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The discrete logarithm problem (DLP) plays an important role in modern cryptography since it cannot be efficiently solved on a classical computer. Currently, the DLP based on the hyperelliptic curve of genus 2 (HCDLP) is widely used in industry and also a research field of hot interest. At the same time, quantum computing, a new paradigm for computing based on quantum mechanics, provides the ability to solve certain hard problems that cannot be efficiently solved on classical computers. In this paper, we consider the problem of solving the HCDLP in the paradigm of quantum computing. We propose a quantum algorithm for solving the HCDLP by applying the framework of quantum algorithm designed by Shor. The key of the algorithm is the realization of divisor addition. We solve the key problem and get analytical results for divisor addition by geometric meaning of the group addition. Therefore, the procedure can be efficiently realized on a quantum computer using the basic modular arithmetic operations. Finally, we conclude that the HCDLP defined over an n-bit prime field F p can be computed on a quantum computer with at most 13 n + 2 ⌊ log 2 n ⌋ + 10 qubits using 2624 n 3 log 2 n - 2209.2 n 3 + 1792 n 2 log 2 n - 3012.8 n 2 Toffoli gates. For current parameters at comparable classical security levels, there are fewer qubits and Toffoli gates to solve the HCDLP than the ones to solve the DLP based on elliptic curves. [ABSTRACT FROM AUTHOR]

Fairness is an important standard needed to be considered in a secure quantum key agreement (QKA) protocol. However, it found that most of the quantum key agreement protocols in the travelling model are not fair, i.e., some of the dishonest participants can collaborate to predetermine the final key without being detected. Thus, how to construct a fair and secure key agreement protocol has obtained much attention. In this paper, a new fair multiparty QKA protocol that can resist the collusive attack is proposed. More specifically, we show that in a client-server scenario, it is possible for the clients to share a key and reveal nothing about what key has been agreed upon to the server. The server prepares quantum states for clients to encode messages to avoid the participants' collusive attack. This construction improves on previous work, which requires either preparing multiple quantum resources by clients or two-way quantum communication. It is proven that the protocol does not reveal to any eavesdropper, including the server, what key has been agreed upon, and the dishonest participants can be prevented from collaborating to predetermine the final key. [ABSTRACT FROM AUTHOR]