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AIP conference proceedings

Computing quantum hashing in the model of quantum branching programs

Farid Mansurovich Ablayev, Marat Ablayev, Alexander Vasiliev

Published Jan 1, 2018• 3 citations
Publisher

Abstract

We investigate the branching program complexity of quantum hashing. We consider a quantum hash function that maps elements of a finite field into quantum states. We require that this function is preimage-resistant and collision-resistant.We consider two complexity measures for Quantum Branching Programs (QBP): a number of qubits and a number of compu-tational steps. We show that the quantum hash function can be computed efficiently. Moreover, we prove that such QBP construction is optimal. That is, we prove lower bounds that match the constructed quantum hash function computation.

DOI
10.1063/1.5025458

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We investigate the branching program complexity of quantum hashing. We consider a quantum hash function that maps elements of a finite field into quantum states. We require that this function is preimage-resistant and collision-resistant.We consider two complexity measures for Quantum Branching Programs (QBP): a number of qubits and a number of compu-tational steps. We show that the quantum hash function can be computed efficiently. Moreover, we prove that such QBP construction is optimal. That is, we prove lower bounds that match the constructed quantum hash function computation.
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Quantum hashing is a useful technique in different computational and cryptographic scenarios in the quantum world. A set of random parameters is required to construct a quantum hashing scheme. For instance, random walks on expander graphs (expanders) are known to be efficient randomness generators in many areas of computer science. We analyze a scheme based on expanders. Collision resistance and preimage resistance of this scheme are considered. We show that quantum hashing based on expanders is collision-resistant (in quantum sense), and the considered scheme’s accessible information is O(1)O(1)O(1) .
Cryptographic hash functions are important building blocks of many security systems. A preimage attack on hash functions tries to find a message that has a specific hash value. A cryptographic hash function should resist attacks on its preimage. The current paper presents a new quantum algorithm for hash preimage attacks, which can break the preimage resistance with circuit complexity O(2n/3) using O(2n/3) running times, where n is the input bit length of the hash function. The quantum circuit complexity is apparently reduced from O(2n/2) to O(2n/3) at the cost of O(2n/3) running times requirements compared with the existing algorithms.
We develop quantum fingerprinting technique for constructing quantum branching pro-grams (QBPs), which are considered as circuits with an ability to use classical bits as control variables. We demonstrate our approach constructing optimal quantum ordered binary decision diagram (QOBDD) for MODm Boolean function. The construction of our technique also allows to extend the recent result of Ambainis and Nahimovs it is based on. In addition we show how our technique works for encoding quantum information for the equality problem in the simultaneous message passing model. 1