On the weight distributions of several classes of cyclic codes from APN monomials

Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}_{p}$ are presented via an examination of the condition for the cyclic codes $\mathcal{C}_{(1,d)}$ and $\mathcal{C}_{(1,e)}$, which have parity-check polynomials $m_1(x)m_d(x)$ and $m_1(x)m_e(x)$ respectively, to have the same weight distribution, where $m_i(x)$ is the minimal polynomial of $\pi^{-i}$ over $\mathbb{F}_{p}$ for a primitive element $\pi$ of $\mathbb{F}_{p^m}$. %For $p=3$, the duals of five classes of the proposed cyclic codes are optimal in the sense that they meet certain bounds on linear codes. Furthermore, for $p\equiv 3 \pmod{4}$ and positive integers $e$ such that there exist integers $k$ with $\gcd(m,k)=1$ and $\tau\in\{0,1,\cdots, m-1\}$ satisfying $(p^k+1)\cdot e\equiv 2 p^{\tau}\pmod{p^m-1}$, the value distributions of the two exponential sums T(a,b)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where $s=(p^m-1)/2$, are settled. As an application, the value distribution of $S(a,b,c)$ is utilized to investigate the weight distribution of the cyclic codes $\mathcal{C}_{(1,e,s)}$ with parity-check polynomial $m_1(x)m_e(x)m_s(x)$. In the case of $p=3$ and even $e$ satisfying the above condition, the duals of the cyclic codes $\mathcal{C}_{(1,e,s)}$ have the optimal minimum distance

Experimental relativistic zero-knowledge proofs

Protecting secrets is a key challenge in our contemporary information-based era. In common situations, however, revealing secrets appears unavoidable, for instance, when identifying oneself in a bank to retrieve money. In turn, this may have highly undesirable consequences in the unlikely, yet not unrealistic, case where the bank's security gets compromised. This naturally raises the question of whether disclosing secrets is fundamentally necessary for identifying oneself, or more generally for proving a statement to be correct. Developments in computer science provide an elegant solution via the concept of zero-knowledge proofs: a prover can convince a verifier of the validity of a certain statement without facilitating the elaboration of a proof at all. In this work, we report the experimental realisation of such a zero-knowledge protocol involving two separated verifier-prover pairs. Security is enforced via the physical principle of special relativity, and no computational assumption (such as the existence of one-way functions) is required. Our implementation exclusively relies on off-the-shelf equipment and works at both short (60 m) and long distances (400 m) in about one second. This demonstrates the practical potential of multi-prover zero-knowledge protocols, promising for identification tasks and blockchain-based applications such as cryptocurrencies or smart contracts.Comment: 8 pages, 3 figure

Optimal ternary cyclic codes with minimum distance four and five

Cyclic codes are an important subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics. In this paper, two families of optimal ternary cyclic codes are presented. The first family of cyclic codes has parameters [3m−1,3m−1−2m,4] and contains a class of conjectured cyclic codes and several new classes of optimal cyclic codes. The second family of cyclic codes has parameters [3m−1,3m−2−2m,5] and contains a number of classes of cyclic codes that are obtained from perfect nonlinear functions over F3m, where m > 1 and is a positive integer