Bernstein-Vazirani Algorithm Research Articles

Finding Zero Correlation Linear Hulls Using Quantum Bernstein-vazirani Algorithm

Abstract Facing the threat of quantum computing to cryptography schemes, it is urgent to study the applications of quantum algorithms to cryptanalysis and evaluate the security of cryptographic schemes in a quantum computing environment. To do this, it is necessary to analyze the ability of cryptanalysis methods, such as integral and linear attacks, when combined with quantum algorithms. In this article, we investigate the properties of block ciphers with SPN structure and their security against a quantum adversary and apply the Bernstein-Vazirani algorithm to zero correlation linear attacks. Based on the relation between linear approximates of SPN ciphers and the dual cipher’s linear structures, we proposed a quantum algorithm that can find linear hulls with zero correlation of SPN ciphers. We show the validity of the proposed quantum algorithm and estimate the corresponding computational complexity. Implementing our quantum algorithm only costs polynomial qubits and quantum gates.

Quantum Attacks on MIBS Block Cipher Based on Bernstein–Vazirani Algorithm

Because of the substantial progress in quantum computing technology, the safety of traditional cryptologic schemes is facing serious challenges. In this study, we explore the quantum safety of the lightweight cipher MIBS and propose quantum key-recovery attacks on the MIBS cipher by utilizing Grover’s algorithm and Bernstein–Vazirani algorithm. We first construct linear-structure functions based on the 5-round MIBS cipher according to the characteristics of the linear transformations, and then we obtain a quantum distinguisher of the 5-round MIBS cipher by applying Bernstein–Vazirani algorithm to the constructed functions. Finally, utilizing this distinguisher and Grover’s algorithm, we realize a 7-round key-recovery attack on the MIBS cipher, and then we expand the attack to more rounds of MIBS based on a similar idea. The quantum attack on the 7-round MIBS requires 156 qubits and has a time complexity of 210.5. An 8-round attack requires 179 qubits and has a time complexity of 222. Compared with existing quantum attacks, our attacks have better time complexity when attacking the same number of rounds.

Homomorphic encryption of the k=2 Bernstein–Vazirani algorithm

Abstract We introduce a class of circuits that solve a particular case of the Bernstein–Vazirani recursive problem for second-level recursion. This class of circuits allows for the implementation of the oracle using a number of T-gates that grows linearly with the number of qubits in the problem. We find an application of this scheme to quantum homomorphic encryption (QHE), which is an important cryptographic technology useful for delegated quantum computing, allowing a remote server to perform quantum computations on encrypted quantum data, so that the server cannot know anything about the client’s data. Liang’s QHE schemes are suitable for circuits with a polynomial number of gates T / T † . Thus, the simplified circuits we have constructed can be evaluated homomorphically in an efficient manner.

Impact of unreliable devices on stability of quantum computations

Noisy intermediate-scale quantum (NISQ) devices are valuable platforms for testing the tenets of quantum computing, but these devices are susceptible to errors arising from de-coherence, leakage, cross-talk and other sources of noise. This raises concerns regarding the stability of results when using NISQ devices since strategies for mitigating errors generally require well-characterized and stationary error models. Here, we quantify the reliability of NISQ devices by assessing the necessary conditions for generating stable results within a given tolerance. We use similarity metrics derived from device characterization data to derive and validate bounds on the stability of a 5-qubit implementation of the Bernstein-Vazirani algorithm. Simulation experiments conducted with noise data from IBM washington, spanning January 2022 to April 2023, revealed that the reliability metric fluctuated between 41% and 92%. This variation significantly surpasses the maximum allowable threshold of 2.2% needed for stable outcomes. Consequently, the device proved unreliable for consistently reproducing the statistical mean in the context of the Bernstein-Vazirani circuit.

