Shor's Algorithm Research Articles

Quantum Computing Mathematical Foundations and Practical Implications

Quantum computing is a rapidly advancing field with the potential to revolutionize computation. This paper provides an overview of quantum computing, emphasizing its mathematical foundations and practical implications. We discuss key concepts from quantum mechanics that form the basis of quantum computing, such as superposition and entanglement, and explore quantum algorithms like Shor's algorithm and Grover's algorithm. The paper also examines the practical implications of quantum computing in cryptography, optimization, and machine learning, highlighting quantum key distribution, quantum annealing, and quantum neural networks. Furthermore, we discuss the challenges and future directions of quantum computing, including error correction, scalability, and achieving quantum supremacy. Addressing these challenges will pave the way for realizing the full potential of quantum computing and unlocking new possibilities in computation and simulation.

CAPACITIVE COMPLEXITY OF DETERMINING GCD IN THE SHOR S ALGORITHM

The article analyzes the results of finding the period r of the function y = axmodM (a is a random number) which is used in the Shor's factorization algorithm for quantum computers. The module M is the product of two primes p and q. The article analyzes the solutions r obtained for various a, for which the capacitive complexity H of finding the greatest common divisor GCD(ar/2 + 1, M) is the least. A digital quantum computer is a classic processor and its digital quantum coprocessor. A digital quantum coprocessor with hundreds and thousands of digital qubits can be implemented in one programmable logic integrated circuit FPGA. In the Shor’s algorithm, the factorization problem of the number M reduces to the problem of determining the period r of the function y. It is known that GCD(ar/2 + 1, M) can be a divisor of the number M The task of the quantum coprocessor in implementing the Shor’s algorithm is to find the period r. After that it is necessary to find the GCD. Since for random a the problem of finding the period r has many solutions, these solutions can be compared by the value of one of the arguments when finding the GCD - the number ar / 2 . In this case, H = (rlog2a)/2 is taken for analysis. It approximately represents the bit depth of binary codes that a classic computer will have to process when determining the GCD. H can vary over a wide range from tens to thousands of bits even for small values of M. In this research the period r, which ensures the least complexity of the subsequent task of finding the GCD, is most often a solution for a = 3 and a = 2, but it can also occur often with other values of a. To clarify the revealed patterns, especially for large M, it is necessary to conduct additional research.

Enabling accuracy-aware Quantum compilers using symbolic resource estimation

Approximation errors must be taken into account when compiling quantum programs into a low-level gate set. We present a methodology that tracks such errors automatically and then optimizes accuracy parameters to guarantee a specified overall accuracy while aiming to minimize the implementation cost in terms of quantum gates. The core idea of our approach is to extract functions that specify the optimization problem directly from the high-level description of the quantum program. Then, custom compiler passes optimize these functions, turning them into (near-)symbolic expressions for (1) the total error and (2) the implementation cost (e.g., total quantum gate count). All unspecified parameters of the quantum program will show up as variables in these expressions, including accuracy parameters. After solving the corresponding optimization problem, a circuit can be instantiated from the found solution. We develop two prototype implementations, one in C++ based on Clang/LLVM, and another using the Q# compiler infrastructure. We benchmark our prototypes on typical quantum computing programs, including the quantum Fourier transform, quantum phase estimation, and Shor's algorithm.

Privacy-Preserving Location-Based Services Query Scheme Against Quantum Attacks

Location-based service (LBS) provides more and more conveniences to people. However, it also brings potential threats of offending users’ privacy. How to protect users’ privacy in LBS schemes has aroused increasing research interests in recent years. Most of the existing privacy-preserving LBS schemes are based on the hardness of traditional number-theoretic problems such as the integer factorization or the discrete logarithm problems. However, with the development of large scale quantum computers, these traditional problems can be easily solved by Shor's algorithms, hence the security of these LBS schemes is greatly threatened. In this paper, we solve this problem by constructing a privacy-preserving LBS scheme against quantum attacks from an LWE-based key-homomorphic pseudorandom functions (PRF). In our scheme, due to the key-homomorphic property of the PRF, an LBS user only has to compute one PRF value of the target location and the remaining computation is outsourced to a cloud server, which releases the user from heavy computation burden. In addition, by dividing the key encrypting LBS data into two parts and assigning the two parts to the cloud sever and each user respectively, our scheme avoids the threats of key abuse and information leaking of LBS data. Moreover, we use this PRF to realize an authenticated protocol, which protects the communications between the LBS users and the cloud server. We stress that the security of our scheme is based only on the security of the LWE-based key-homomorphic PRF, hence our scheme is the first LBS scheme secure against quantum attacks.

