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Quantum Information Processing

Discrete-time interacting quantum walks and quantum Hash schemes

Dan Li, Jie Zhang, Fen-Zhuo Guo, Wei Huang, Qiaoyan Wen, Hui Chen

Published May 23, 2012• 82 citations
Publisher
DOI
10.1007/s11128-012-0421-8

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A method of constructing a quantum hash function (QHF) based on continuous-time quantum walk is proposed, in which the time of quantum walk is controlled by the binary string message, namely, a binary string message as the input of the constructed QHF and the probability value of finally walking to the nodes on the cycle is used as the output of the QHF. Numerical simulation and performance analysis show that our QHF has obvious advantages in the efficiency of the QHF compared with the existing QHF based on the discrete-time quantum walk, that is, the efficiency of our scheme is nearly 4 times faster in computing the hash value of the same size file compared with the most efficient scheme. Besides, the QHF has also better collision resistance compared with the existing QHF based on the discrete-time quantum walk.
Quantum walk based hash functions have attracted a lot of attention in recent years because of its faster execution time and robust resistance against attacks compared to classical hash functions. It has been observed that the underlying graph and the way message controls the quantum walk iteration steps play a crucial role for the robustness of the hash function. We propose a quantum hash function based on the discrete-time quantum walk on a Hanoi network--a one dimensional periodic lattice with extra long-range edges of a specific form--which is highly collision resistant. The message bits of our scheme control the flow of probability amplitude through the extra long-range edges and the conditional shift operators. Our method even works for messages with small bit-lengths, contrary to most of the quantum walk based hash functions defined on a cycle, which usually work for messages with bit-lengths more than the length of the cycle.
In this paper, we develop a generic controlled alternate quantum-walk model (called CQWMP) by combining parity-dependent quantum walks with distinct arbitrary memory lengths and propose a hash function (called QHFM-P) based on this model. The statistical properties of the proposed scheme are stable with respect to the coin parameters of the underlying controlled quantum walks; with certain parameter values, the collision resistance property of QHFM-P is better than that of the state-of-the-art hash functions based on discrete quantum walks. Moreover, the proposed hash function can also maintain near-ideal statistical performance when the input message is of small length. Additionally, we derive a type of inappropriate initial states of hash functions based on one-dimensional one-particle quantum walks (with ordinary shift operator) on cycles, with which all messages will be mapped to the same hash value, regardless of the angles adopted by the coin parameters.