Distributed Bernstein–Vazirani algorithm

Distributed Bernstein–Vazirani algorithm

A generalization of Bernstein–Vazirani algorithm with multiple secret keys and a probabilistic oracle

A probabilistic version of the Bernstein-Vazirani problem (which is a generalization of the original Bernstein-Vazirani problem) and a quantum algorithm to solve it are proposed. The problem involves finding one or more secret keys from a set of multiple secret keys (encoded in binary form) using a quantum oracle. From a set of multiple unknown keys, the proposed quantum algorithm is capable of (a) obtaining any key (with certainty) using a single query to the probabilistic oracle and (b) finding all keys with a high probability (approaching 1 in the limiting case). In contrast, a classical algorithm will be unable to find even a single bit of a secret key with certainty (in the general case). Owing to the probabilistic nature of the oracle, a classical algorithm can only be useful in obtaining limiting probability distributions of $ 0 $ and $ 1 $ for each bit-position of secret keys (based on multiple oracle calls) and this information can further be used to infer some estimates on the distribution of secret keys based on combinatorial considerations. For comparison, it is worth noting that a classical algorithm can be used to exactly solve the original Bernstein-Vazirani problem (involving a deterministic oracle and a single hidden key containing $n$ bits) with a query complexity of $\mathcal{O}(n)$. An interesting class of problems similar to the probabilistic version of the Bernstein-Vazirani problem can be construed, where quantum algorithms can provide efficient solutions with certainty or with a high degree of confidence and classical algorithms would fail to do so.

Demonstration of Algorithmic Quantum Speedup

Despite the development of increasingly capable quantum computers, an experimental demonstration of a provable algorithmic quantum speedup employing today's non-fault-tolerant devices has remained elusive. Here, we unequivocally demonstrate such a speedup within the oracular model, quantified in terms of the scaling with the problem size of the time-to-solution metric. We implement the single-shot Bernstein-Vazirani algorithm, which solves the problem of identifying a hidden bitstring that changes after every oracle query, using two different 27-qubit IBM Quantum superconducting processors. The speedup is observed on only one of the two processors when the quantum computation is protected by dynamical decoupling but not without it. The quantum speedup reported here does not rely on any additional assumptions or complexity-theoretic conjectures and solves a bona fide computational problem in the setting of a game with an oracle and a verifier.

Quantum Related-Key Attack Based on Simon’s Algorithm and Its Applications

With the development of quantum technology, quantum computing has an increasingly significant impact on cryptanalysis. Several quantum algorithms, such as Simon’s algorithm, Grover’s algorithm, the Bernstein–Vazirani algorithm, Shor’s algorithm, and the Grover-meets-Simon algorithm, have been proposed successively. However, almost all cryptanalysis is based on the quantum chosen-plaintext attack (qCPA) model. This paper focuses on a powerful cryptanalytic model, quantum related-key attack (qRKA), and proposes a strategy of qRKAs against symmetric ciphers using Simon’s algorithm. We construct a periodic function to efficiently recover the secret key of symmetric ciphers if the attacked symmetric ciphers satisfy Simon’s promise, and present the complexity analysis on specific symmetric ciphers. Then, we apply qRKA to the Even–Mansour cipher and SoEM construction, recover their secret keys, and show their complexity comparison in the distinct attack models. This work is of great significance for the qRKA cryptanalysis of existing provably secure cryptographic schemes and the design of future quantum secure cryptographic schemes.

Demonstration and Implementation of Quantum Computing in Cryptanalysis

Quantum computing has long been a hot topic in both fields of physics and computer science. As a matter of fact, most of the earliest quantum algorithms are related to cryptology on account of its potential advantage over conventional computers. In this paper, Bernstein-Vazirani algorithm, a linear cryptanalysis method performing on quantum computers, is selected as a typical algorithm to demonstration and implementation of the quantum computing in cryptanalysis. To be specific, this study analyzes the method itself, and realizes the method with Qiskit so as to compare it with conventional methods. With the comparison, the superiority of quantum computing in certain fields as well as some current disadvantages to be dealt with are clarified. Thus, it is hoped to present the huge potential of quantum computing in its processing ability and its property of superposition beginning with the field of cryptanalysis. Besides, a direction of future improvement of the technology is also proposed. Overall, these results shed light on guiding further exploration of quantum computing.

