Monte Carlo methods for index computation ()

@article{Pollard1978MonteCM,
  title={Monte Carlo methods for index computation ()},
  author={John M. Pollard},
  journal={Mathematics of Computation},
  year={1978},
  volume={32},
  pages={918-924}
}
  • J. Pollard
  • Published 1 September 1978
  • Mathematics, Computer Science
  • Mathematics of Computation
We describe some novel methods to compute the index of any integer relative to a given primitive root of a prime p. Our flrst method avoids the use of stored tables and apparently requires O(p 1/2) operations. Our second algorithm, which may be regarded as a method of catching kangaroos, is applicable when the index is known to lie in a certain interval; it requires O(w/2) operations for an interval of width w, but does not have complete certainty of success. It has several possible areas of… 

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References

SHOWING 1-10 OF 17 REFERENCES

On factorisation, with a suggested new approach

This paper gives a brief survey of methods based mainly on Fermat's Theorem, for testing and establishing primality of large integers. It gives an extension of the Fermat-Lucas-Lehmer Theorems which

A New Factorization Technique Using Quadratic Forms

The paper presents a practical method for factoring an arbitrary N by represent- ing N or XN by one of at most three quadratic forms: XN = x- - Dy2, X = 1,-1, 2, D = -1, ?2, ?3, ?6. These three forms

Theorems on factorization and primality testing

  • J. Pollard
  • Computer Science
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1974
This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality and uses a multi-tape Turing machine for this purpose.

A method of factoring and the factorization of

The continued fraction method for factoring integers, which was introduced by D. H. Lehmer and R. E. Powers, is discussed along with its computer implementation. The power of the method is

Cycle distributions in random nets.

  • A. Rapoport
  • Mathematics
    The Bulletin of mathematical biophysics
  • 1948
Characteristics of random nets are derived from assumptions concerning the distribution of connections and it is shown that in the single aggregate with random connections, the cycle saturation varies inversely as the square root of the number of neurons; in the dense two-chain net it varies in proportional proportion to the neuron density.

Steady states in random nets.

A neural net is taken to consist of a semi-infinite chain of neurons with connections distributed according to a certain probability frequency of the lengths of the axones, and the statistical properties of the net determine a certain steady state output.

The Art of Computer Programming

The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.

Sorting and Searching

The first revision of this third volume is a survey of classical computer techniques for sorting and searching. It extends the treatment of data structures in Volume 1 to consider both large and

New Directions in Cryptography

This paper suggests ways to solve currently open problems in cryptography, and discusses how the theories of communication and computation are beginning to provide the tools to solve cryptographic problems of long standing.

A monte carlo method for factorization

A novel factorization method involving probabilistic ideas is described briefly, and it is suggested that this method should be considered as a viable alternative to traditional factorization methods.