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Russian Mathematics

Binary quantum hashing

Alexander Vasiliev

Published Jul 21, 2016• 5 citations
Publisher

Abstract

We propose a binary quantum hashing technique that allows to present binary inputs by quantum states. We prove the cryptographic properties of the quantum hashing, including its collision resistance and preimage resistance. We also give an efficient quantum algorithm that performs quantum hashing, and altogether this means that this function is quantum one-way. The proposed construction is asymptotically optimal in the number of qubits used.

DOI
10.3103/s1066369x16090073

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We propose a binary quantum hashing technique that allows to present binary inputs by quantum states. We prove the cryptographic properties of the quantum hashing, including its collision resistance and preimage resistance. We also give an efficient quantum algorithm that performs quantum hashing, and altogether this means that this function is quantum one-way. The proposed construction is asymptotically optimal in the number of qubits used.
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We present a version of quantum hash function based on non-binary discrete functions. The proposed quantum procedure is "classical-quantum", that is, it takes a classical bit string as an input and produces a quantum state. The resulting function has the property of a one-way function (pre-image resistance), in addition it has the properties analogous to classical cryptographic hash second pre-image resistance and collision resistance. This function can be naturally used in a quantum digital signature protocol.
In the letter we define the notion of a quantum resistant ( -resistant) hash function which consists of a combination of pre-image (one-way) resistance ( ε -resistance) and collision resistance ( δ -resistance) properties. We present examples and discussion that supports the idea of quantum hashing. We present an explicit quantum hash function which is ‘balanced’, one-way resistant and collision resistant and demonstrate how to build a large family of quantum hash functions. Balanced quantum hash functions need a high degree of entanglement between the qubits. We use a phase transformation technique to express quantum hashing constructions, which is an effective way of mapping hash states to coherent states in a superposition of time-bin modes. The phase transformation technique is ready to be implemented with current optical technology.
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.
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) .