On Polynomial Representations of Boolean Functions Related to Some Number Theoretic Problems

  • Erion Plaku
  • Igor E. Shparlinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2245)


We say a polynomial P over ℤM strongly M-represents a Boolean function F if F(x) ≡P(x) (mod M) for all x∈ {0,1}n. Similarly, P one-sidedly M-represents F if F(x) = 0 ⇔ P(x) ≡ 0 (modM) for all x ∈ {0,1}n. Lower bounds are obtained on the degree and the number of monomials of polynomials over ℤ M, which strongly or one-sidedly M-represent the Boolean function deciding if a given n- bit integer is square-free. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function.


Lower Bound Boolean Function Great Common Divisor Polynomial Representation Real Polynomial 
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  1. 1.
    E. Allender, M. Saks and I. E. Shparlinski, ‘A lower bound for primality’, J. of Comp. and Syst. Sci., 62 (2001), 356–366.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. Barrington, R. Beigel, and S. Rudich, ‘Representing Boolean functions as polynomials modulo composite integers’, Comp. Compl., 4, (1994), 367–382.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Barrington and G. Tardos, ‘A lower bound on the MOD 6 degree of the OR function’ Comp. Compl., 7 (1998), 99–108.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Bernasconi, C. Damm and I. E. Shparlinski, ‘On the average sensitivity of testing square-free numbers’, Proc. 5th Intern. Computing and Combinatorics Conf., Tokyo, 1999, Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 1999, v.1627, 291–299.Google Scholar
  5. 5.
    A. Bernasconi, C. Damm and I. E. Shparlinski, ‘The average sensitivity of squarefreeness’, Comp. Compl. 9 (2000), 39–51.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Bernasconi, C. Damm and I. E. Shparlinski, ‘Circuit and decision tree complexity of some number theoretic problems’, Inform. and Comp., 168 (2001), 113–124.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. Bernasconi and I. E. Shparlinski, ‘Circuit complexity of testing square-free numbers’, Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 1563 (1999), 47–56.Google Scholar
  8. 8.
    J. Bruck, ‘Harmonic analysis of polynomial threshold functions’, SIAM J. Discr. Math., 3 (1990), 168–177.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Bruck and R. Smolensky, ‘Polynomial threshold functions, AC 0 functions, and spectral norms’, SIAM J. Comp., 21 (1992), 33–42.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    N. H. Bshouty, Y. Mansour, B. Schieber and P. Tiwari, ‘Fast exponentiation using the truncation operations’, Comp. Compl., 2 (1992), 244–255.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Bürgisser, M. Clausen and M. A. Shokrollahi, Algebraic complexity theory, Springer-Verlag, Berlin, 1996.Google Scholar
  12. 12.
    D. Coppersmith and I. E. Shparlinski, ‘On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping’, J. Cryptology, 13 (2000), 339–360.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    H. Davenport, Multiplicative number theory, Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000.Google Scholar
  14. 14.
    C. Gotsman and N. Linial, ‘Spectral properties of threshold functions’, Combinatorica, 14 (1994), 35–50.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    F. Green, ‘A complex-number Fourier technique for lower bounds on the mod-m degree’, Comp. Compl., 9 (2000), 16–38.zbMATHCrossRefGoogle Scholar
  16. 16.
    D. Grigoriev, ‘Lower bounds in the algebraic computational complexity’, Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 118 (1982), 25–82 (in Russian).Google Scholar
  17. 17.
    D. R. Heath-Brown, ‘The least square-free number in an arithmetic progression’, J. Reine Angew. Math., 332 (1982), 204–220.zbMATHMathSciNetGoogle Scholar
  18. 18.
    A. G. Khovanski, Fewnomials, Amer. Math. Soc., Providence, RI, 1997.Google Scholar
  19. 19.
    M. Krause and Pudlák, ‘On computing Boolean functions by sparse real polynomials’, Proc. 36th IEEE Symp. on Foundations of Comp. Sci., 1995, 682–691.Google Scholar
  20. 20.
    F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, North-Holland, Amsterdam, 1977.Google Scholar
  21. 21.
    Y. Mansour, B. Schieber and P. Tiwari, ‘A lower bound for integer greatest common divisor computations’, J. Assoc. Comp. Mach., 38 (1991), 453–471.zbMATHMathSciNetGoogle Scholar
  22. 22.
    Y. Mansour, B. Schieber and P. Tiwari, ‘Lower bounds for computation with the floor operations’, SIAM J. Comp., 20 (1991), 315–327.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    J. Meidânis, ‘Lower bounds for arithmetic problems’, Inform. Proc. Letters, 38 (1991), 83–87.Google Scholar
  24. 24.
    N. Nisan and M. Szegedy, ‘On the degree of Boolean functions as real polynomials’, Comp. Compl., 4 (1994), 301–313.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    J.-J. Risler, ‘Khovansky’s theorem and complexity theory’, Rocky Mountain J. Math., 14 (1984), 851–853.zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    J.-J. Risler, ‘Additive complexity of real polynomials’, SIAM J. Comp., 14 (1985), 178–183.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    V. Roychowdhry, K.-Y. Siu and A. Orlitsky, ‘Neural models and spectral methods’, Theoretical advances in neural computing and learning, Kluwer Acad. Publ., Dordrecht, 1994, 3–36.Google Scholar
  29. 29.
    I. E. Shparlinski, Number theoretic methods in cryptography: Complexity lower bounds, Birkhäuser, 1999.Google Scholar
  30. 30.
    S.-C. Tsai, ‘Lower bounds on representing Boolean functions as polynomials in ℤ m’, SIAM J. Discr. Math., 9(1) (1996), 55–62.zbMATHCrossRefGoogle Scholar
  31. 31.
    I. Wegener, The complexity of Boolean functions, Wiley-Teubner Series in Comp. Sci., Stuttgart, 1987.zbMATHGoogle Scholar
  32. 32.
    A. Woods, ‘Subset sum “cubes” and the complexity of prime testing‘, Preprint, 2001, 1–17. Available from

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Erion Plaku
    • 1
  • Igor E. Shparlinski
    • 2
  1. 1.Department of Mathematics and Computer ScienceClarkson UniversityPotsdamUSA
  2. 2.Department of ComputingMacquarie UniversityAustralia

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