Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

ACM SIGACT News

A short note on Shor's factoring algorithm

Harry Buhrman

Published Mar 1, 1996• 8 citations
Publisher

Abstract

This note shows that Shor's algorithm for factoring in polynomial time on a quantum computer can be made to work with zero error probability.
DOI
10.1145/230514.230515

Related papers

A Note on Shor's Quantum Algorithm for Prime Factorization.

Zhengjun Cao

Abstract It’s well known that Shor[1] proposed a polynomial time algorithm for prime factorization by using quantum computers. For a given number n, he gave an algorithm for finding the order r of an element x (mod n) instead of giving an algorithm for factoring n directly. The indirect algorithm is feasible because factor-ization can be reduced to finding the order of an element by using randomization[2]. But a point should be stressed that the order of the number must be even. Actually, the restriction can be removed in a particular case. In this paper, we show that factoring RSA modulus (a product of two primes) only needs to find the order of 2, whether it is even or not. Keywords Shor’s quantum algorithm, RSA modulus. 1

•Jan 1, 2005•1 citation

Using Single Time of Quantum Computer for Shor's Factoring Algorithm

Yousef Alhammadi

Peter Shor's in 1994 developed a factoring algorithm that can factor a composite number$\boldsymbol{n}$in a polynomial time by using a quantum computer in one step of the algorithm. The algorithm may contain a loop that requires calling a quantum computer many times to find the order of selected number$a, a\in Z_{n}$. In this paper, the improvement to the Shor's factoring algorithm guarantees the single use of the quantum computer to find the order, where the rest of the algorithm can be implemented by a classical computer. A numerical example for the original and improved version is presented.

International Conference on IT Innovation and Knowledge Discovery (ITIKD)•Mar 8, 2023•0 citations

Shor's factorization algorithm

E. Desurvire

This chapter describes what is generally considered to be one of the most important and historical contributions to the field of quantum computing, namely Shor's factorization algorithm . As its name indicates, this algorithm makes it possible to factorize numbers, which consists in their decomposition into a unique product of prime numbers. Other classical factorization algorithms previously developed have a complexity or computing time that increases exponentially with the number size, making the task intractable if not hopeless for large numbers. In contrast, Shor's algorithm is able to factor a number of any size in polynomial time , making the factorization problem tractable should a quantum computer ever be realized in the future. Since Shor's algorithm is based on several nonintuitive properties and other mathematical subtleties, this chapter presents a certain level of difficulty. With the previous chapters and tools readily assimilated, and some patience in going through the different preliminary steps required, such a difficulty is, however, quite surmountable. I have sought to make this description of Shor's algorithm as mathematically complete as possible and crack-free, while avoiding some academic considerations that may not be deemed necessary from any engineering perspective. Eventually, Shor's algorithm is described in only a few basic instructions. What is conceptually challenging is to grasp why it works so well, and also to feel comfortable with the fact that its implementation actually takes a fair amount of trial and error. The two preliminaries of Shor's algorithm are the phase estimation and the related order-finding algorithms.

Cambridge University Press eBooks•Feb 19, 2009•0 citations

Shor's Factoring Algorithm and Modern Cryptography. An Illustration of the Capabilities Inherent in Quantum Computers

Gerjuoy, E

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 (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the Shor algorithm; (iii) how the Shor algorithm performs the factoring; and (iv) 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 can. It is made apparent that factoring $N$ generally requires many successive runs of the algorithm. The careful analysis herein reveals, however, that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.

•Nov 25, 2004•50 citations

Shor’s factoring algorithm and modern cryptography. An illustration of the capabilities inherent in quantum computers

Edward Gerjuoy

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.

arXiv:quant-ph/0411184•American Journal of Physics•May 13, 2005•76 citations

Shor's algorithm without partial fractions

Nolan R. Wallach

The purpose of this note was to give a proof that Shor's algorithm for period search is polynomial using only the standard $2^{n}$ quantum Fourier thansform and some simple trigonometry. There is an error that was pointed out to the author by Pavel Wocjan.

arXiv:1208.1987•Aug 9, 2012•0 citations
Abstract It’s well known that Shor[1] proposed a polynomial time algorithm for prime factorization by using quantum computers. For a given number n, he gave an algorithm for finding the order r of an element x (mod n) instead of giving an algorithm for factoring n directly. The indirect algorithm is feasible because factor-ization can be reduced to finding the order of an element by using randomization[2]. But a point should be stressed that the order of the number must be even. Actually, the restriction can be removed in a particular case. In this paper, we show that factoring RSA modulus (a product of two primes) only needs to find the order of 2, whether it is even or not. Keywords Shor’s quantum algorithm, RSA modulus. 1
Peter Shor's in 1994 developed a factoring algorithm that can factor a composite numbern\boldsymbol{n}nin a polynomial time by using a quantum computer in one step of the algorithm. The algorithm may contain a loop that requires calling a quantum computer many times to find the order of selected numbera,a∈Zna, a\in Z_{n}a,a∈Zn​. In this paper, the improvement to the Shor's factoring algorithm guarantees the single use of the quantum computer to find the order, where the rest of the algorithm can be implemented by a classical computer. A numerical example for the original and improved version is presented.
This chapter describes what is generally considered to be one of the most important and historical contributions to the field of quantum computing, namely Shor's factorization algorithm . As its name indicates, this algorithm makes it possible to factorize numbers, which consists in their decomposition into a unique product of prime numbers. Other classical factorization algorithms previously developed have a complexity or computing time that increases exponentially with the number size, making the task intractable if not hopeless for large numbers. In contrast, Shor's algorithm is able to factor a number of any size in polynomial time , making the factorization problem tractable should a quantum computer ever be realized in the future. Since Shor's algorithm is based on several nonintuitive properties and other mathematical subtleties, this chapter presents a certain level of difficulty. With the previous chapters and tools readily assimilated, and some patience in going through the different preliminary steps required, such a difficulty is, however, quite surmountable. I have sought to make this description of Shor's algorithm as mathematically complete as possible and crack-free, while avoiding some academic considerations that may not be deemed necessary from any engineering perspective. Eventually, Shor's algorithm is described in only a few basic instructions. What is conceptually challenging is to grasp why it works so well, and also to feel comfortable with the fact that its implementation actually takes a fair amount of trial and error. The two preliminaries of Shor's algorithm are the phase estimation and the related order-finding algorithms.
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 (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the Shor algorithm; (iii) how the Shor algorithm performs the factoring; and (iv) 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 can. It is made apparent that factoring NNN generally requires many successive runs of the algorithm. The careful analysis herein reveals, however, that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.
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 purpose of this note was to give a proof that Shor's algorithm for period search is polynomial using only the standard 2n2^{n}2n quantum Fourier thansform and some simple trigonometry. There is an error that was pointed out to the author by Pavel Wocjan.