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Efficient Parallel Pseudo-Random Number Generation

  • J. H. Reif
  • J. D. Tygar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 218)

0. Abstract

We present a parallel algorithm for pseudo-random number generation. Given a seed of n ε truly random bits for any ε > 0, our algorithm generates n c pseudo-random bits for any c > 1. This takes poly-log time using n ε′ processors where ε′ = κε for some fixed small constant κ > 1. We show that the pseudo-random bits output by our algorithm can not be distinguished from truly random bits in parallel poly-log time using a polynomial number of processors with probability 1/2 + 1/n O(1) if the multiplicative inverse problem almost always can not be solved in RNC. The proof is interesting and is quite different from previous proofs for sequential pseudo-random number generators.

Our generator is fast and its output is provably as effective for RNC algorithms as truly random bits. Our generator passes all the statistical tests in Knuth[14].

Moreover, the existence of our generator has a number of central consequences for complexity theory. Given a randomized parallel algorithm A (over a wide class of machine models such as parallel RAMs and fixed connection networks) with time bound T(n) and processor bound P(n), we show A can be simulated by a parallel algorithm with time bound T(n) + O((log n)(log log n)), processor bound P(n)n ε′, and only using n ε truly random bits for any ε > 0.

Also, we show that if the multiplicative inverse problem is almost always not in RNC, then RNC is within the class of languages accepted by uniform poly-log depth circuits with unbounded fan-in and strictly sub-exponential size \( \bigcap\limits_{\varepsilon > 0} {2^{n^\varepsilon } } \).

7. Bibliography

  1. [1]
    L. AdlemanTwo Theorems on Random Polynomial Time, Proc. 19th IEEE Symposium on Foundations of Computer Science, Ann Arbor, MI, October 1978, pp. 75–83.Google Scholar
  2. [2]
    R. Anderson, A Parallel Algorithm for the Maximal Path Problem, Proc. 17th ACM Symposium on Theory of Computing, Providence, RI, May 1985, pp. 33–37.Google Scholar
  3. [3]
    W. Alexi, B. Chor, O. Goldreich, and C. Schnorr, RSA/Rabin Bits Are 1/2 + 1/poly(log N) Secure, Proc. 25th IEEE Symposium on Foundations of Computer Science, Singer Island, FL, October 1984, pp. 449–457.Google Scholar
  4. [4]
    P. Beame, S. Cook, and H. Hoover, Small Depth Circuits for Integer Products, Powers, and Division, Proc. 25th IEEE Symposium on Foundations of Computer Science, Singer Island, FL, October 1984, pp. 1–6.Google Scholar
  5. [5]
    L. Blum, M. Blum, and M. Shub, A Simple Secure Pseudo-Random Number Generator, Proc. of CRYPTO-82, Santa Barbra, CA, September 1982, pp. 112–117.Google Scholar
  6. [6]
    M. Blum and S. Micali, How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits, SIAM J. Comp., 13 (1984), pp. 850–864.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Chernoff, A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations, Ann. Math. Statist., 23 (1952), pp. 493–507.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S. Cook, Towards a Complexity Theory of Synchronous Parallel Computation, (Presented at) Inter. Symp. Logic. Alg. (1980).Google Scholar
  9. [9]
    O. Goldreich, S. Goldwasser, and S. Micali, How to Construct Random Functions, Proc. 25th Symposium IEEE Symposium Foundations of Computer Science, Singer Island, FL, October 1984, pp. 464–479.Google Scholar
  10. [10]
    S. Goldwasser, S. Micali, and P. Tong, Why and How to Establish a Private Code on a Public Network, Proc. 23rd IEEE Symposium Foundations of Computer Science, Chicago, IL, October 1982, pp. 134–144.Google Scholar
  11. [11]
    R. Kannan, G. Miller, and L. RudolfSublinear Parallel Algorithms for the Greatest Common Divisor of Two Integers, Proc. 25th IEEE Symposium Foundations of Computer Science, Singer Island, FL, October 1984, pp. 7–11.Google Scholar
  12. [12]
    R. Karp and A. Wigderson, A Fast Parallel Algorithm for the Maximal Independent Set Problem, Proc. 16th ACM Symposium on Theory of Computation, Washington, DC, May 1984, pp. 266–272.Google Scholar
  13. [13]
    R. Karp, E. Upfal, and A. Wigderson, Constructing a Perfect Graph Matching in RNC, Proc. 17th ACM Symposium on Theory of Computing, Providence, RI, May 1985, pp. 22–32.Google Scholar
  14. [14]
    D. Knuth, The Art of Computer Programming, vol. 2: Seminumerical Algorithms, 2nd ed., Addison-Wesley, Reading, MA, 1981.zbMATHGoogle Scholar
  15. [15]
    J. Reif, Logarithmic Depth Circuits for Algebraic Functions, Proc. 24th Symposium IEEE Foundations of Computer Science, Tuscon, AZ October 1983, pp. 138–145. Revised in Technical Report TR-84-18, Center for Research in Computing Technology, Harvard University. To appear in SIAM J. Comp.Google Scholar
  16. [16]
    J. Reif and J. Tygar, The Complexity of Chaotic Iterative Maps. To appear.Google Scholar
  17. [17]
    A. Shamir, On the Generation of Cryptographically Strong Pseudo-Random Sequences, ACM Trans. on Comp. Sys., 1, (1983), pp. 38–44.CrossRefGoogle Scholar
  18. [18]
    A. Shonhage and V. Strassen, Schnelle Multiplication grosser Zahlen, Computing, 7 (1974), pp. 281–292.CrossRefGoogle Scholar
  19. [19]
    L. Valiant, S. Sykum, S. Berkowitz, and C. Rackoff, Fast Parallel Computation of Polynomials Using Few Processors, SIAM J. Comp., 12 (1983), pp. 641–644.zbMATHCrossRefGoogle Scholar
  20. [20]
    U. Vazirani and V. Vazirani, Trapdoor Pseudo-Random Number Generators with Applications to Protocol Design, Proc. 24th IEEE Symposium Foundations of Computer Science, Tuscon, AZ, October 1983, pp. 23–30.Google Scholar
  21. [21]
    Von Zur Gathen, Private communication.Google Scholar
  22. [22]
    A. Yao, Theory and Applications of Trapdoor Functions, Proc. 23rd IEEE Symposium Foundations of Computer Science, Chicago, IL, October 1982, pp. 80–91.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • J. H. Reif
    • 1
  • J. D. Tygar
    • 1
  1. 1.Aiken Computation LaboratoryHarvard UniversityCambridge

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