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International Conference on Computational Intelligence and Security (CIS)

Quantum Algorithm for Polynomial Root Finding Problem

Guodong Sun, Shenghui Su, Maozhi Xu

Published Nov 1, 2014• 14 citations
Publisher

Abstract

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.

DOI
10.1109/cis.2014.40

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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.
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%.
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.
This paper's aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover\cite{grover}. Grover's algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this paper is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover's algorithm or similar techniques, which can be used as subroutines in many quantum algorithms.}{The tools we provide are carefully constructed: they are easy to use, and in many cases they are asymptotically faster than the best tools previously available. The tools we build on include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer.}{After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known deterministic classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible deterministic classical algorithm. These algorithms solve problems from the fields of graph theory and computational geometry, and some employ dynamic programming techniques. We discuss a breadth-first search that is faster than Θ(edges)\Theta(\text{edges})Θ(edges) (the classical limit) in a dense graph, maximum-points-on-a-line in O(N3/2lg⁡N)O(N^{3/2}\lg N)O(N3/2lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.
Abstract: Grover’s quantum search algorithm is one of the most widely studied and has produced results in some search applications faster than their classical counterpart by a square-root. This paper modifies Grover’s algorithm to solve nonlinear equations over Galois Finite field GF(q) in O( mn 2) iteration, while the best classical general solution takes O(2 nm) iteration. The modification is done by using a register for each variable and represent it by n qubits. The paper also introduces the implementation of the suggested algorithm by using the simulator QCL 5.1.
This thesis’ aim is to explore improvements to, and applications of, a fundamental quantum algorithm invented by Grover. Grover’s algorithm is a basic tool that can be applied to a large number of problems in computer science, creating quantum algorithms that are polynomially faster than fastest known and fastest possible classical algorithms that solve the same problems. Our goal in this thesis is to make these techniques readily accessible to those without a strong background in quantum physics: we achieve this by providing a set of tools, each of which makes use of Grover’s algorithm or similar techniques, that can be used as subroutines in many quantum algorithms. The tools we provide are carefully constructed: they are easy to use, and they are asymptotically faster than the best tools previously available. The tools that we supersede include algorithms by Boyer, Brassard, Hoyer and Tapp, Buhrman, Cleve, de Witt and Zalka and Durr and Hoyer. After creating our tools, we create several new quantum algorithms, each of which is faster than the fastest known classical algorithm that accomplishes the same aim, and some of which are faster than the fastest possible classical algorithm. These algorithms come from graph theory, computational geometry and dynamic programming. We discuss a breadth-first search that is faster than (edges) (the classical limit) in a dense graph, maximum-points-on-a-line in (N3/2 lgN) (faster than the fastest classical algorithm known), as well as several other algorithms that are similarly illustrative of solutions in some class of problem. Through these new algorithms we illustrate the use of our tools, working to encourage their use and the study of quantum algorithms in general.
The development of large quantum computers will have dire consequences for cryptography. Most of the symmetric and asymmetric cryptographic algorithms are vulnerable to quantum algorithms. Grover’s search algorithm gives a square root time boost for the searching of the key in symmetric schemes like AES and 3DES. The security of asymmetric algorithms like RSA, Diffie Hellman, and ECC is based on the mathematical hardness of prime factorization and discrete logarithms. The best classical algorithms available take exponential time. Shor’s factoring algorithm can solve the problems in polynomial time. Major breakthroughs in quantum computing will render all the present-day widely used asymmetric cryptosystems insecure. This paper analyzes the vulnerability of the classical cryptosystems in the context of quantum computers, discusses various post-quantum cryptosystem families, discusses the status of the NIST post-quantum cryptography standardization process, provides a performance comparison of post-quantum cryptography (PQC) algorithms on different platforms, and finally provides a couple of future research directions in this field.