On the Design of Cryptographic Primitives

The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed

Statistical Zero Knowledge and quantum one-way functions

One-way functions are a very important notion in the field of classical cryptography. Most examples of such functions, including factoring, discrete log or the RSA function, can be, however, inverted with the help of a quantum computer. In this paper, we study one-way functions that are hard to invert even by a quantum adversary and describe a set of problems which are good such candidates. These problems include Graph Non-Isomorphism, approximate Closest Lattice Vector and Group Non-Membership. More generally, we show that any hard instance of Circuit Quantum Sampling gives rise to a quantum one-way function. By the work of Aharonov and Ta-Shma, this implies that any language in Statistical Zero Knowledge which is hard-on-average for quantum computers, leads to a quantum one-way function. Moreover, extending the result of Impagliazzo and Luby to the quantum setting, we prove that quantum distributionally one-way functions are equivalent to quantum one-way functions. Last, we explore the connections between quantum one-way functions and the complexity class QMA and show that, similarly to the classical case, if any of the above candidate problems is QMA-complete then the existence of quantum one-way functions leads to the separation of QMA and AvgBQP.Comment: 20 pages; Computational Complexity, Cryptography and Quantum Physics; Published version, main results unchanged, presentation improve