Distribution of the absolute indicator of random Boolean functions

On the Differential-Linear Connectivity Table of Vectorial Boolean Functions

Vectorial Boolean functions are crucial building-blocks in symmetric ciphers. Different known attacks on block ciphers have resulted in diverse cryptographic criteria for vectorial Boolean functions, such as differential uniformity and nonlinearity. Very recently, Bar-On et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers $E_0$ and $E_1$ involved in differential-linear attacks. This new notion leads to significant improvements of differential-linear attacks on seve…

The "Most informative boolean function" conjecture holds for high noise

We prove the "Most informative boolean function" conjecture of Courtade and Kumar for high noise $ε\ge 1/2 - δ$, for some absolute constant $δ> 0$. Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by flipping each coordinate of $X$ independently with probability $ε$, then, provided $ε\ge 1/2 - δ$, for any boolean function $f$ holds $I(f(X);Y) \le 1 - H(ε)$. This conjecture was previously known to hold only for balanced functions.

几乎每个布尔函数都具有中心阈值次数

Almost every Boolean function has central threshold degree

该研究证明几乎每个n元布尔函数的阈值次数为⌊n/2⌋或⌈n/2⌉，例外概率极低，通过关联多项式符号表示与Reed-Muller码恢复完成证明，为布尔函数阈值次数研究提供关键结果。

On the entropy of a noisy function

Let $0 < ε< 1/2$ be a noise parameter, and let $T_ε$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of $T_ε f$ by the average entropy of conditional expectations of $f$, given sets of roughly $(1-2ε)^2 \cdot n$ variables. In information-theoretic terms, we prove the following strengthening of "Mrs. Gerber's lemma": Let $X$ be a random binary vector of length $n$, and let $Z$ be a noise vector, corresponding to a binary symmetric channel with crossover probability $ε$. Then, setting $v = (1-…