Electric circuits for universal quantum gates and quantum Fourier transformation

Universal quantum computation may be realized based on quantum walk, by formulating it as a scattering problem on a graph. In this paper, we simulate quantum gates through electric circuits, following a recent report that a one-dimensional $LC$ electric circuit can simulate a Schr\"{o}dinger equation and hence a quantum walk. Especially, we propose a physical realization of a set of universal quantum gates consisting of the CNOT, Hadamard and $\pi /4$ phase-shift gates with the use of telegrapher wires and mixing bridges. Furthermore, we construct the $\pi /2^{n}$ phase-shift gate for an arbitrary integer $n$, which is an essential element to perform the quantum Fourier transformation and prime factorization based on the Shor algorithm. Our results will open a way to universal quantum computation based on electric circuits.

Integrated Analysis of Performance and Resources in Large-Scale Quantum Computing

To see the feasibility of a large-scale quantum computing, it is required to accurately analyze the performance and the quantum resource. However, most of the analysis reported so far have focused on the statistical examination, i.e., simply calculating the performance and resource based on individual data, and even worse usually only a few components have been considered. In this work, to achieve more exact analysis, we propose an integrated analysis method for a practical quantum computing model with three components (\textit{algorithm}, \textit{error correction} and \textit{device}) under a realistic quantum computer system architecture. To implement the above method, we develop a quantum computing framework composed of three functional layers: compile, system and building block. This framework can support, for the first time, the mapping of quantum algorithm from physical qubit level to system architecture level with a given fault-tolerant scheme. Therefore, the proposed method can measure the effect of dynamic situation when the quantum computer practically runs. By using our method, we found that Shor algorithm to factorize 512-bit integer requires $8.78\times 10^ 5$ hours. We also show how the proposed method can be used for analyzing optimal concatenation level and code distance of fault-tolerant quantum computing.

Ironwood meta key agreement and authentication protocol

<p style='text-indent:20px;'>Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now showing promise in providing quantum-resistant cryptographic primitives, and may provide suitable alternatives for those looking to address known quantum attacks. In this paper, we introduce a new protocol called a <i>Meta Key Agreement and Authentication Protocol</i> (MKAAP) that has some characteristics of a public-key solution and some of a shared-key solution. Specifically, it has the deployment benefits of a public-key system, allowing two entities that have never met before to authenticate without requiring real-time access to a third-party, but does require secure provisioning of key material from a trusted key distribution system (similar to a symmetric system) prior to deployment. We then describe a specific MKAAP instance, the Ironwood MKAAP, discuss its security, and show how it resists certain quantum attacks such as Shor's algorithm or Grover's quantum search algorithm. We also show Ironwood implemented on several "internet of things" (IoT devices), measure its performance, and show how it performs significantly better than ECC using fewer device resources.

EXPLAINING THE BASICS OF QUANTUM MECHANICS FOR COMPUTING

Quantum computers operate on the principles of quantum logic, a fundamentally distinct paradigm from classical Boolean logic. This disparity leads to quantum computation's enhanced efficiency compared to classical computing. In this comprehensive review, we demystify the fundamental concepts of quantum computation, covering the creation of elementary gates and networks. We highlight the capabilities of quantum algorithms by examining the simple Deutsch problem and, in straightforward terms, dissect the renowned Shor algorithm for factoring large numbers into primes. Furthermore, we delve into the realm of physical quantum computer implementations, with a particular focus on tlo he linear ion trap approach. Here, we shed light on the primary challenge hindering the realization of practical quantum computers: the issue of decoherence. Nonetheless, we demonstrate that this hurdle can be overcome through the application of quantum error correction methods

Security Enhancement of RSA Algorithm using Increased Prime Number Set

In this era of digital age a lot of secret and non-secret data is transmitted over the internet. Cryptography is one of the many techniques to secure data on network. It is one of the techniques that can be used to ensure information security and data privacy. It is used to secure data in rest as well as data in transit. RSA in the most commonly used cryptographic algorithm and it is also used for the creation on Digital Certificates. RSA algorithm is now not considered to be as secure due to advancement in technology and newer attack vectors. This paper proposed an algorithm for security enhancement of RSA algorithm by increasing prime numbers count. Proposed algorithm has been implemented to encrypt and decrypt the data and execution results for encryption and decryption time have been compared for increased prime numbers count. This proposed algorithm of RSA can be used to replace the existing RSA algorithm in digital signature certificates as well as in all other places where the base RSA algorithm is currently being used. In the proposed technique, as the number of prime number count increases, prime factor calculation becomes difficult. If the attacker has encryption key (e) and Product of prime numbers (N) then it is not easy to find out the prime number combinations and hence decryption key (d) will be more secure by using proposed algorithm. This will be more difficult because given a number n, it is easy to find two numbers whose product is equal to n using Shor's algorithm and Grover’s Search Algorithm but it is not very difficult and time taking to exactly determine m numbers whose product is equal to n.

