Shor's Algorithm Research Articles

QSMAH: A novel quantum-based secure cryptosystem using mutual authentication for healthcare in the internet of things

Healthcare in IoT provides many benefits, such as real-time data transfer and decision-making based on the information received from the patient's Body Sensor Nodes (BSNs). Many healthcare centers have adopted IoT-based devices such as heart monitoring implants and Electrocardiograms (ECG) for continuous patient monitoring in and out of hospitals. However, due to open Internet connectivity and lack of authentication, these devices may lead to several life-threatening risks. Recent advancements in Quantum Computing threatened the security of classical cryptographic primitives such as the RSA algorithm. The security and privacy of patient data are of utmost importance and are to be protected from existing vulnerabilities and futuristic Quantum attacks. Motivated by the abovementioned issues, we proposed the QSMAH protocol which ensures secure Key Agreement (KA) and Mutual Authentication (MA) based on Quantum Cryptography. Unlike classical authentication schemes, in our protocol, Quantum Teleportation and Quantum Entanglement are effectively utilized for secure data transfer among entities. The security of the proposed QSMAH protocol relies on a Quantum Key and Greenberger–Horne–Zeilinger (GHZ) states to achieve strong authentication. An extensive formal security analysis using BAN logic is provided to prove the goal of our protocol. The simulation of the proposed protocol is provided using Automated Validation of Internet Security Protocols and Applications (AVISPA). The results reflect that the security of our protocol cannot be tampered with by Quantum Shor's and Grover's algorithms. The security analysis proves that QSMAH is also resistant to classical attacks and futuristic Quantum attacks on cryptographic schemes.

Shor's Factoring Algorithm and Modular Exponentiation Operators

We provide a pedagogical presentation of Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor such large numbers. Shor's algorithm therefore has profound implication for public-key encryption such as RSA and Diffie–Hellman key exchange. We assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. Shor's algorithm contains a number of moving parts, and can be rather daunting at first. The literature is replete with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. They require a thicket of appendices and assume a knowledge of quantum algorithms and classical mathematics with which the reader might not be familiar. We therefore start with first principle derivations of the quantum Fourier transform (QFT) and quantum phase estimation (QPE), which are the essential building blocks of Shor's algorithm. We then go on to develop the theory of modular exponentiation (ME) operators, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also delve into the number theory that establishes the link between factorization and the period of the modular exponential function. We then apply the QPE algorithm to obtain Shor's factoring algorithm. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger integers, developing a systematic procedure that will find the ME operators for any semi-prime N=p×q (where q and p are prime). Finally, we factor the composite numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected.Quanta 2023; 12: 41–130.

Synthesis of Hidden Subgroup Quantum Algorithms and Quantum Chemical Dynamics.

We describe a general formalism for quantum dynamics and show how this formalism subsumes several quantum algorithms, including the Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Shor algorithms as well as the conventional approach to quantum dynamics based on tensor networks. The common framework exposes similarities among quantum algorithms and natural quantum phenomena: we illustrate this connection by showing how the correlated behavior of protons in water wire systems that are common in many biological and materials systems parallels the structure of Shor's algorithm.

Making existing software quantum safe: A case study on IBM Db2

The software engineering community is facing challenges from quantum computers (QCs). In the era of quantum computing, Shor's algorithm running on QCs can break asymmetric encryption algorithms that classical computers practically cannot. Though the exact date when QCs will become "dangerous" for practical problems is unknown, the consensus is that this future is near. Thus, the software engineering community needs to start making software ready for quantum attacks and ensure quantum safety proactively. We argue that the problem of evolving existing software to quantum-safe software is very similar to the Y2K bug. Thus, we leverage some best practices from the Y2K bug and propose our roadmap, called 7E, which gives developers a structured way to prepare for quantum attacks. It is intended to help developers start planning for the creation of new software and the evolution of cryptography in existing software. In this paper, we use a case study to validate the viability of 7E. Our software under study is the IBM Db2 database system. We upgrade the current cryptographic schemes to post-quantum cryptographic ones (using Kyber and Dilithium schemes) and report our findings and lessons learned. We show that the 7E roadmap effectively plans the evolution of existing software security features towards quantum safety, but it does require minor revisions. We incorporate our experience with IBM Db2 into the revised 7E roadmap. The U.S. Department of Commerce's National Institute of Standards and Technology is finalizing the post-quantum cryptographic standard. The software engineering community needs to start getting prepared for the quantum advantage era. We hope that our experiential study with IBM Db2 and the 7E roadmap will help the community prepare existing software for quantum attacks in a structured manner.

