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Theorems on factorization and primality testing

Published online by Cambridge University Press:  24 October 2008

J. M. Pollard
J. M. Pollard
Affiliation:
(Mathematics Department, Plessey Telecommunications Research, Taplow Court, Taplow, Maidenhead, Berkshire)
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Extract

1. Introduction. 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. One way of making these problems precise uses a multi-tape Turing machine (e.g. (1), although we require a version with an input tape). At the start of the calculation n is written in radix notation on one of the tapes, and the machine is to stop after writing out the factors in radix notation or after writing one of two symbols denoting ‘prime’ or ‘composite’. There are, of course, other definitions which could be used; but the differences between these are unimportant for our purpose.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 3 , November 1974 , pp. 521 - 528
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

REFERENCES

(1)Hartmanis, J. and Stearns, R. E.On the computational complexity of algorithms. Trans. Amer. Math. Soc. 117 (1965), 285–306.Google Scholar
(2)Lehmer, D. H. Computer technology applied to the theory of numbers. Studies in Number Theory (LeVeque, W. J., Editor) (Prentice-Hall, Englewood Cliffs N.J. 1969), 117–151.Google Scholar
(3)Burgess, D. A.On character sums and primitive roots. Proc. London Math. Soc. 12 (1962), 179–192.Google Scholar
(4)Berlekamp, E. R.Factoring polynomials over large finite fields. Math. Comp. 24 (1970), 713–735.CrossRefGoogle Scholar
(5)Shanks, D. Class number, a theory of factorisation, and genera. 1969 Number Theory Institute, Proc. Sympos. Pure Math. 20, Amer. Math. Soc., Providence, Rhode Island (1970), 415–440.Google Scholar
(6)Lehman, R. S. Factoring large integers. To appear in Math. Comp.Google Scholar
(7)Pollard, J. M.An algorithm for testing the primality of any integer. Bull. London Math. Soc. 3 (1971), 337–340.CrossRefGoogle Scholar
(8)Good, I. J.The interaction algorithm and practical Fourier analysis. J. Roy. Statist. Soc. Ser. B 20 (1958), 361–372.Google Scholar
(9)Cooley, J. W. and Tukey, J. W.An algorithm for the machine computation of complex Fourier series. Math. Comp. 19 (1965), 297–301.Google Scholar
(10)Schonnage, A. and Strassen, V.Schnelle Multiplikation grosser Zahlen. Computing(Arch. Elektron Rechnen), 7 (1971), 281–292.CrossRefGoogle Scholar
(11)Pollard, J. M.The fast Fourier transform in a finite field. Math. Comp. 25 (1971), 365–374.Google Scholar
(12)Knuth, D. E.The art of computer programming, Volume 2 (revised edition 1971), Semi-numerical algorithms. Addison-Wesley, New York, 1971.Google Scholar
(13)Norton, Karl K. Numbers with small prime factors and the least kth power non-residue. Memoirs of the Amer. Math. Soc. 106, Amer. Math. Soc., Providence, Rhode Island, 1971.CrossRefGoogle Scholar
(14)Brillhart, John and Selfridge, J. L.Some factorizations of n ± 1 and related results. Math. Comp. 21 (1967), 87–96.Google Scholar
(15)Borning, Alan. Some results for k! ± 1 and 2.3.5 … xp + 1. Math. Comp. 26 (1972), 567–570.Google Scholar