Doubly-Focused Enumeration of Pseudosquares and Pseudocubes

  • Kjell Wooding
  • Hugh C. Williams
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)


This paper offers numerical evidence for a conjecture that primality proving may be done in (logN)3 + o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences—doubly-focused enumeration— is examined. This technique, first described by D. J. Bernstein, allowed us to obtain record-setting sieve computations in software on general purpose computers.


Arithmetic Progression Residue Class Primality Test Chinese Remainder Theorem General Purpose Computer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Annals of Mathematics 160(2), 781–793 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bach, E.: Explicit bounds for primality testing and related problems. Mathematics of Computation 55(191), 355–380 (1990)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bernstein, D.J.: More news from the Rabin-Williams front. In: Conference slides from Mathematics of Public Key Cryptography (MPKC). University of Illinois at Chicago (2003), Available from:
  4. 4.
    Bernstein, D.J.: Doubly Focused Enumeration Of Locally Square Polynomial Values. In: van der Poorten, A., Stein, A. (eds.) High Primes and Misdemeanors—Lectures in honour of the 60th birthday of Hugh Cowie Williams, pp. 69–76 (2004)Google Scholar
  5. 5.
    Bernstein, D.J.: Proving primality after Agrawal-Kayal-Saxena (unpublished, 2003), Available from:
  6. 6.
    Bernstein, D.J.: Proving primality in Essentially Quartic Random Time. Mathematics of Computation (to appear, 2004), Available from:
  7. 7.
    Berrizbeitia, P.: Sharpening PRIMES is in P for a large family of numbers (preprint, 2002), Available from:
  8. 8.
    Berrizbeitia, P., Müller, S., Williams, H.C.: Pseudocubes and Primality Testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Bronson, N.D., Buell, D.A.: Congruential sieves on FPGA computers, Mathematics of computation. In: Gautschi, W. (ed.) 1943–1993: a half-century of computational mathematics: Mathematics of Computation 50th Anniversary Symposium, Vancouver, British Columbia, August 9–13, pp. 547–551 (1994)Google Scholar
  10. 10.
    Cheng, Q.: Primality Proving via One Round in ECPP and One Iteration in AKS. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 338–348. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Gropp, W., Lusk, E., Skjellum, A.: Using MPI: Portable Parallel Programming with the Message Passing Interface. MIT Press, Cambridge (1994)Google Scholar
  12. 12.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)MATHGoogle Scholar
  13. 13.
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, Heidelberg (1990)MATHGoogle Scholar
  14. 14.
    Kraitchik, M.: Récherches sur la Théorie des Nombres. Tome I Gauthier-Villars (1924)Google Scholar
  15. 15.
    Lehmer, D.H.: The Sieve Problem for All-Purpose Computers. Mathematical Tables and Other Aids to Computation 7(41), 6–14 (1953)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lehmer, D.H.: The mechanical combination of linear forms. American Mathematical Monthly 35(4), 114–121 (1928)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lehmer, D.H., Lehmer, E., Shanks, D.: Integer Sequence having Prescribed Quadratic Character. Mathematics of Computation 24(110), 433–451 (1970)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lukes, R.F.: A Very Fast Electronic Number Sieve. Ph.D. Thesis. University of Manitoba (1995)Google Scholar
  19. 19.
    Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Mathematics of Computation 65(213), 361–372 (1996)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ousterhout, J.K.: Tcl and the Tk Toolkit. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  21. 21.
    Schinzel, A.: On Pseudosquares. New Trends in Probability and Statistics 4, 213–220 (1997)MathSciNetGoogle Scholar
  22. 22.
    Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing 7, 281–292 (1971)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kjell Wooding
    • 1
  • Hugh C. Williams
    • 1
  1. 1.Centre for Information Security and CryptographyUniversity of CalgaryCalgary, AlbertaCanada

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