On Tractable Exponential Sums

  • Jin-Yi Cai
  • Xi Chen
  • Richard Lipton
  • Pinyan Lu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6213)


We consider the problem of evaluating certain exponential sums. These sums take the form
$$\sum_{x_1,x_2, \ldots, x_n \in \mathbb{Z}_N } e^{\frac{2 \pi i}{N} {f(x_1,x_2, \ldots, x_n)} },$$
where each x i is summed over a ring ℤ N , and f(x 1, x 2,..., x n ) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and logN when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree ≥ 3 we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem — a complete classification of each problem to be either computable in polynomial time or #P-hard — for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks; for the hardness result we prove group-theoretic necessary conditions for tractability.


Partition Function Polynomial Time Prime Power Quadratic Polynomial Basic Building Block 
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.
    Lang, S.: Algebraic Number Theory. Addison-Wesley, Reading (1970)zbMATHGoogle Scholar
  2. 2.
    Hua, L.: Introduction to Number Theory. Springer, Heidelberg (1982)Google Scholar
  3. 3.
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  4. 4.
    Goldberg, L., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. In: Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science, pp. 493–504 (2009)Google Scholar
  5. 5.
    Cai, J.Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. In: Proceedings of the 37th International Colloquium on Automata, Languages and Programming (2010)Google Scholar
  6. 6.
    Grabmeier, J., Kaltofen, E., Weispfenning, V.: Computer Algebra Handbook. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (2003)zbMATHGoogle Scholar
  8. 8.
    Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  9. 9.
    Ehrenfeucht, A., Karpinski, M.: The computational complexity of (XOR, AND)-counting problems. University of Bonn (Technical Report 8543-CS) (1990)Google Scholar
  10. 10.
    Bulatov, A., Grohe, M.: The complexity of partition functions. Theoretical Computer Science 348(2), 148–186 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cai, J.-Y., Chen, X., Lipton, R., Lu, P.: On tractable exponential sums. arXiv (1005.2632) (2010)Google Scholar
  12. 12.
    Lovász, L.: Operations with structures. Acta Mathematica Hungarica 18, 321–328 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. In: Proceedings of the 9th International Conference on Random Structures and Algorithms, pp. 260–289 (2000)Google Scholar
  14. 14.
    Freedman, M., Lovász, L., Schrijver, A.: Reflection positivity, rank connectivity, and homomorphism of graphs. Journal of the American Mathematical Society 20, 37–51 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dyer, M., Goldberg, L., Paterson, M.: On counting homomorphisms to directed acyclic graphs. Journal of the ACM 54(6) (2007)Google Scholar
  16. 16.
    Valiant, L.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Valiant, L.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Xi Chen
    • 2
  • Richard Lipton
    • 3
  • Pinyan Lu
    • 4
  1. 1.University of Wisconsin-MadisonUSA
  2. 2.University of Southern CaliforniaUSA
  3. 3.Georgia Institute of TechnologyUSA
  4. 4.Microsoft Research AsiaUSA

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