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Recently, a new concept called multiplicative differential cryptanalysis and the corresponding $c$-differential uniformity were introduced by Ellingsen et al.~\cite{Ellingsen2020}, and then some low differential uniformity functions were…
It is shown that a class of optical physical unclonable functions (PUFs) can be learned to arbitrary precision with arbitrarily high probability, even in the presence of noise, given access to polynomially many challenge-response pairs and…
Boolean functions are mathematical objects used in diverse applications. Different applications also have different requirements, making the research on Boolean functions very active. In the last 30 years, evolutionary algorithms have been…
Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ is perfectly nonlinear over $\mathbb{F}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in\mathbb{N}$. First, we study singularities of…
The notion of o-polynomial comes from finite projective geometry. In 2011 and later, it has been shown that those objects play an important role in symmetric cryptography and coding theory to design bent Boolean functions, bent vectorial…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…
We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dam's oracle interrogation is…
Polynomial Networks (PNs) have demonstrated promising performance on face and image recognition recently. However, robustness of PNs is unclear and thus obtaining certificates becomes imperative for enabling their adoption in real-world…
Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Card-based protocols to compute various Boolean functions have been developed. As each…
We give a large family of almost perfect nonlinear (APN) permutations of finite vector spaces of every odd dimension divisible by three. We also give APN functions that are not bijective on even dimensions and related highly nonlinear…
We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomial functions of odd degree are not exceptional,…
Logical models have been successfully used to describe regulatory and signaling networks without requiring quantitative data. However, existing data is insufficient to adequately define a unique model, rendering the parametrization of a…
Boolean functions have very nice applications in cryptography and coding theory, which have led to a lot of research focusing on their applications. The objective of this paper is to construct binary linear codes with few weights from the…
The approximate non-deterministic degree of a Boolean function $f$, denoted $\mathsf{ndeg}_\epsilon(f)$ (written $\mathsf{N}_\epsilon(f)$ for brevity), is the minimum degree of a real polynomial $p$ such that $0 \le |p(x)| \le \epsilon$…
We study the relation among some security parameters for vectorial Boolean functions which prevent attacks on the related block cipher. We focus our study on a recently-introduced security criterion, called weak differential uniformity,…
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions…
In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…
It is already shown that a Boolean function for a NP-complete problem can be computed by a polynomial-sized circuit if its variables have enough number of automorphisms. Looking at this previous study from the different perspective gives us…
The approximate degree of a Boolean function is the minimum degree of real polynomial that approximates it pointwise. For any Boolean function, its approximate degree serves as a lower bound on its quantum query complexity, and generically…
We study combinatorial properties of plateaued functions $F \colon \mathbb{F}_p^n \rightarrow \mathbb{F}_p^m$. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on…