Entanglement and coherence in the Bernstein-Vazirani algorithm

Quantum algorithms can outperform their classical counterparts in various tasks, the most prominent example being Shor's algorithm for efficient prime factorization on a quantum computer. It is clear that one of the reasons for the speedup is the superposition principle of quantum mechanics, which allows a quantum processor to be in a superposition of different states at the same time. While such a superposition can lead to entanglement across different qubits of the processors, there also exist quantum algorithms that outperform classical ones using superpositions of individual qubits without entangling them. As an example, the Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. While the classical version of the algorithm requires multiple calls of the oracle to learn the bit string, a single query of the oracle is enough in the quantum case. In this article, we analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. For this, we introduce and study its probabilistic version, where the goal is to guess the bit string after a single call of the oracle. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state. We further demonstrate that a large amount of entanglement in the initial state prevents the algorithm from achieving optimal performance. We also apply our methods to quantum computation with mixed states, proving that pseudopure states achieve optimal performance for a given purity in the Bernstein-Vazirani algorithm. We further investigate quantum resources in the one clean qubit model, showing that the model can exhibit speedup over any known classical algorithm even with an arbitrarily little amount of multipartite entanglement, general quantum correlations, and coherence.

A programmable qudit-based quantum processor

Controlling and programming quantum devices to process quantum information by the unit of quantum dit, i.e., qudit, provides the possibilities for noise-resilient quantum communications, delicate quantum molecular simulations, and efficient quantum computations, showing great potential to enhance the capabilities of qubit-based quantum technologies. Here, we report a programmable qudit-based quantum processor in silicon-photonic integrated circuits and demonstrate its enhancement of quantum computational parallelism. The processor monolithically integrates all the key functionalities and capabilities of initialisation, manipulation, and measurement of the two quantum quart (ququart) states and multi-value quantum-controlled logic gates with high-level fidelities. By reprogramming the configuration of the processor, we implemented the most basic quantum Fourier transform algorithms, all in quaternary, to benchmark the enhancement of quantum parallelism using qudits, which include generalised Deutsch-Jozsa and Bernstein-Vazirani algorithms, quaternary phase estimation and fast factorization algorithms. The monolithic integration and high programmability have allowed the implementations of more than one million high-fidelity preparations, operations and projections of qudit states in the processor. Our work shows an integrated photonic quantum technology for qudit-based quantum computing with enhanced capacity, accuracy, and efficiency, which could lead to the acceleration of building a large-scale quantum computer.

QKD Based on Symmetric Entangled Bernstein-Vazirani.

This paper introduces a novel entanglement-based QKD protocol, that makes use of a modified symmetric version of the Bernstein-Vazirani algorithm, in order to achieve secure and efficient key distribution. Two variants of the protocol, one fully symmetric and one semi-symmetric, are presented. In both cases, the spatially separated Alice and Bob share multiple EPR pairs, each one qubit of the pair. The fully symmetric version allows both parties to input their tentative secret key from their respective location and acquire in the end a totally new and original key, an idea which was inspired by the Diffie-Hellman key exchange protocol. In the semi-symmetric version, Alice sends her chosen secret key to Bob (or vice versa). The performance of both protocols against an eavesdroppers attack is analyzed. Finally, in order to illustrate the operation of the protocols in practice, two small scale but detailed examples are given.

Implementation of the Bernstein-Vazirani Quantum Algorithm Using the Qiskit Framework

Abstract This paper describes the basics of quantum computing and then focuses on the implementation of the Bernstein-Vazirani algorithm, which can be seen as an extension of the Deutsch-Josza problem (that solves the question on whether a function is balanced or not). The idea behind the B-V algorithm is that someone can find a secret number (sequence of bits) using only one measurement, unlike the classical counter-part, that requires n measurements, where n is the number of bits of the secret number. The implementation of this algorithm, using the Python programming language, along with the Qiskit framework (an open-source library for quantum operations from IBM), illustrates how to create and simulate a circuit for such an algorithm. The circuit is dynamically generated for the required number (which in practice is received from a different source) and is used to measure the probability of each qubit. The algorithm can also be extended for different types of data and can be used for signal or image processing, as well as applications in cryptography.

