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We review and compare three algebraic methods to compute the nonlinearity of Boolean functions. Two of them are based on Gröbner basis techniques: the first one is defined over the binary field, while the second one over the rationals. The third method improves the second one by avoiding the Gröbner basis computation. We also estimate the complexity of the algorithms, and, in particular, we show that the third method reaches an asymptotic worst-case complexity of O (n 2 n) operations over the integers, that is, sums and doublings. This way, with a different approach, the same asymptotic complexity of established algorithms, such as those based on the fast Walsh transform, is reached. [ABSTRACT FROM AUTHOR]

We show how a small subgroup confinement-like attack may be mounted on the Bitcoin addresses generation protocol, by inspecting a special subgroup of the group associated to point multiplication. This approach does not undermine the system security but highlights the importance of using fair random sources during the private key selection. [ABSTRACT FROM AUTHOR]

Blockchain technology has attracted a lot of research interest in the last few years. Originally, their consensus algorithm was Hashcash, which is an instance of the so-called Proof-of-Work. Nowadays, there are several competing consensus algorithms, not necessarily PoW. In this paper, we propose an alternative proof of work algorithm which is based on the solution of consecutive discrete logarithm problems over the point group of elliptic curves. At the same time, we sketch a blockchain scheme, whose consensus is reached via our algorithm. 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. [ABSTRACT FROM AUTHOR]