We provide a historical overview of proof-of-work techniques and the fields in which it plunges its roots. We are interested in PoW-techniques applied to blockchain technology and therefore we survey the state-of-the-art protocols employing these methods for consensus algorithms, emphasizing the differences between the efficient hashcash systems and the promising bread pudding protocols. Afterwards, the consensus mechanisms are discussed and some interesting known attacks to these algorithms are collected and classified according to their underlying ideas.

We describe first-degree prime ideals of biquadratic extensions in terms of the first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. The correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of specially-shaped principal ideals in their respective rings, with some exceptions that we explicitly characterize.

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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.

We characterize the possible groups E(Z/NZ) arising from elliptic curves over Z/NZ in terms of the groups E(F p ), with p varying among the prime divisors of N. This classification is achieved by showing that the infinity part of any elliptic curve over Z/p e Z is a Z/p e Z-torsor.As a first consequence, when E(Z/NZ) is a p-group, we provide an explicit and sharp bound on its rank. As a second consequence, when N = p e is a prime power and the projected curve E(F p ) has trace one, we provide an isomorphism attack to the ECDLP, which works only by means of finite rings arithmetic rather than involved methods.