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Lobachevskii Journal of Mathematics

Identification of Quantum Hashes: Numerical Experiment

Farid Mansurovich Ablayev, Mansur Ziatdinov

Published Feb 1, 2023• 0 citations
Publisher

Abstract

The problem of reliable identification of quantum hash states, which are images of quantum hash functions, is considered. It is shown that this can be done with a high probability for a linear number of $$O(n/(1-\epsilon^{2}))$$ experiments when hashing words of length $$n$$ with $$\epsilon$$ -resistance to collisions. The results of a numerical experiment on the implementation of the procedure are presented, demonstrating the result of the above. We provide such an experiment to simulate quantum hashing for binary words of length 5.

DOI
10.1134/s1995080223020038

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The problem of reliable identification of quantum hash states, which are images of quantum hash functions, is considered. It is shown that this can be done with a high probability for a linear number of O(n/(1−ϵ2))O(n/(1-\epsilon^{2}))O(n/(1−ϵ2)) experiments when hashing words of length nnn with ϵ\epsilonϵ -resistance to collisions. The results of a numerical experiment on the implementation of the procedure are presented, demonstrating the result of the above. We provide such an experiment to simulate quantum hashing for binary words of length 5.
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.
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.
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Hash functions are a basic cryptographic primitive. Certain hash functions try to prove security against collision and preimage attacks by reductions to known hard problems. These hash functions usually have some additional properties that allow for that reduction. Hash functions which are additive or multiplicative are vulnerable to a quantum attack using the hidden subgroup problem algorithm for quantum computers. Using a quantum oracle to the hash, we can reconstruct the kernel of the hash function, which is enough to find collisions and second preimages. When the hash functions are additive with respect to the group operation in an Abelian group, there is always an efficient implementation of this attack. We present concrete attack examples to provable hash functions, including a preimage attack to ⊕\oplus⊕-linear hash functions and for certain multiplicative homomorphic hash schemes.