Novel One Time Signatures (NOTS): A Compact Post-Quantum Digital Signature Scheme

The future of the hash based digital signature schemes appears to be very bright in the upcoming quantum era because of the quantum threats to the number theory based digital signature schemes. The Shor's algorithm is available to allow a sufficiently powerful quantum computer to break the building blocks of the number theory based signature schemes in a polynomial time. The hash based signature schemes being quite efficient and provably secure can fill in the gap effectively. However, a draw back of the hash based signature schemes is the larger key and signature sizes which can prove a barrier in their adoption by the space critical applications, like the blockchain. A hash based signature scheme is constructed using a one time signature (OTS) scheme. The underlying OTS scheme plays an important role in determining key and signature sizes of a hash based signature scheme. In this article, we have proposed a novel OTS scheme with minimized key and signature sizes as compared to all of the existing OTS schemes. Our proposed OTS scheme offers an 88% reduction in both key and signature sizes as compared to the popular Winternitz OTS scheme. Furthermore, our proposed OTS scheme offers an 84% and an 86% reductions in the signature and the key sizes respectively as compared to an existing compact variant of the WOTS scheme, i.e. WOTS + .

Public-key cryptosystem based on quantum BCH codes and its quantum digital signature

There is a security threat to the traditional cryptography because of the emergence of quantum computation. Quantum computers can break through many cryptosystems based on mathematical puzzles. The post-cryptography has been established for the sake of the security. It has been proved that McEliece public-key cryptosystem based on error-correcting codes can resist quantum attacks. In this paper, several new generating algorithms of the quantum Bose-Chaudhuri-Hocquenghem (BCH) codes have been proposed, and the quantum BCH codes can correct the errors including bit-flip and phase-flip. At the same time, the quantum McEliece public-key cryptosystem and quantum Niederreiter public key cryptosystem have been designed by us on the basis of quantum BCH codes in this paper, and their encryption and decryption procedures are created in detail. Our cryptosystem not only retains the advantages of postquantum computation, but also can encrypt or decrypt in quantum state. From the perspective of computational complexity theory, both cryptosystems can resist the attacks of Shor algorithm and Grover algorithm very well. The Niederreiter classical digital signature is designed by us according to the structure characteristics of quantum BCH codes.

Quantum Computing: A Comprehensive Review

Quantum computing has emerged as a revolutionary paradigm that promises to solve computational problems beyond the capabilities of classical computers. This comprehensive review paper explores the fundamentals, models, algorithms, technologies, challenges, and practical applications of quantum computing. The paper begins with an introduction to quantum computing, highlighting its defining features and significance. It then discusses the fundamentals of quantum computing, including qubits, quantum gates, quantum entanglement, and quantum parallelism. The paper also examines different quantum computing models, such as the circuit model, adiabatic model, quantum annealing, and topological quantum computing. It further explores quantum algorithms, including Shor's algorithm, Grover's algorithm, the quantum phase estimation algorithm, and the quantum approximate optimization algorithm (QAOA). Additionally, the paper delves into quantum error correction, fault-tolerant quantum computation, and error detection and correction methods. It discusses the challenges faced in quantum computing, such as decoherence, scalability, qubit connectivity, quantum software development, and quantum supremacy. The paper also reviews current quantum computing platforms, applications in cryptography, optimization, and machine learning, and future prospects and challenges in the field.

LCPPA: Lattice‐based conditional privacy preserving authentication in vehicular communication

AbstractVehicular ad hoc network (VANET) plays an extremely important role in intelligent transportation systems. It enables vehicles equipped with smart devices to communicate with vehicles or available road side units or infrastructure. It has been observed that the security of data and privacy of user are two key challenges for vehicular communication to enable sharing of resources among unfamiliar vehicles. The existing classical cryptography‐based schemes are not secure in the quantum computing world due to Shor's algorithm. To address the challenges in post‐quantum era, we have presented a framework based on short integer solution problem in some lattice that attains all effective identification and authenticity of the user to get authorized access, along with revocation to services. The integrity of proposed framework ensures that information is accurate and can be trusted. This paper ensures conditional autonomous driving and efficient vehicular traffic flow in a VANET. As a result, the presented protocol is suitable for practical applications even in the post‐quantum environment, which can be utilized for reliable vehicular communication.

