The CREW PRAM complexity of modular inversion

  • Joachim van zur Gathen
  • Igor Shparlinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)


One of the long-standing open questions in the theory of parallel computation is the parallel complexity of the integer gcd and related problems, such as modular inversion. We present a lower bound Ω(log n) for the CREW PRAM complexity for inversion modulo certain n-bit integers, including all such primes. For infinitely many moduli, our lower bound matches asymptotically the known upper bound. We obtain a similar lower bound for computing a specified bit in a large power of an integer. Our main tools are certain estimates for exponential sums in finite fields.


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  1. 1.
    L. M. Adleman and K. Kompella, ‘Using smoothness to achieve parallelism', Prac. 20th ACM Symp. on Theory of Comp., (1988), 528–538.Google Scholar
  2. 2.
    P. W. Beame, S. A. Cook and H. J. Hoover, ‘Log depth circuits for division and related problems', SIAM J. Comp., 15 (1986) 994–1003.CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    S. A. Cook, C. Dwork and R. Reischuk, ‘Upper and lower time bounds for parallel random access machines without simultaneous writes', SIAM J. Comp., 15 (1986), 87–97.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Coppersmith and I. E. Shparlinski, ‘On polynomial approximation and the parallel complexity of the discrete logarithm and breaking the Diffie-Hellman cryptosystem', Research Report RC 20724, IBM T. J. Watson Research Centre, 1997, 1–103.Google Scholar
  5. 5.
    M. Dietzfelbinger, M. Kutylowski and R. Reischuk, ‘Exact time bounds for computing Boolean functions on PRAMs without simultaneous writes', J. Comp. and Syst. Sci., 48 (1994), 231–254.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Dietzfelbinger, M. Kutyłowski and R. Reischuk, ‘Feasible time-optimal algorithms for Boolean functions on exclusive-write parallel random access machine', SIAM J. Comp., 25 (1996), 1196–1230.CrossRefzbMATHGoogle Scholar
  7. 7.
    F. E. Fich, ‘The complexity of computation on the parallel random access machine', Handbook of Theoretical Comp. Sci., Vol.A, Elsevier, Amsterdam, 1990, 757–804.Google Scholar
  8. 8.
    E. Fich and M. Tompa, ‘The parallel complexity of exponentiating polynomials over finite fields', J. ACM, 35 (1988), 651–667.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Gao, J. von zur Gathen and D. Panario, ‘Gauss periods and fast exponentiation in finite fields', Lecture Notes in Comp. Sci., 911 (1995), 311–322.Google Scholar
  10. 10.
    J. von zur Gathen, ‘Computing powers in parallel', SIAM J. Comp., 16 (1987), 930–945.zbMATHCrossRefGoogle Scholar
  11. 11.
    J. von zur Gathen, ‘Inversion in finite fields using logarithmic depth', J. Symb. Comp., 9 (1990), 175–183.zbMATHCrossRefGoogle Scholar
  12. 12.
    J. von zur Gathen, ‘Efficient and optimal exponentiation in finite fields', Comp. Complexity, 1 (1991), 360–394.zbMATHCrossRefGoogle Scholar
  13. 13.
    J. von zur Gathen, ‘Processor-efficient exponentiation in finite fields', Inform. Proc. Letters, 41 (1992), 81–86.zbMATHCrossRefGoogle Scholar
  14. 14.
    J. von zur Gathen and G. Seroussi, ‘Boolean circuits versus arithmetic circuits', Inform, and Comp., 91 (1991), 142–154.CrossRefzbMATHGoogle Scholar
  15. 15.
    L.-K. Hua, Introduction to number theory, Springer-Verlag, 1982.Google Scholar
  16. 16.
    D. Ismailov, ‘On a method of Hua Loo-Keng of estimating complete trigonometric sums', Adv. Math. (Benijing), 23 (1992), 31–49.Google Scholar
  17. 17.
    R. Kannan, G. Miller and L. Rudolph, ‘Sublinear parallel algorithm for computing the greatest common divisor of two integers', SIAM J. Comp., 16 (1987), 7–16.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    R. Lidl and H. Niederreiter, Finite fields, Addison-Wesley, MA, 1983.zbMATHGoogle Scholar
  19. 19.
    B. E. Litow and G. I. Davida, ‘O(log(n)) parallel time finite field inversion', Lect. Notes in Comp. Science, 319 (1988), 74–80.MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Mnuk, ‘A div (n) depth Boolean circuit for smooth modular inverse', Inform. Proc. Letters, 38 (1991), 153–156.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    I. Parberry and P. Yuan Yan, ‘Improved upper and lower time bounds for parallel random access machines without simultaneous writes', SIAM J. Comp., 20 (1991), 88–99.CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    J. B. Rosser and L. Schoenfeld, ‘Approximate formulas for some functions of prime numbers', Ill. J. Math. 6 (1962), 64–94.MathSciNetzbMATHGoogle Scholar
  23. 23.
    I. E. Shparlinski, Computational and algorithmic problems in finite fields, Kluwer Acad. Publ., Dordrecht, The Netherlands, 1992.zbMATHGoogle Scholar
  24. 24.
    I. E. Shparlinski, ‘Number theoretic methods in lower bounds of the complexity of the discrete logarithm and related problems', Preprint, 1997, 1–168.Google Scholar
  25. 25.
    I. E. Shparlinski and S. A. Stepanov, ‘Estimates of exponential sums with rational and algebraic functions', Automorphic Functions and Number Theory, Vladivostok, 1989, 5–18 (in Russian).Google Scholar
  26. 26.
    S. B. Steckin, ‘An estimate of a complete rational exponential sum', Proc. Math. Inst. Acad. Sci. of the USSR, Moscow, 143 (1977), 188–207 (in Russian).MathSciNetGoogle Scholar
  27. 27.
    I. Wegener, The complexity of Boolean functions, Wiley Interscience Publ., 1987.Google Scholar
  28. 28.
    A. Weil, Basic number theory, Springer-Verlag, NY, 1974.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Joachim van zur Gathen
    • 1
  • Igor Shparlinski
    • 2
  1. 1.FB Mathematik-InformatikUniversität-GH PaderbornPaderbornGermany
  2. 2.School of Mathematics, Physics, Computing and ElectronicsMacquarie UniversitySydneyAustralia

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