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}
}
  • P. Shor
  • Published 30 August 1995
  • Computer Science
  • SIAM J. Comput.
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.

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References

SHOWING 1-10 OF 119 REFERENCES

Algorithms for quantum computation: discrete logarithms and factoring

  • P. Shor
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.

Quantum Cryptanalysis of Hidden Linear Functions (Extended Abstract)

It is shown that any cryptosystem based on what is referred to as a ‘hidden linear form’ can be broken in quantum polynomial time and the notion of ‘junk bits’ is introduced which are helpful when performing classical computations that are not injective.

Quantum Computers, Factoring, and Decoherence

Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm, a problem of practical significance for cryptographic applications.

Good quantum error-correcting codes exist.

  • CalderbankShor
  • Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1996
The techniques investigated in this paper can be extended so as to reduce the accuracy required for factorization of numbers large enough to be difficult on conventional computers appears to be closer to one part in billions.

Quantum Computation and Shor's Factoring Algorithm

The authors give an exposition of Shor's algorithm together with an introduction to quantum computation and complexity theory, and discuss experiments that may contribute to its practical implementation.

The quantum challenge to structural complexity theory

  • A. BerthiaumeG. Brassard
  • Physics, Computer Science
    [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference
  • 1992
There are cryptographic tasks that are demonstrably impossible to implement with unlimited computing power probabilistic interactive turning machines, yet they can be implemented even in practice by quantum mechanical apparatus.

Quantum Circuit Complexity

  • A. Yao
  • Computer Science
    FOCS
  • 1993
It is shown that any function computable in polynomial time by a quantum Turing machine has aPolynomial-size quantum circuit, and this result enables us to construct a universal quantum computer which can simulate a broader class of quantum machines than that considered by E. Bernstein and U. Vazirani (1993), thus answering an open question raised by them.

Quantum computation

This article gives an introduc tion to quantum computing and briefly looks at a few results in quantum computation, not the least of which is Shor's polynomial-time factoring algorithm.

On the power of quantum computation

  • Daniel R. Simon
  • Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
This work presents here a problem of distinguishing between two fairly natural classes of function, which can provably be solved exponentially faster in the quantum model than in the classical probabilistic one, when the function is given as an oracle drawn equiprobably from the uniform distribution on either class.

Oracle Quantum Computing

Oracles are constructed relative to which there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynometric time if errors are not tolerated.
...