The Maximum Likelihood Decoding Problem (MLD) and the Multivariate Quadratic System Problem (MQ) are known to be NP-hard. In this paper we present a polynomial-time reduction from any instance of MLD to an instance of MQ, and viceversa.
In this paper we extend the Multidimensional Byzantine Agreement (MBA) Protocol arXiv:2105.13487v2, a leaderless Byzantine agreement for vectors of arbitrary values, into the emph{Cob} protocol, that works in Asynchronous Gossiping (AG) networks. This generalization allows the consensus process to be run by an incomplete network of nodes provided with (non-synchronized) same-speed clocks. Not all nodes are active in every step, so the network size does not hamper the efficiency, as long as the gossiping broadcast delivers the messages to every node in reasonable time. These network assumptions model more closely real-life communication channels, so the Cob protocol may be applicable to a variety of practical problems, such as blockchain platforms implementing sharding. The Cob protocol has the same Bernoulli-like distribution that upper bounds the number of steps required as the MBA protocol, and we prove its correctness and security assuming a supermajority of honest nodes in the network.
Shors shockingly fast quantum algorithm for solving the period-finding problem is a threat for the most common public-key primitives, as it can be efficiently applied to solve both the Integer Factorisation Problem and the Discrete Logarithm Problem. In other words, many once-secure protocols have to be replaced by still-secure alternatives. Instead of relying, for example, on the RSA protocol, the Diffie-Hellman key-exchange or the (Elliptic Curve) Digital Signature Algorithm, many researchers moved their attention to the design and analysis of primitives which are yet to be broken by quantum algorithms. The urgency of the threat imposed by quantum computers led the U.S. National Institute of Standards and Technology (NIST) to open calls for both Post-Quantum Public-Keys Exchange Algorithms and Post-Quantum Digital Signature Algorithms. In this brief survey we focus on the round 3 finalists and alternate candidates for Digital Signatures: CRYSTALS-DILITHIUM, FALCON, Rainbow, SPHINCS+, GeMSS, Picnic.
In this paper we will present the Multidimensional Byzantine Agreement (MBA) Protocol, a leaderless Byzantine agreement protocol defined for complete and synchronous networks that allows a network of nodes to reach consensus on a vector of relevant information regarding a set of observed events. The consensus process is carried out in parallel on each component, and the output is a vector whose components are either values with wide agreement in the network (even if no individual node agrees on every value) or a special value $bot$ that signals irreconcilable disagreement. The MBA Protocol is probabilistic and its execution halts with probability 1, and the number of steps necessary to halt follows a Bernoulli-like distribution. The design combines a Multidimensional Graded Consensus and a Multidimensional Binary Byzantine Agreement, the generalization to the multidimensional case of two protocols by Micali and Feldman. We prove the correctness and security of the protocol assuming a synchronous network where less than a third of the nodes are malicious.
A $(t,n)-$ threshold signature scheme enables distributed signing among $n$ players such that any subgroup of size $t$ can sign, whereas any group with fewer players cannot. Our goal is to produce signatures that are compatible with an existing centralized signature scheme: the key generation and signature algorithm are replaced by a communication protocol between the parties, but the verification algorithm remains identical to that of a signature issued using the centralized algorithm. Starting from the threshold schemes for the ECDSA signature due to R. Gennaro and S. Goldfeder, we present the first protocol that supports multiparty signatures with an offline participant during the Key Generation Phase, without relying on a trusted third party. Following well-established approaches, we prove our scheme secure against adaptive malicious adversaries.
We introduce the concept of spread of a code, and we specialize it to the case of maximum weight spectrum (MWS) codes. We classify two newly-defined sub-families of MWS codes according to their weight distributions, and completely describe their fundamental parameters. We focus on one of these families, the strictly compact MWS codes, proving their optimality as MWS codes and linking them to known codes.
The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singletons defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.
We lay the foundations for a blockchain scheme, whose consensus is reached via a proof of work algorithm based on the solution of consecutive discrete logarithm problems over the point group of elliptic curves. In the considered architecture, the curves are pseudorandomly determined by block creators, chosen to be cryptographically secure and changed every epoch. Given the current state of the chain and a prescribed set of transactions, the curve selection is fully rigid, therefore trust is needed neither in miners nor in the scheme proposers.
The main problem faced by smart contract platforms is the amount of time and computational power required to reach consensus. In a classical blockchain model, each operation is in fact performed by each node, both to update the status and to validate the results of the calculations performed by others. In this short survey we sketch some state-of-the-art approaches to obtain an efficient and scalable computation of smart contracts. Particular emphasis is given to sharding, a promising method that allows parallelization and therefore a more efficient management of the computational resources of the network.
