cryptographic function designers

Abstract: A subset $S \subseteq \mathbb{F}_2^n$ is called a Sidon set if no four distinct points of $S$ have zero sum. It is shown that if $n \geq 7$ and $n \equiv 3 \mod 4$, then $|S|\leq 2^{\frac{n+1}{2}}-3$. As a consequence, for any even $t \geq 4$, there does not exist a binary linear $[2^t-3,2^t-2t-2,5]$-code, strengthening a nonexistence result of Brouwer and Tolhuizen from 1993. As a second consequence, we also show that for an even integer $n \geq 4$, every almost perfect nonlinear (APN) function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ has nonlinearity at least $3$.

Abstract: We prove the following inequality for Boolean functions: $2\sum_{x\in F_2^n}f(x)wt(x)\geq wt(f)(n-deg(f))$. Using this inequality, we establish the Fourier Entropy-Influence (FEI) conjecture for Boolean plateaued functions. In particular, we show that the sharp FEI constant for the class of plateaued functions is 4. We also prove the FEI conjecture for partially bent functions and show that the corresponding sharp constant is 2. Finally, we derive several estimates for the p-biased distribution on the Boolean hypercube. Keywords: Fourier entropy, total influence, average sensitivity, plateaued function, algebraic degree, Reed-Muller code, p-biased distribution.