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arXiv:2311.03723

Generalized Hybrid Search and Applications to Blockchain and Hash Function Security

Alexandru Cojocaru, Juan A. Garay, Fang Song

Published Nov 7, 2023• 0 citations
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Abstract

In this work we first examine the hardness of solving various search problems by hybrid quantum-classical strategies, namely, by algorithms that have both quantum and classical capabilities. We then construct a hybrid quantum-classical search algorithm and analyze its success probability. Regarding the former, for search problems that are allowed to have multiple solutions and in which the input is sampled according to arbitrary distributions we establish their hybrid quantum-classical query complexities -- i.e., given a fixed number of classical and quantum queries, determine what is the probability of solving the search task. At a technical level, our results generalize the framework for hybrid quantum-classical search algorithms proposed by Rosmanis. Namely, for an arbitrary distribution $D$ on Boolean functions, the probability an algorithm equipped with $τ_c$ classical and $τ_q$ quantum queries succeeds in finding a preimage of $1$ for a function sampled from $D$ is at most $ν_D \cdot(2\sqrt{τ_c} + 2τ_q + 1)^2$, where $ν_D$ captures the average (over $D$) fraction of preimages of $1$. As applications of our hardness results, we first revisit and generalize the security of the Bitcoin protocol called the Bitcoin backbone, to a setting where the adversary has both quantum and classical capabilities, presenting a new hybrid honest majority condition necessary for the protocol to properly operate. Secondly, we examine the generic security of hash functions against hybrid adversaries. Regarding our second contribution, we design a hybrid algorithm which first spends all of its classical queries and in the second stage runs a ``modified Grover'' where the initial state depends on the distribution $D$. We show how to analyze its success probability for arbitrary target distributions and, importantly, its optimality for the uniform and the Bernoulli distribution cases.

