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In this paper we explore a context of application of Cob, a recently introduced Byzantine Fault Tolerant consensus protocol. Cob proves to be a leaderless consensus protocol which carries out the consensus process in parallel on each component of a list of events to be observed and recorded. We show how Cob can be used to define a consensus layer for scalable and sustainable blockchains. This layer is used to design consensus protocols based on sharding as a mean to achieve scalability, and on the fragmentation of time in time-slots (which get assigned to nodes that are instructed to create new blocks) as a mean to reduce the amount of computation and communication necessary for the maintenance of the distributed ledger. We explain why Cob is a viable candidate to implement such consensus layer through the introduction of an auxiliary blockchain that we name Synchronization Chain., 15 pages, 2 figures, 1 table

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., Comment: 15 pages, 0 figures

We analyse the security of some variants of the CFS code-based digital signature scheme. We show how the adoption of some code-based hash-functions to improve the efficiency of CFS leads to the ability of an attacker to produce a forgery compatible to the rightful user's public key.

A $(t,n)$-threshold signature scheme enables distributed signing among $n$ players such that any subset of size at least $t$ can sign, whereas any subset with fewer players cannot. The goal is to produce threshold digital signatures that are compatible with an existing centralized signature scheme. Starting from the threshold scheme for the ECDSA signature due to Battagliola et al., we present the first protocol that supports EdDSA multi-party signatures with an offline participant during the key-generation phase, without relying on a trusted third party. Under standard assumptions we prove our scheme secure against adaptive malicious adversaries. Furthermore we show how our security notion can be strengthen when considering a rushing adversary. We discuss the resiliency of the recovery in the presence of a malicious party. Using a classical game-based argument, we prove that if there is an adversary capable of forging the scheme with non-negligible probability, then we can build a forger for the centralized EdDSA scheme with non-negligible probability., 27 pages. arXiv admin note: substantial text overlap with arXiv:2007.04036

A statistical estimation algorithm of the weight distribution of a linear code is shown, based on using its generator matrix as a compression function on random bit strings.

We provide a combinatorial construction for linear codes attaining the maximum possible number of distinct weights. We then introduce the related problem of determining the existence of linear codes with an arbitrary number of distinct non-zero weights, and we completely determine a solution in the binary case.

We consider the simplified Algebraic Normal Form (sANF) of Boolean functions vanishing on Hamming spheres centred at zero and the associated sANF vector. We show that this vector is periodic, leading to an efficient computation of the sANF and to specific formulas for particular cases. Moreover, we explicitly provide a connection to the binary M{\"o}bius transform of the elementary symmetric functions. We conclude by presenting a method based on polynomial evaluation to bound the minimum distance of binary nonlinear codes. The same method can be used to compute the minimum distance and the weight distribution of binary linear codes.

Most bounds on the size of codes hold for any code, whether linear or nonlinear. Notably, the Griesmer bound holds only in the linear case. In this paper we identify code parameters (q, d, k) for which the Griesmer bound holds also in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by showing explicit counterexamples. On the other hand, we are also able to provide some versions of the Griesmer bound holding for all systematic codes.

Most bounds on the size of codes hold for any code, whether linear or nonlinear. Notably, the Griesmer bound, holds only in the linear case. In this paper we characterize a family of systematic nonlinear codes for which the Griesmer bound holds. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by showing explicit counterexamples. On the other hand, we are also able to provide (weaker) versions of the Griesmer bound holding for all systematic codes.

We consider a bound on the bias reduction of a random number generator by processing based on binary linear codes. We introduce a new bound on the total variation distance of the processed output based on the weight distribution of the code generated by the chosen binary matrix. Starting from this result we show a lower bound for the entropy rate of the output of linear binary extractors., 10 pages, 2 figures. submitted to IEEE