録

APN functions provide the optimal resistance to differential attacks. In 2022, Li et al. [IEEE TIT 68(7), 4761-4769 (2022)] constructed an infinite family of quadratic APN functions over $\mathbb{F}_{2^{2m}}$ with $\gcd(3,m)=1$ in the bivariate form $F(x,y)=(x^{3}+xy^{2}+y^{3}+xy,x^{5}+x^{4}y+y^{5}+xy+x^{2}y^{2})$. In this work, we theoretically prove that functions in a more general form $F'(x,y)=(x^{2^k+1}+xy^{2^k}+y^{2^k+1}+\sum_{i=0}^{k-1}(xy)^{2^i}, x^{2^{2k}+1}+x^{2^{2k}}y+y^{2^{2k}+1}+\sum_{i=0}^{k-1}(xy+(xy)^{2^k})^{2^i})$ are CCZ-inequivalent to APN power functions on $\mathbb{F}_{2^{2m}}$ with $\gcd(3k,m)=1$.