DOI
10.48550/arxiv.2311.03723

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Scientific Reports•Nov 26, 2025•1 citation
In this work we first examine the hardness of solving various search problems by hybrid quantum-classical strategies, namely, by algorithms that have both quantum and classical capabilities. We then construct a hybrid quantum-classical search algorithm and analyze its success probability. Regarding the former, for search problems that are allowed to have multiple solutions and in which the input is sampled according to arbitrary distributions we establish their hybrid quantum-classical query complexities -- i.e., given a fixed number of classical and quantum queries, determine what is the probability of solving the search task. At a technical level, our results generalize the framework for hybrid quantum-classical search algorithms proposed by Rosmanis. Namely, for an arbitrary distribution DDD on Boolean functions, the probability an algorithm equipped with τcτ_cτc​ classical and τqτ_qτq​ quantum queries succeeds in finding a preimage of 111 for a function sampled from DDD is at most νD⋅(2τc+2τq+1)2ν_D \cdot(2\sqrt{τ_c} + 2τ_q + 1)^2νD​⋅(2τc​​+2τq​+1)2, where νDν_DνD​ captures the average (over DDD) fraction of preimages of 111. As applications of our hardness results, we first revisit and generalize the security of the Bitcoin protocol called the Bitcoin backbone, to a setting where the adversary has both quantum and classical capabilities, presenting a new hybrid honest majority condition necessary for the protocol to properly operate. Secondly, we examine the generic security of hash functions against hybrid adversaries. Regarding our second contribution, we design a hybrid algorithm which first spends all of its classical queries and in the second stage runs a ``modified Grover'' where the initial state depends on the distribution DDD. We show how to analyze its success probability for arbitrary target distributions and, importantly, its optimality for the uniform and the Bernoulli distribution cases.
Bitcoin and its underlying blockchain protocol have recently received significant attention in the context of building distributed systems and foundations of the consensus problem. At the same time, the rapid development in quantum computing makes the threats to cryptography more and more concerning. In this work, we revisit the formal security of the core of the Bitcoin consensus protocol, called the Bitcoin backbone (Eurocrypt 2015), in the presence of quantum adversaries -- i.e. adversaries equipped with quantum computers. We show that the security of the Bitcoin backbone holds under a quantum analogue of the ``honest majority'' assumption that we develop. The critical ingredient of proving security of the blockchain is to analyzing the quantum query complexity of a Chain-of-Proofs-of-Work search problem. This problem in turn reduces to a problem we call multi-solution Bernoulli search, for which we establish its quantum query complexity. This can be viewed as an extension of a threshold direct product theorem to an average-case unstructured search problem. Our proof, adding to active recent efforts, simplifies and generalizes the powerful recording technique due to Zhandry (Crypto 2019). Our analysis indicates that the security of the Bitcoin backbone protocol is guaranteed provided that the number of adversarial quantum queries is bounded so that each quantum query is worth O(p−1/2)O(p^{-1/2})O(p−1/2) classical ones, where ppp is the probability of success of a single classical query to the protocol's underlying hash function. Perhaps surprisingly, the wait time for safe settlement of transactions in the case of quantum adversaries matches (up to a constant) the safe settlement time in the classical case and thus does not result in any further overhead.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most qqq oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision εεε requires Θ(1qε2)Θ(\frac{1}{q ε^2})Θ(qε21​) queries; (ii) (Unbiased) Amplitude estimation up to a precision εεε requires Θ(p(1−p)qε2)Θ(\frac{p(1-p)}{q ε^2})Θ(qε2p(1−p)​) queries; (iii) Search among NNN items requires Θ(Nq)Θ(\frac{N}{q})Θ(qN​) queries; (iv) Two level AND-OR tree (AND of mmm ORs, with nnn inputs to each OR) requires Θ(nmq)Θ(\frac{nm}{q})Θ(qnm​) queries. All of these bounds are optimal up to a constant factor, for all qqq from 1, corresponding to the classical complexity, to Q(f)Q(f)Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.
The paper considers the problem of finding a given substring in a text. It is known that the complexity of a classical search query in an unordered database is linear in the length of the text and a given substring. At the same time, Grover’s quantum search provides a quadratic speed-up in the complexity of the query and gives the correct result with a high probability. We propose a hybrid classical–quantum algorithm (hybrid random–quantum algorithm, to be more precise) that implements Grover’s search to find a given substring in a text. As expected, the algorithm works (a) with a high probability of obtaining the correct result and (b) with a quadratic query acceleration compared to the classical one. What is new is that our algorithm uses the uniform hash family functions technique. As a result, our algorithm is much more memory efficient (in terms of the number of qubits used) compared to previously known quantum algorithms.
The rapid advancement of quantum computing presents a fundamental challenge to modern cryptographic security, particularly in the domain of hash functions that ensure data integrity, authentication, and blockchain security. Traditional crypto graphic hash functions such as SHA-256, SHA-3, and BLAKE2 rely on computational hardness assumptions that become obsolete in the presence of large-scale quantum computers. Shor’s algorithm can efficiently break RSA and ECC-based cryptosys tems, while Grover’s algorithm reduces the security of traditional hash functions by square root complexity, significantly weakening their preimage and collision resistance. This quantum threat necessitates the development of post-quantum secure hashing techniques that remain resilient against both classical and quantum adversaries. This paper proposes Quantum Hashing, a novel cryptographic framework that integrates quantum entanglement, lattice-based cryptography, and hybrid quantum classical hashing to construct post-quantum secure hash functions. We introduce a formal model for Quantum Collision Resistance (QCR) and provide entropy-based ran domness enhancement to ensure unpredictable hash outputs. Unlike classical hashing approaches, our framework leverages the hardness of lattice problems (e.g., Shortest Vector Problem, Learning with Errors) to withstand quantum attacks while incorpo rating Quantum Key Distribution (QKD) mechanisms to enhance entropy and key management. Furthermore, we evaluate the security of Quantum Hashing under various attack models, comparing its resistance against Grover’s search and collision attacks. We benchmark its performance against NIST Post-Quantum Cryptography (PQC) final ists, including CRYSTALS-DILITHIUM, SPHINCS+, and Falcon, demonstrating that our approach offers superior resilience while maintaining computational feasibility. Additionally, we present an implementation of Quantum Hashing using Qiskit, show casing its practical applicability in quantum circuits and quantum-secure blockchain architectures. Our findings highlight that Quantum Hashing provides a scalable, entropy-efficient, and post-quantum resilient cryptographic primitive suitable for next-generation cryptographic applications. This work paves the way for secure post-quantum digital signatures, blockchain consensus mechanisms, and zero-knowledge proof systems that require tamper-resistant hashing in a quantum computing era.
This thesis presents two main projects focused on hybrid quantum search algorithms. The first, Particle-Guided Grover's Search, applies Bayesian inference to quantum search when the number of solutions is a priori unknown. It begins with a prior distribution that encodes our initial "beliefs" about the true number of solutions in the search space, and recursively updates these beliefs using Bayesian inference. Measurement outcomes of the quantum search that do not yield a solution are interpreted as "evidence", guiding the Bayesian update to determine the optimal number of Grover iterations for subsequent rounds. This approach leverages Particle Filtering -- a Bayesian filtering technique -- to avoid explicitly storing the entire prior distribution at each step. Instead of requiring O(N) memory, the method draws a set of "particles" from the prior and updates their associated weights as quantum search failures accumulate. Particle filtering also offers the advantage of incorporating information from all past failures, rather than only the most recent one. The algorithm's performance is benchmarked against a prior work. The second project, Runtime--Coherence Trade-offs for Hybrid SAT Solvers, is focused on a class of hybrid k-SAT solvers. These solvers are based on a well-known random walk technique called Schoening's algorithm, which uses two sources of randomness to perform a local search over the space of all assignments for a k-SAT problem. In this project, we assume that these sources of randomness are encoded into a long bitstring, allowing the random walk to be interpreted as a deterministic function mapping each bitstring to either 0 or 1, depending on whether the walk successfully finds a satisfying assignment. This functional perspective enables us to pose the question: Given a bound on the coherence time of a quantum computer, which portion of the bitstring should we search coherently, and which portion should be left to classical search? We address this question by introducing a family of hybrid algorithms called "partial Groverizations", which strategically combine quantum and classical resources. We analyze the runtime performance of this family as a function of the available quantum coherence time. The results of this work have been published in IEEE Transactions on Quantum Engineering.
The advent of quantum computing poses substantial risks to conventional cryptographic mechanisms, particularly hash-based authentication and zero-knowledge identification (ZKI) protocols, which are susceptible to quantum algorithms such as Grover's and Shor's. This study presents a comprehensive benchmarking framework for evaluating the quantum resistance of cryptographic hash functions through performance metrics (execution time, memory utilization), statistical characteristics (entropy, bit-level randomness, avalanche effect), and security attributes (collision and preimage resistance) across diverse input sizes and edge conditions. A novel hybrid hashing strategy, integrating SHA-512 and BLAKE3 in a defense-in-depth configuration, is introduced to enhance post-quantum resilience. Its efficacy is validated via Grover's algorithm simulations, demonstrating a methodology for evaluating the increased computational workload for quantum search algorithms relative to conventional hash functions. The framework incorporates visualization utilities and structured reporting modules, enabling systematic assessment and practical implementation of quantum-resistant cryptographic solutions within ZKI systems. Findings indicate that while classical hashes such as SHA-256 and SHA-512 exhibit theoretically diminished security in quantum threat scenarios, the proposed hybrid method acts as a practical risk mitigation strategy with acceptable computational overhead, providing a viable pathway for safeguarding authentication systems against quantum-capable adversaries.