# Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

@article{Shor1995PolynomialTimeAF, title={Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer}, author={Peter W. Shor}, journal={SIAM J. Comput.}, year={1995}, volume={26}, pages={1484-1509} }

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. [] Key Method Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

## 8,682 Citations

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

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
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It is shown that factoring RSA modulus (a product of two primes) only needs to find the order of 2, whether it is even or not.

### Quantum algorithms for computing short discrete logarithms and factoring RSA integers

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The quantum algorithm for computing short discrete logarithms is generalized to allow for various tradeoffs between the number of times that the algorithm need be executed, and the complexity of the algorithm and the requirements it imposes on the quantum computer.

### Quantum algorithms for computing short discrete logarithms and factoring RSA integers

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This paper generalizes the quantum algorithm for computing short discrete logarithms to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other hand.

### Implementation of Shor's Algorithm and Reliability of Quantum Computing Devices

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Shor's Algorithm is presented and an implementation, a way to factor number 21 is described and some reliability issues of quantum devices were considered in order to explore the potentiality of Shor's algorithm.

### Discrete Logarithm (1994; Shor)

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The discrete logarithm problem in Zp, p prime as well as in the group of points of an elliptic curve over a finite field, is believed to be intractable for randomised classical computers. That is…

### On computing ord N ( 2 ) and its application

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This paper proposes a new quantum algorithm for factoring RSA modulus without the two drawbacks and shows that the cost of the algorithm mainly depends on the calculation of ordN (2).

### A simplification of the Shor quantum factorization algorithm employing a quantum Hadamard transform

- Computer Science, PhysicsDefense + Security
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Some modifications of the Shor quantum factorization algorithm are proposed by employing some simplification to the stage employing the quantum Fourier transform to reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.

### Quantum Factoring Algorithm: Resource Estimation and Survey of Experiments

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This study investigates the details of quantum circuits used in several factoring experiments and indicates that some of the circuits have been constructed under the condition that the order of an element modulo a target composite is known in advance.

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

- Computer Science
- 2005

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 theShor algorithm performs the factoring; and the precise sense in which a quantum computer employing Shor’s algorithm can be said to accomplish the factored of very large numbers with less computational effort than a classical computer.

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