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RSA Encryption and the Shor Factoring Algorithm

Alice Flarend, Bob Hilborn

Published Apr 15, 2022• 0 citations
Publisher

Abstract

Abstract This chapter is the most formidable in the book, but it provides a deep and complete understanding of the classic Shor factoring algorithm. Alice and Bob set the context for the algorithm by explaining how the difficulty of factoring large numbers plays a critical role in securing encrypted communications channels, using RSA encryption as an example. The Shor algorithm, if implemented on a sufficiently large quantum computer, could defeat those encryption schemes by finding those factors relatively quickly, a task that would take years or more on even the fastest classical supercomputers. Alice and Bob lead us through some number theory to establish the connection between factoring and finding the period of a periodic function. They then show how the clever use of quantum algorithms—this is Shor’s main contribution—leads to a fast way to find the required period using quantum Fourier transforms.

DOI
10.1093/oso/9780192857972.003.0013

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Abstract This chapter is the most formidable in the book, but it provides a deep and complete understanding of the classic Shor factoring algorithm. Alice and Bob set the context for the algorithm by explaining how the difficulty of factoring large numbers plays a critical role in securing encrypted communications channels, using RSA encryption as an example. The Shor algorithm, if implemented on a sufficiently large quantum computer, could defeat those encryption schemes by finding those factors relatively quickly, a task that would take years or more on even the fastest classical supercomputers. Alice and Bob lead us through some number theory to establish the connection between factoring and finding the period of a periodic function. They then show how the clever use of quantum algorithms—this is Shor’s main contribution—leads to a fast way to find the required period using quantum Fourier transforms.
Abstract This chapter sets the background for the famous Shor algorithm that can allow a sufficiently powerful quantum computer to easily break the current classical RSA encryption method. The RSA algorithm relies on the impossibility of even the most powerful classical computers to factor large numbers in any practical length of time. Before going through the details of the RSA encryption method, the number theory of the modulus or mod function is explained with many examples including exponential ones. The connection between factoring and the periodicity of mod functions is emphasized because it is needed for the RSA and Shor algorithms. The in-depth explanation of the RSA algorithm makes use of number theory to explain how the encryption key can be made public while other information is kept secret to ensure secure communication.
Abstract Expanding on the factoring and periodicity in Chapter 14, the Shor algorithm is explained in detail, with a warning that this is one of the toughest chapters in the book. While the steps of Shor algorithm may look just like the RSA in reverse, the quantum computer is needed to find the periodicity of the factors and thereby the prime factors themselves. Fourier analysis is the tool for this. The quantum Fourier transform is described, including the quantum circuit, after first showing what the classical Fourier transform accomplishes. Extensive examples of all the steps are given.
“Shor's Algorithm: Factoring and Cryptanalysis” focuses on Shor's Algorithm, particularly its application to factoring and cryptanalysis. It begins with an overview of Shor's algorithm, detailing the classical extraction of periods from measured values and the quantum gates required for the Quantum Fourier Transform (QFT). Further, this chapter explores the applications and limitations of Shor's algorithm, including its impact on RSA encryption. A literature review follows, discussing Shor's algorithm in the context of quantum computation, its effectiveness on RSA, and the effects of imperfections on its factorization process. The section also includes an experimental study of Shor's algorithm using IBM Q, followed by a practical implementation of factoring numbers N=21 and N=35. Further, it covers resource estimation, quantum simulations, and different types of simulators in IBM Qiskit, including state vector, QASM, Matrix Product State (MPS), and stabilizer simulators, providing insights into their use cases and the limits of classical simulation.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (conventional) computers. In 1994 Peter Shor showed that for sufficiently large N, a quantum computer could perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the nonexpert, the RSA encryption protocol; the various quantum computer manipulations constituting the Shor algorithm; how the Shor algorithm performs the factoring; and the precise sense in which a quantum computer employing Shor’s algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer. It is made apparent that factoring N generally requires many successive runs of the algorithm. Our analysis reveals that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.
The present study explores the groundbreaking possibilities of Shor's algorithm in cracking RSA encryption, hence enhancing the security consequences of this extensively employed cryptographic protocol. Due to Shor's algorithm's exponential speed advantage over classical algorithms, RSA encryption—a popular public key cryptosystem—is vulnerable to quantum attacks. To keep its security, RSA depends on the difficulty of factoring huge semi-primes. RSA's security presumptions are called into question by Shor's algorithm's quick factorization on a quantum computer, which has prompted research into post-quantum cryptography solutions to guarantee secure communication in the quantum age. Even with parallel computation, the speed and storage capacity of classical computing are constrained. A key idea in quantum computing is quantum parallelism, which allows quantum systems to carry out several computations at once. In contrast to classical portions. The scope of this paper is to understand the fundamentals of Quantum Computing, the current state of Shor’s algorithm implementation, and efficiency, particularly its application in compromising public key cryptosystems like RSA. The project aims to achieve how factorization is done through classical methods and further, how is this done through Shor’s algorithm and QFT techniques.