Performance Analysis of a Repetition Cat Code Architecture: Computing 256-bit Elliptic Curve Logarithm in 9Hours with 126 133 Cat Qubits.

Cat qubits provide appealing building blocks for quantum computing. They exhibit a tunable noise bias yielding an exponential suppression of bit flips with the average photon number and a protection against the remaining phase errors can be ensured by a simple repetition code. We here quantify the cost of a repetition code and provide valuable guidance for the choice of a large scale architecture using cat qubits by realizing a performance analysis based on the computation of discrete logarithms on an elliptic curve with Shor's algorithm. By focusing on a 2D grid of cat qubits with neighboring connectivity, we propose to implement 2-qubit gates via lattice surgery and Toffoli gates with off-line fault-tolerant preparation of magic states through projective measurements and subsequent gate teleportations. All-to-all connectivity between logical qubits is ensured by routing qubits. Assuming a ratio between single- and two-photon losses of 10^{-5} and a cycle time of 500ns, we show concretely that such an architecture can compute a 256-bit elliptic curve logarithm in 9 h with 126 133 cat qubits and on average 19 photons by cat state. We give the details of the realization of Shor's algorithm so that the proposed performance analysis can be easily reused to guide the choice of architecture for others platforms.

Design Space Exploration for Efficient Quantum Most-Significant Digit-First Arithmetic

Quantum computing has been considered as an emerging approach in addressing problems which are not easily solvable using classical computers. In parallel to the physical implementation of quantum processors, quantum algorithms have been actively developed for real-life applications to show quantum advantages, many of which benefit from quantum arithmetic algorithms and their efficient implementations. As one of the most important operations, quantum addition has been adopted in Shor's algorithm and quantum linear algebra algorithms. Although various least-significant digit-first quantum adders have been introduced in previous work, interest in investigating the efficient implementation of most-significant digit-first addition is growing. In this work, we propose a novel design method for most-significant digit-first addition with several quantum circuit optimisations to reduce the number of quantum bits ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> qubits), quantum gates, and circuit depth. An open-source library of different arithmetic operators based on our proposed method is presented, where all circuits are implemented on IBM Qiskit SDK. Extensive experiments demonstrate that our proposed design, together with the optimisation techniques, reduces T-depth by up-to 4.0×, T-count by 3.5×, and qubit consumption by 1.2×.

A formally certified end-to-end implementation of Shor’s factorization algorithm

Quantum computing technology may soon deliver revolutionary improvements in algorithmic performance, but it is useful only if computed answers are correct. While hardware-level decoherence errors have garnered significant attention, a less recognized obstacle to correctness is that of human programming errors-"bugs." Techniques familiar to most programmers from the classical domain for avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to the quantum domain because of its unique characteristics. To address this problem, we have been working to adapt formal methods to quantum programming. With such methods, a programmer writes a mathematical specification alongside the program and semiautomatically proves the program correct with respect to it. The proof's validity is automatically confirmed-certified-by a "proof assistant." Formal methods have successfully yielded high-assurance classical software artifacts, and the underlying technology has produced certified proofs of major mathematical theorems. As a demonstration of the feasibility of applying formal methods to quantum programming, we present a formally certified end-to-end implementation of Shor's prime factorization algorithm, developed as part of a framework for applying the certified approach to general applications. By leveraging our framework, one can significantly reduce the effects of human errors and obtain a high-assurance implementation of large-scale quantum applications in a principled way.

Comparison of Performances for Quantum and Conventional Algorithms: Shor’s algorithm and Boson Sampling

Quantum computing has been extensive popularity discussed recently. The foundation of quantum study is the investigation of how quantum computing shows its superiority compared with conventional algorithms. On this basis, this paper focuses on the comparison of the performances of quantum and conventional algorithms about the same relevant area. Two classic problems, factorization of large integer and boson sampling, are chosen to show the specific difference of related quantum and conventional algorithms towards the same problem. This paper specifically compares the performance of Shor's algorithm and other advanced traditional integer factorization algorithms for prime factorization problems and compares the processing capabilities of traditional algorithms and Gaussian Boson sampling for sampling problems. According to the analysis, for the same research problem, quantum algorithms show strong advantages over traditional algorithms in many performances, e.g., time efficiency, algorithm adaptability. It leads to a deeper understanding of quantum superiority and a further explanation of the significance of the development of quantum computing.

