Advertisement

Algorithmica

, Volume 64, Issue 3, pp 454–480 | Cite as

Computing Sparse Multiples of Polynomials

  • Mark Giesbrecht
  • Daniel S. Roche
  • Hrushikesh Tilak
Article
  • 245 Downloads

Abstract

We consider the problem of finding a sparse multiple of a polynomial. Given fF[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple hF[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=ℚ and t is constant, we give an algorithm which requires polynomial-time in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.

Keywords

Sparse polynomial Sparsest multiple 

Notes

Acknowledgements

The authors would like to thank John May, Arne Storjohann, and the anonymous referees for their careful reading and useful observations on earlier versions of this work.

References

  1. 1.
    Adleman, L.M., McCurley, K.S.: Open problems in number-theoretic complexity. II. In: Algorithmic Number Theory, Ithaca, NY, 1994. Lecture Notes in Computer Science, vol. 877, pp. 291–322. Springer, Berlin (1994) CrossRefGoogle Scholar
  2. 2.
    Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Symposium on the Theory of Computing (STOC’01), pp. 601–610 (2001) Google Scholar
  3. 3.
    Aumasson, J.P., Finiasz, M., Meier, W., Vaudenay, S.: TCHo: a hardware-oriented trapdoor cipher. In: ACISP’07: Proceedings of the 12th Australasian Conference on Information Security and Privacy, pp. 184–199. Springer, Berlin/Heidelberg (2007) Google Scholar
  4. 4.
    Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.: On the inherent intractability of certain coding problems. IEEE Trans. Inf. Theory 24(3), 384–386 (1978) zbMATHCrossRefGoogle Scholar
  5. 5.
    Brent, R.P., Zimmermann, P.: Algorithms for finding almost irreducible and almost primitive trinomials. In: Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, p. 212. Fields Institute (2003) Google Scholar
  6. 6.
    Didier, F., Laigle-Chapuy, Y.: Finding low-weight polynomial multiples using discrete logarithm. In: Proc. IEEE International Symposium on Information Theory (ISIT 2007), pp. 1036–1040 (2007) CrossRefGoogle Scholar
  7. 7.
    Egner, S., Minkwitz, T.: Sparsification of rectangular matrices. J. Symb. Comput. 26(2), 135–149 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    El Aimani, L., von zur Gathen, J.: Finding low weight polynomial multiples using lattices. Cryptology ePrint Archive, Report 2007/423 (2007). http://eprint.iacr.org/2007/423.pdf
  9. 9.
    Emiris, I.Z., Kotsireas, I.S.: Implicitization exploiting sparseness. In: Geometric and Algorithmic Aspects of Computer-Aided Design and Manufacturing. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 67, pp. 281–297 (2005) Google Scholar
  10. 10.
    Giesbrecht, M., Roche, D.S., Tilak, H.: Computing sparse multiples of polynomials. In: Cheong, O., Chwa, K.Y., Park, K. (eds.) Algorithms and Computation. Lecture Notes in Computer Science, vol. 6506, pp. 266–278. Springer, Berlin/Heidelberg (2010) CrossRefGoogle Scholar
  11. 11.
    Guruswami, V., Vardy, A.: Maximum-likelihood decoding of Reed-Solomon codes is NP-hard. IEEE Trans. Inf. Theory 51(7), 2249–2256 (2005) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Herrmann, M., Leander, G.: A practical key recovery attack on basic TCHo. In: Public Key Cryptography, pp. 411–424 (2009) Google Scholar
  13. 13.
    Lenstra, A.K., Lenstra, H.W. Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lenstra, H.W. Jr.: Finding small degree factors of lacunary polynomials. In: Number Theory in Progress, vol. 1, Zakopane-Kościelisko, 1997, pp. 267–276. de Gruyter, Berlin (1999) Google Scholar
  15. 15.
    Meijer, A.R.: Groups, factoring, and cryptography. Math. Mag. 69(2), 103–109 (1996) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Regev, O.: A simply exponential algorithm for SVP (Ajtai-Kumar-Sivakumar). Lecture notes: http://www.cs.tau.ac.il/~odedr/teaching/lattices_fall_2004/, Scribe: Michael Khanevsky (2004)
  17. 17.
    Risman, L.J.: On the order and degree of solutions to pure equations. Proc. Am. Math. Soc. 55(2), 261–266 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sadjadpour, H., Sloane, N., Salehi, M., Nebe, G.: Interleaver design for turbo codes. IEEE J. Sel. Areas Commun. 19(5), 831–837 (2001) CrossRefGoogle Scholar
  20. 20.
    Shoup, V.: Searching for primitive roots in finite fields. Math. Comput. 58(197), 369–380 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Storjohann, A.: Algorithms for matrix canonical forms. PhD thesis, Swiss Federal Institute of Technology Zürich (2000) Google Scholar
  22. 22.
    Tilak, H.: Computing sparse multiples of polynomials. Master’s thesis, University of Waterloo (2010) Google Scholar
  23. 23.
    Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theory 43(6), 1757–1766 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, Chap. 14, pp. 367–380. Cambridge University Press, New York (2003) Google Scholar
  25. 25.
    von zur Gathen, J., Shparlinski, I.: Constructing elements of large order in finite fields. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science, vol. 1719, p. 730. Springer, Berlin/Heidelberg (1999) Google Scholar
  26. 26.
    Wang, Y.: On the least primitive root of a prime. Acta Math. Sin. 9, 432–441 (1959) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC (outside the USA) 2012

Authors and Affiliations

  • Mark Giesbrecht
    • 1
  • Daniel S. Roche
    • 2
  • Hrushikesh Tilak
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.United States Naval AcademyAnnapolisUSA

Personalised recommendations