Time-optimal implementations of quantum algorithms

In this paper, we briefly discuss the methodology for simulating a quantum computer which performs Shor's algorithm on a 7-qubit system to factorise 15. Using this simulation and the overlooked quantum brachistochrone method, we devised a Monte Carlo algorithm to calculate the expected time a theoretical quantum computer could perform this calculation under the same energy conditions as current working quantum computers. We found that, experimentally, a nuclear magnetic resonance quantum computer would take $1.59 \pm 0.04$ s to perform our simulated computation, whereas the expected optimal time under the same energy conditions is $0.955 \pm 0.004$ ms. Moreover, we found that the expected time is inversely proportional to the energy variance of our qubit states (as expected). Finally, we propose this theoretical method for analysing the time-efficiency of future quantum computing experiments.

Efficient magic state factories with a catalyzed|CCZ⟩to2|T⟩transformation

We present magic state factory constructions for producing|CCZ⟩states and|T⟩states. For the|CCZ⟩factory we apply the surface code lattice surgery construction techniques described in \cite{fowler2018} to the fault-tolerant Toffoli \cite{jones2013, eastin2013distilling}. The resulting factory has a footprint of12d×6d(wheredis the code distance) and produces one|CCZ⟩every5.5dsurface code cycles. Our|T⟩state factory uses the|CCZ⟩factory's output and a catalyst|T⟩state to exactly transform one|CCZ⟩state into two|T⟩states. It has a footprint25%smaller than the factory in \cite{fowler2018} but outputs|T⟩states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the caseθ=22.5∘produces a particularly efficient circuit for producing|T⟩states. Compared to using the12d×8d×6.5d|T⟩factory of \cite{fowler2018}, our|CCZ⟩factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm \cite{shor1994} and the chemistry algorithm of Babbush et al. \cite{babbush2018}. Assuming a physical gate error rate of10−3, our CCZ factory can produce∼1010states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to∼1012by increasing the factory footprint∼20%.

Integer Factorization – Cryptology Meets Number Theory

Integer factorization is one of the oldest mathematical problems. Initially, the interest in factorization was motivated by curiosity about be­haviour of prime numbers, which are the basic building blocks of all other integers. Early factorization algorithms were not very efficient. However, this dramatically has changed after the invention of the well-known RSA public-key cryptosystem. The reason for this was simple. Finding an efficient fac­toring algorithm is equivalent to breaking RSA. The work overviews development of integer factoring algorithms. It starts from the classical sieve of Eratosthenes, covers the Fermat algorithm and explains the quadratic sieve, which is a good representative of modern fac­toring algorithms. The progress in factoring is illustrated by examples of RSA challenge moduli, which have been factorized by groups of mathemati­cians and cryptographers. Shor's quantum factorization algorithm with poly­nomial complexity is described and the impact on public-key encryption is discussed.

Quantum entanglement involved in Grover’s and Shor’s algorithms: the four-qubit case

In this paper, we study the nature of entanglement in quantum Grover's and Shor's algorithms. So far, the authors who have been interested in this problem have approached the question quantitatively by introducing entanglement measures (numerical ones most of the time). One can ask a different question: what about a qualitative measure of entanglement ? In other words, we try to find what are the different entanglement SLOCC classes that can be generated by these two algorithms. We treat in this article the case of pure four-qubit systems.

Optimising Matrix Product State Simulations of Shor's Algorithm

We detail techniques to optimise high-level classical simulations of Shor's quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on r, the solution to the underlying order-finding problem of Shor's algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shor's algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.

Rashba control to minimize circuit cost of quantum Fourier algorithm in ballistic nanowires

The presented work provides a new prospective of quantum computer hardware development through ballistic nanowires with Rashba effect. We address Rashba effect as a possible mechanism to realize novel qubit gates in nanowires. We apply our results to the design of a new quantum circuit for Fourier algorithm, which shows an improved quantum cost in terms of the number of gates. The current system can be successfully implemented to increase the production of quantum computer experimentally thanks to the absence of impurities. The introduced circuit will enhance the scope of experimental and theoretical research for the realization of other quantum algorithms, such as phase estimation and Shor's algorithms.

Factoring using 2n + 2 qubits with Toffoli based modular multiplication

We describe an implementation of Shor's quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli ba...