Quantum finite difference solvers for physics simulation

AbstractPhysics systems are becoming increasingly complex and require more and more computing time. Quantum computing, which has shown its efficiency on some problems, such as the factorisation of a number with Shor's algorithm, may be the solution to reduce these computation times. Here, the authors propose two quantum numerical schemes for the simulation of physics phenomena, based on the finite difference method. The aim is to see if quantum versions of standard numerical schemes offer an advantage over their classical counterparts, either in accuracy, stability or computation time. First, the authors will present the different phenomena studied as well as the classical solution methods chosen. The authors will then describe the implementation of the quantum numerical schemes and present some results obtained on the different physics phenomena beforehand and then compare both approaches, classical and quantum.

Protection from A Quantum Computer Cyber-Attack

Encryption is required to address the ongoing issue of information security within communication networks. In the second part of the 20th century, quantum computing was first utilized to crack encryption protocols with the SHOR algorithm's development. The transformation from traditional to computational has the possibility of jeopardizing the protection of the current transaction system, even though recent improvements in QC functionality have increased the privacy and security, integrity, and accessibility of networks by shielding them from attacks like eavesdropping. This research adopts a comprehensive literature review approach to clarify the effects of quantum computers on information security. In addition, the study provides a summary of current efforts to guard against quantum attacks. Examining all suggested information security and privacy safeguards, this study examines the additional issue that quantum computers would severely damage information security.

Distributed Shor's algorithm

Shor's algorithm is one of the most important quantum algorithm proposed by Peter Shor [Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124--134]. Shor's algorithm can factor a large integer with certain probability and costs polynomial time in the length of the input integer. The key step of Shor's algorithm is the order-finding algorithm, the quantum part of which is to estimate $s/r$, where $r$ is the ``order" and $s$ is some natural number that less than $r$. {{Shor's algorithm requires lots of qubits and a deep circuit depth, which is unaffordable for current physical devices.}} In this paper, to reduce the number of qubits required and circuit depth, we propose a quantum-classical hybrid distributed order-finding algorithm for Shor's algorithm, which combines the advantages of both quantum processing and classical processing. {{ In our distributed order-finding algorithm, we use two quantum computers with the ability of quantum teleportation separately to estimate partial bits of $s/r$.}} The measuring results will be processed through a classical algorithm to ensure the accuracy of the results. Compared with the traditional Shor's algorithm that uses multiple control qubits, our algorithm reduces nearly $L/2$ qubits for factoring an $L$-bit integer and reduces the circuit depth of each computer.

Delegating signing rights in a multivariate proxy signature scheme

In the context of digital signatures, the proxy signature holds a significant role of enabling an original signer to delegate its signing ability to another party (i.e., proxy signer). It has significant practical applications. Particularly it is useful in distributed systems, where delegation of authentication rights is quite common. For example, key sharing protocol, grid computing, and mobile communications. Currently, a large portion of existing proxy signature schemes are based on the hardness of problems like integer factoring, discrete logarithms, and/or elliptic curve discrete logarithms. However, with the rising of quantum computers, the problem of prime factorization and discrete logarithm will be solvable in polynomial-time, due to Shor's algorithm, which dilutes the security features of existing ElGamal, RSA, ECC, and the proxy signature schemes based on these problems. As a consequence, construction of secure and efficient post-quantum proxy signature becomes necessary. In this work, we develop a post-quantum proxy signature scheme Mult-proxy, relying on multivariate public key cryptography (MPKC), which is one of the most promising candidates of post-quantum cryptography. We employ a 5-pass identification protocol to design our proxy signature scheme. Our work attains the usual proxy criterion and a one-more-unforgeability criterion under the hardness of the Multivariate Quadratic polynomial (MQ) problem. It produces optimal size proxy signatures and optimal size proxy shares in the field of MPKC.

Machine Learning Through Quantum Computing: Validation of a New and Sustainable Approach

The onset of quantum fever in India is ever growing and fascinating. Being a common chapter to subjects like Computer Science, physics and engineering, the basics of so called "quantum advantage" or "quantum supremacy" foundation is being laid. The usage of quantum computers is keeping on increasing and is no more a theoretical utopia since real quantum computers are now available which have been proved to serve results faster than regular modern-day computers. The workings demand well established such as Shor's and Grover's Algorithms. This paper is a short epiphany of the degree and heights of usage of quantum computers in the field of machine learning. An exponential rise has taken place in understanding machine learning through quantum computing due to its vast ranges through which it can be applied. For instance, if a classical artificial neuron can produce results up to N dimensions, then a corresponding quantum perception has the ability of processing results in about 2N dimensions. This paper rather excavates the above-mentioned possibilities and tries for understanding new prospects, keeping in mind that quantum machine intelligence will probably be the pioneering research field in the years to come.

