Transcendental logarithm problem is a new problem which can be used to build signature schemes. Although no polynomial time algorithm or sub-exponential time algorithm has been found to solve this problem, whether it is still an intractable problem with quantum computers is a question. In this paper, we solve the transcendental logarithm problem with improved Grover's quantum search algorithm. In allusion to some characteristics of the transcendental logarithm problem, the average number of the Grover iterations can be reduced to lower the time complexity. The algorithm calls the oracle operator fewer times than before. According to our theoretic analysis and simulation data, cryptosystems based on transcendental logarithm problem can be improved through increasing the length of the key or modifying the original problem by adding suitable parameters to lower the number of solutions.
Transcendental logarithm problem is a new problem which can be used to build signature schemes. Although no polynomial time algorithm or sub-exponential time algorithm has been found to solve this problem, whether it is still an intractable problem with quantum computers is a question. In this paper, we solve the transcendental logarithm problem with improved Grover's quantum search algorithm. In allusion to some characteristics of the transcendental logarithm problem, the average number of the Grover iterations can be reduced to lower the time complexity. The algorithm calls the oracle operator fewer times than before. According to our theoretic analysis and simulation data, cryptosystems based on transcendental logarithm problem can be improved through increasing the length of the key or modifying the original problem by adding suitable parameters to lower the number of solutions.
Quantum computation is a new computing model based on fundamental quantum mechanical principle. Grover's algorithm finds the solution for a searching problem in the square root time of exhaustive search. Brassard, Hoyer, Tapp's algorithm counts the number of solutions for a searching problem. Through exploiting the two quantum algorithms, we propose a quantum algorithm for solving a new cryptography problem -- polynomial root finding problem, which could be used to design a cryptosystem. The algorithm will take O(rootM/t) steps for finding one of the t solutions to the problem, where M is the modular of the equation. The success rate of the algorithm is a constant and the cost of the algorithm depends on the calculations of modular exponentiation and the number of iterations.
The K-SAT problem is a fundamental challenge in computational theory, widely used in artificial intelligence, cryptography, and optimization. Classical algorithms struggle with exponential time complexity, making them inefficient for large-scale instances. In this research, we propose an efficient quantum computing approach based on Grover’s algorithm to enhance the search process for satisfiable solutions in K-SAT problems. Our implementation leverages quantum superposition and amplitude amplification to explore multiple possible solutions simultaneously, reducing the search complexity from O (2ⁿ) in classical methods to O(√2ⁿ) in quantum computing. We design a quantum oracle that encodes the K-SAT clauses and integrates it into Grover’s iterative search framework. The performance is evaluated through Qiskit simulations, demonstrating a significant improvement in search efficiency compared to classical brute-force techniques. The results highlight the potential of quantum algorithms in solving complex combinatorial problems with enhanced speed and accuracy. This study contributes to the development of quantum-accelerated optimization methods, paving the way for real-world applications in machine learning, cryptanalysis, and large-scale data processing.
Root finding problem is a hard problem in computational number theory. To improve the efficiency of solving the problem and expand the scope of applications of quantum computing,the quantum algorithms for solving the root finding problem are analyzed in this paper. Based on the two important quantum algorithms—Shor's algorithm and Grover's algorithm, two quantum mechanical algorithms are proposed to solve the root finding problem,which are named RF-Shor algorithm and RFGrover algorithm. According to the analysis,the RF-Shor algorithm can find all the solutions to the root finding problem with a probability closed to 100%,with a consumption of polynomial quantum gates; the RF-Grover algorithm does not involve any classical method to improve the search efficiency and one of the k solutions to the root finding problem can be obtained in O(M/k) steps with a probability of at least50%.
Quantum search is one of the most important algorithmic developments in quantum computing because it demonstrates how quantum principles can improve the efficiency of solving unstructured search problems. Grover’s algorithm provides a quadratic speed-up over classical linear search by reducing query complexity from O(N) to O(√N), where N represents the size of the search space. This review examines the principles, performance, and applications of Grover’s algorithm, with emphasis on its theoretical foundation and practical relevance. The discussion covers the roles of superposition, quantum interference, oracle operations, diffusion operators, and amplitude amplification in increasing the probability of measuring a desired solution. The manuscript also reviews the computational performance of quantum search in comparison with classical search methods and considers its relevance to database searching, combinatorial optimisation, cryptography, machine learning, bioinformatics, quantum chemistry, and materials science. The review highlights that Grover’s algorithm is not limited to database retrieval, as many computationally intensive problems can be reformulated as search or optimisation tasks. However, practical implementation remains constrained by current quantum hardware limitations, including decoherence, quantum noise, limited qubit availability, circuit depth, and the difficulty of designing efficient oracles. Despite these challenges, continuing progress in quantum hardware, error correction, cloud-based quantum platforms, and hybrid quantum-classical architectures suggests that quantum search will remain an important area of research. Overall, Grover’s algorithm represents a foundational quantum algorithm with significant theoretical value and potential long-term application in computational domains involving large and unstructured search spaces.
Abstract The article proposes a modification of Grover’s quantum oracle quantum search algorithm, which makes it easier to search the database. The algorithm is implemented in the Python programming language using the Reggeti Forest cloud quantum service. The authors of the article use the mean flipping method, which solves the search problem during the iterative order √ (2 ^ n). Development offers great potential for the practical application of the Grover quantum algorithm, as it is characterized by higher performance and speed when performing research. Theoretically, the algorithm provides quadratic acceleration compared to conventional computers. It is not an exponential acceleration, but it remains important for large data carriers. The quantum parallelism of the Grover search algorithm is based on a simultaneous change in the amplitudes of all the inputs. This is done through a superposition of states, which is a purely quantum concept. In addition, the research is carried out globally, which indicates a significant improvement in optimization procedures. Grover’s algorithm, on the other hand, is sensitive to the number of iterations. The more iterations, the smaller the amplitude of the correct answer, so the wrong choice of this parameter can digest the solution. In addition, the operation of the algorithm is limited in the case of the introduction of noise into a quantum system, which is real in modern quantum computers.
Grover quantum search algorithm access the unsorted database O (N~(1/2)) times,however,the probability of acquiring so- lutions usually falls with the increase of the number of solutions,the reason of which is analyzed and an improved algorithm based on en- larging searching space is proposed with a new parameter i to make the algorithm controllable in order to trade off between success proba- bility and iterative steps.Simulated experiments of the reciprocal problem showcase that this proposed algorithm can find solutions with higher probability than traditional Grover algorithm at the condition of equivalent iterative steps,and the success probability can even be increased if the iterative steps is unlimited.Thus,it can conclude that our proposed algorithm is more efficient and effective.The validity of the improved algorithm is manifested by experiment.