Modeling noisy quantum circuits using experimental characterization

Noisy intermediate-scale quantum (NISQ) devices offer unique platforms to test and evaluate the behavior of quantum computing. However, validating circuits on NISQ devices is difficult due to fluctuations in the underlying noise sources and other nonreproducible behaviors that generate computational errors. Here we present a test-driven approach that decomposes a noisy, application-specific circuit into a series of bootstrapped experiments on a NISQ device. By characterizing individual subcircuits, we generate a composite noise model for the original quantum circuit. We demonstrate this approach to model applications of Greenberger-Horne-Zeilinger(GHZ)-state preparation and the Bernstein-Vazirani algorithm on a family of superconducting transmon devices. We measure the model accuracy using the total variation distance between predicted and experimental results, and we demonstrate that the composite model works well across multiple circuit instances. Our approach is shown to be computationally efficient and offers a trade-off in model complexity that can be tailored to the desired predictive accuracy.

Quantum algorithms for learning the algebraic normal form of quadratic Boolean functions

Quantum algorithms for the analysis of Boolean functions have received a lot of attention over the last few years. The algebraic normal form (ANF) of a linear Boolean function can be recovered by using the Bernstein–Vazirani (BV) algorithm. No research has been carried out on quantum algorithms for learning the ANF of general Boolean functions. In this paper, quantum algorithms for learning the ANF of quadratic Boolean functions are studied. We draw a conclusion about the influences of variables on quadratic functions, so that the BV algorithm can be run on them. We study the functions obtained by inversion and zero-setting of some variables in the quadratic function and show the construction of their quantum oracle. We introduce the concept of “club” to group variables that appear in quadratic terms and study the properties of clubs. Furthermore, we propose a bunch of algorithms for learning the full ANF of quadratic Boolean functions. The most efficient algorithm, among those we propose, provides an O(n) speedup over the classical one, and the number of queries is independent of the degenerate variables.

A Hybrid Quantum-Classical Approach to Mitigating Measurement Errors in Quantum Algorithms

When noisy intermediate scalable quantum (NISQ) devices are applied in information processing, all of the stages through preparation, manipulation, and measurement of multipartite qubit states contain various types of noise that are generally hard to be verified in practice. In this work, we present a scheme to deal with unknown quantum noise and show that it can be used to mitigate errors in measurement readout with NISQ devices. Quantum detector tomography that identifies a type of noise in a measurement can be circumvented. The scheme applies single-qubit operations only, that are with relatively higher precision than measurement readout or two-qubit gates. A classical post-processing is then performed with measurement outcomes. The scheme is implemented in quantum algorithms with NISQ devices: the Bernstein-Vazirani algorithm and a quantum amplitude estimation algorithm in IBMQ yorktown and IBMQ essex. The enhancement in the statistics of the measurement outcomes is presented for both of the algorithms with NISQ devices.

A quantum related-key attack based on the Bernstein–Vazirani algorithm

Due to the powerful computing capability of quantum computers, cryptographic researchers have applied quantum algorithms to cryptanalysis and obtained many interesting results in recent years. In this paper, we study related-key attack in the quantum setting and propose a specific related-key attack, which can recover the key of block ciphers efficiently as long as the attacked block ciphers satisfy certain condition. The attack algorithm employs the Bernstein–Vazirani algorithm as a subroutine and requires the attacker to query the encryption oracle with quantum superpositions. We give a condition under which the attack will succeed and prove that any block cipher either satisfies the condition or has a distinguishing attack. As a specific example of its application, we use the attack algorithm to extract the private key of the Even–Mansour cipher. The results of this study show the power of related-key attack when combined with quantum algorithms and provide guidance for the design of quantum-secure block ciphers.

Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomography

We propose a simple scheme to reduce readout errors in experiments on quantum systems with finite number of measurement outcomes. Our method relies on performing classical post-processing which is preceded by Quantum Detector Tomography, i.e., the reconstruction of a Positive-Operator Valued Measure (POVM) describing the given quantum measurement device. If the measurement device is affected only by an invertible classical noise, it is possible to correct the outcome statistics of future experiments performed on the same device. To support the practical applicability of this scheme for near-term quantum devices, we characterize measurements implemented in IBM's and Rigetti's quantum processors. We find that for these devices, based on superconducting transmon qubits, classical noise is indeed the dominant source of readout errors. Moreover, we analyze the influence of the presence of coherent errors and finite statistics on the performance of our error-mitigation procedure. Applying our scheme on the IBM's 5-qubit device, we observe a significant improvement of the results of a number of single- and two-qubit tasks including Quantum State Tomography (QST), Quantum Process Tomography (QPT), the implementation of non-projective measurements, and certain quantum algorithms (Grover's search and the Bernstein-Vazirani algorithm). Finally, we present results showing improvement for the implementation of certain probability distributions in the case of five qubits.

Quantum learning Boolean linear functions w.r.t. product distributions

The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein–Vazirani algorithm (Bernstein and Vazirani, in: Kosaraju (ed) Proceedings of the twenty-fifth annual ACM symposium on theory of computing, ACM, New York, 1993. https://doi.org/10.1145/167088.167097). A similar strategy can be applied in the case of noisy quantum training data, as was observed in Grilo et al. (Learning with errors is easy with quantum samples, 2017). However, extensions of these learning algorithms beyond the uniform distribution have not yet been studied. We employ the biased quantum Fourier transform introduced in Kanade et al. (Learning dnfs under product distributions via mu -biased quantum Fourier sampling, 2018) to develop efficient quantum algorithms for learning Boolean linear functions on n bits from quantum examples w.r.t. a biased product distribution. Our first procedure is applicable to any (except full) bias and requires mathcal {O}(ln (n)) quantum examples. The number of quantum examples used by our second algorithm is independent of n, but the strategy is applicable only for small bias. Moreover, we show that the second procedure is stable w.r.t. noisy training data and w.r.t. faulty quantum gates. This also enables us to solve a version of the learning problem in which the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem. Whereas classically, varOmega (n) examples are necessary independently of the bias, we are able to establish a quantum sample complexity lower bound of varOmega (ln (n)) only under an assumption of large bias. Nevertheless, this allows for a discussion of the performance of our suggested learning algorithms w.r.t. sample complexity. With our analysis, we contribute to a more quantitative understanding of the power and limitations of quantum training data for learning classical functions.

On Quantum Chosen-Ciphertext Attacks and Learning with Errors

Large-scale quantum computing poses a major threat to classical public-key cryptography. Recently, strong “quantum access” security models have shown that numerous symmetric-key cryptosystems are also vulnerable. In this paper, we consider classical encryption in a model that grants the adversary quantum oracle access to encryption and decryption, but where we restrict the latter to non-adaptive (i.e., pre-challenge) queries only. We formalize this model using appropriate notions of ciphertext indistinguishability and semantic security (which are equivalent by standard arguments) and call it QCCA 1 in analogy to the classical CCA 1 security model. We show that the standard pseudorandom function ( PRF )-based encryption schemes are QCCA 1 -secure when instantiated with quantum-secure primitives. Our security proofs use a strong bound on quantum random-access codes with shared randomness. Revisiting plain IND − CPA -secure Learning with Errors ( LWE ) encryption, we show that leaking only a single quantum decryption query (and no other leakage or queries of any kind) allows the adversary to recover the full secret key with constant success probability. Information-theoretically, full recovery of the key in the classical setting requires at least a linear number of decryption queries. Our results thus challenge the notion that LWE is unconditionally “just as secure” quantumly as it is classically. The algorithm at the core of our attack is a new variant of the well-known Bernstein–Vazirani algorithm. Finally, we emphasize that our results should not be interpreted as a weakness of these cryptosystems in their stated security setting (i.e., post-quantum chosen-plaintext secrecy). Rather, our results mean that, if these cryptosystems are exposed to chosen-ciphertext attacks (e.g., as a result of deployment in an inappropriate real-world setting) then quantum attacks are even more devastating than classical ones.