On the Walsh spectra of quadratic APN functions
For cryptographers designing block ciphers, this work provides new theoretical constraints on the linearity of optimal differential-resistant functions.
The paper establishes novel connections between the Walsh spectra of quadratic APN functions and vector space partitions/blocking sets, proving that such functions can have at most one component with amplitude larger than 2^{3n/4} and providing the first nontrivial upper bound on the number of bent components.
APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function $F$ operating on $n=2k$ bits is uniquely associated with a vector space partition of $\mathbb{F}_2^n$ and a specific blocking set in the corresponding projective space $PG(n-1,2)$. These connections allow us to prove a variety of results on the Walsh spectrum of $F$. We prove for instance that $F$ can have at most one component function of amplitude larger than $2^{3n/4}$. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and provide conditions for a function to be CCZ-equivalent to a permutation based on its number of bent components.