Quasi-Shor Algorithms for Global Benchmarking of Universal Quantum Processors

This work generalizes Shor’s algorithm into quasi-Shor algorithms by replacing the modular exponentiation with alternative unitary operations. By using the quantum circuits to generate Bell states as the unitary operations, a specific example called the Bell–Shor algorithm was constructed. The system density matrices in the quantum circuits with four distinct input states were calculated in ideal conditions and illustrated through chromatic graphs to witness the evolution of quantum states in the quantum circuits. For the real part of the density matrices, it was revealed that the number of zero elements dramatically declined to only a few points after the operation of the inverse quantum Fourier transformation. Based on this property, a protocol constituting a pair of error metrics Γa and Γb is proposed for the global benchmarking of universal quantum processors by looking at the locations of the zero entries and normalized average values of non-zero entries. The protocol has polynomial resource requirements with the scale of the quantum processor. The Bell–Shor algorithm is capable of being a feasible setting for the global benchmarking of universal quantum processors.

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.

Error-mitigated deep-circuit quantum simulation of open systems: Steady state and relaxation rate problems

Deep-circuit quantum computation, like Shor's algorithm, is undermined by error accumulation, and near-future quantum techniques are far from adequate for full-fledged quantum error correction. Instead of resorting to shallow-circuit quantum algorithms, recent theoretical research suggests that digital quantum simulation (DQS) of closed quantum systems are robust against the accumulation of Trotter errors, as long as local observables are concerned. In this paper, we investigate digital quantum simulation of open quantum systems. First, we prove that the deviation in the steady state obtained from digital quantum simulation depends only on the error in a single Trotter step, which indicates that error accumulation may not be disastrous. By numerical simulation of the quantum circuits for the DQS of the dissipative XYZ model, we then show that the correct results can be recovered by quantum error mitigation as long as the error rate in the DQS is below a sharp threshold. We explain this threshold behavior by the existence of a dissipation-driven quantum phase transition. Finally, we propose a new error-mitigation technique based on the scaling behavior in the vicinity of the critical point of a quantum phase transition. Our results expand the territory of near-future available quantum algorithms and stimulate further theoretical and experimental efforts in practical quantum applications.

Design and implementation of ECC combined with OPT encryption algorithm

【Objective】 Combining ECC and OPT, a novel elliptic curve algorithm was proposed, which can effectively resist the reverse calculation of quantum computer and ensure the security of private key. 【Method】 On the basis of elliptic curve encryption algorithm, OPT algorithm was introduced to protect the private key. 【Result】 The signature and verification of messages can be successfully completed by the repeated algorithm on Python platform. Finally, the algorithm is proved to be safe under Shor algorithm and quantum computer attack through algorithm theory. 【Conclusion】 ECC combined with OPT encryption algorithm can effectively resist reversible computation of functional quantum devices, prevent eavesdroppers from forging signatures, and ensure the security of modern public key cryptosystems.

Implementing a fast unbounded quantum fanout gate using power-law interactions

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a $D$-dimensional lattice in time logarithmic in the number of qubits for interactions with $\alpha \le D$. As a corollary, we show that power-law systems with $\alpha \le D$ are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques.

Deterministic algorithms for the hidden subgroup problem

The hidden subgroup problem (HSP) plays a crucial role in the field of quantum computing, since several celebrated quantum algorithms including Shor's algorithm have a uniform description in the framework of HSP. The problem is as follows: for a finite group G and a finite set X, given a function f:G→X and the promise that for any g1,g2∈G,f(g1)=f(g2) iff g1H=g2H for a subgroup H≤G, the goal of the decision version is to determine whether H is trivial, and the goal of the search version is to find H. Nayak (2021) asked whether there exist deterministic algorithms with O(|G||H|) query complexity for HSP. We answer this problem for Abelian groups, which also extends the main results of Ye et al. (2021), since here the algorithms do not rely on any prior knowledge of H.

Coherence as a Resource for Shor’s Algorithm

Shor's factoring algorithm provides a superpolynomial speedup over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the protocol, which depend on rigorously defined measures of coherence as a dynamical resource. We compare these bounds with the classical limit of the protocol and conclude that within the fixed structure that we consider, coherence is the quantum resource that determines its performance by bounding the success probability from below and above. Therefore, we shine new light on the fundamental role of coherence in quantum computation.