Efficient Quantum Algorithm for Hidden Quadratic and Cubic Polynomial Function Graphs

We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field F_p. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be multivariate polynomial functions of higher degree. For a fixed number of indeterminates and bounded total degree the Hidden Polynomial Function Graph Problem is hard on a classical computer as its black box query complexity is polynomial in p. In contrast, this problem can be reduced to a quantum state identification problem so that the resulting quantum query complexity does not depend on p. For univariate polynomials we construct a von Neumann measurement for distinguishing the states. We relate the success probability and the implementation of this measurement to certain classical problems involving polynomial equations. We present an efficient algorithm for hidden quadratic and cubic function graphs by establishing that the success probability of the measurement is lower bounded by a constant and that it can be implemented efficiently.Comment: (v2) formulated the Hidden Polynomial Function Graph Problem for multivariate polynomials, added results on quantum query complexity, simplified POVM substantially, (v3) derived quantum algorithm for cubic case; 16 page

Cryptocurrencies without Proof of Work

We study decentralized cryptocurrency protocols in which the participants do not deplete physical scarce resources. Such protocols commonly rely on Proof of Stake, i.e., on mechanisms that extend voting power to the stakeholders of the system. We offer analysis of existing protocols that have a substantial amount of popularity. We then present our novel pure Proof of Stake protocols, and argue that they help in mitigating problems that the existing protocols exhibit

Bitcoin Beacon

We examine a protocol $\pi_{\text{beacon}}$ that outputs unpredictable and publicly verifiable randomness, meaning that the output is unknown at the time that $\pi_{\text{beacon}}$ starts, yet everyone can verify that the output is close to uniform after $\pi_{\text{beacon}}$ terminates. We show that $\pi_{\text{beacon}}$ can be instantiated via Bitcoin under sensible assumptions; in particular we consider an adversary with an arbitrarily large initial budget who may not operate at a loss indefinitely. In case the adversary has an infinite budget, we provide an impossibility result that stems from the similarity between the Bitcoin model and Santha-Vazirani sources. We also give a hybrid protocol that combines trusted parties and a Bitcoin-based beacon