Designs, Codes and Cryptography

, Volume 75, Issue 3, pp 483–495 | Cite as

New cube root algorithm based on the third order linear recurrence relations in finite fields

  • Gook Hwa Cho
  • Namhun Koo
  • Eunhye Ha
  • Soonhak Kwon


In this paper, we present a new cube root algorithm in the finite field \(\mathbb {F}_{q}\) with \(q\) a power of prime, which extends the Cipolla–Lehmer type algorithms (Cipolla, Un metodo per la risolutione della congruenza di secondo grado, 1903; Lehmer, Computer technology applied to the theory of numbers, 1969). Our cube root method is inspired by the work of Müller (Des Codes Cryptogr 31:301–312, 2004) on the quadratic case. For a given cubic residue \(c \in \mathbb {F}_{q}\) with \(q \equiv 1 \pmod {9}\), we show that there is an irreducible polynomial \(f(x)\) with root \(\alpha \in \mathbb {F}_{q^{3}}\) such that \(Tr\left( \alpha ^{\frac{q^{2}+q-2}{9}}\right) \) is a cube root of \(c\). Consequently we find an efficient cube root algorithm based on the third order linear recurrence sequences arising from \(f(x)\). The complexity estimation shows that our algorithm is better than the previously proposed Cipolla–Lehmer type algorithms.


Finite field Cube root Linear recurrence relation Tonelli–Shanks algorithm Cipolla–Lehmer algorithm Adleman–Manders–Miller algorithm 

Mathematics Subject Classification

11T06 11Y16 68W40 



The authors would like to thank the anonymous referees for the insightful and valuable comments on this paper. The preliminary version of this paper was presented at 10th Algorithmic Number Theory Symposium (ANTS X) Poster Session, July 9–13, 2012. No proceeding will be published for the poster session. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2013R1A1A2060698).


  1. 1.
    Adleman L., Manders K., Miller G.: On taking roots in finite fields. In: Proceedings of the 18th IEEE Symposium on Foundations on Computer Science (FOCS), pp. 175–177, (1977).Google Scholar
  2. 2.
    Ahmadi O., Hankerson D., Menezes A.: Formulas for cube roots in \(F_{3^m}\). Discret. Appl. Math. 155(3), 260–270 (2007).Google Scholar
  3. 3.
    Ahmadi O., Rodriguez-Henriquez F.: Low complexity cubing and cube root computation over \(F_{3^m}\) in polynomial basis. IEEE Trans. Comput. 59, 1297–1308 (2010).Google Scholar
  4. 4.
    Atkin A.O.L.: Probabilistic primality testing, summary by F. Morain. Inria Res. Rep. 1779, 159–163 (1992).Google Scholar
  5. 5.
    Barreto P.S., Voloch J.F.: Efficient computation of roots in finite fields. Des. Codes Cryptogr. 39, 275–280 (2006).Google Scholar
  6. 6.
    Bernstein D.: Faster square roots in annoying finite fields, preprint., (2001).
  7. 7.
    Boneh D., Franklin M.: Identity based encryption from the Weil pairing, Crypto 2001. Lect. Notes Comput. Sci. 2139, 213–229 (2001).Google Scholar
  8. 8.
    Cipolla M.: Un metodo per la risolutione della congruenza di secondo grado, Rendiconto dell’Accademia Scienze Fisiche e Matematiche, Napoli, Ser. 3, vol. IX, pp. 154–163 (1903).Google Scholar
  9. 9.
    Damgård I.B., Frandsen G.S.: Efficient algorithm for the gcd and cubic residuosity in the ring of Eisenstein integers. J. Symb. Comput. 39, 643–652 (2005).Google Scholar
  10. 10.
    Dickson L.E.: Criteria for the irreducibility of functions in a finite field. Bull. Am. Math. Soc. 13(1), 1–8 (1906).Google Scholar
  11. 11.
    Dudeanu A., Oancea G., Iftene S.: An \(x\)-coordinate point compression method for elliptic curves over \(\mathbb{F}_p\). In: Proceedings of the 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2010), Washington DC, USA, pp. 65–71 (2010).Google Scholar
  12. 12.
    Duursma I., Lee H.: Tate pairing implementation for hyperelliptic curves \(y^2=x^p-x+d\), Asiacrypt 2003. Lect. Notes Comput. Sci. 2894, 111–123 (2003).Google Scholar
  13. 13.
    Gong G., Harn L.: Public key cryptosystems based on cubic finite field extensions. IEEE Trans. Inf. Theory 45, 2601–2605 (1999).Google Scholar
  14. 14.
    Han D., Choi D., Kim H.: Improved computation of square roots in specific finite fields. IEEE Trans. Comput. 58, 188–196 (2009).Google Scholar
  15. 15.
    Kong F., Cai Z., Yu J., Li D.: Improved generalized atkin algorithm for computing square roots in finite fields. Inf. Process. Lett. 98(1), 1–5 (2006).Google Scholar
  16. 16.
    Lang S.: Algebra, Springer, New York (2005).Google Scholar
  17. 17.
    Lehmer, D.H.: Computer technology applied to the theory of numbers. In: Leveque W.J. (ed.) Studies in Number Theory, pp. 117–151. Pretice-Hall, Englewood Cliffs (1969).Google Scholar
  18. 18.
    Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).Google Scholar
  19. 19.
    Lindhurst S.: An analysis of Shanks’s algorithm for computing square roots in finite fields. CRM Proc. Lect. Notes 19, 231–242 (1999).Google Scholar
  20. 20.
    Menezes A.J., Blake I.F., Gao X., Mullin R.C., Vanstone S.A., Yaghoobian T.: Applications of Finite Fields. Springer, Berlin (1992).Google Scholar
  21. 21.
    Menezes A.J., van Oorschot P.C., Vanstone S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996).Google Scholar
  22. 22.
    Müller S.: On the computation of square roots in finite fields. Des. Codes Cryptogr. 31, 301–312 (2004).Google Scholar
  23. 23.
    Nishihara N., Harasawa R., Sueyoshi Y., Kudo A.: A remark on the computation of cube roots in finite fields, preprint. (2009).
  24. 24.
    Panario D., Thomson D.: Efficient \(p\)th root computations in finite fields of characteristic \(p\). Des. Codes Cryptogr. 50, 351–358 (2009).Google Scholar
  25. 25.
    Peralta R.C.: A simple and fast probabilistic algorithm for computing square roots modulo a prime number. IEEE Trans. Inf. Theory 32, 846–847 (1986).Google Scholar
  26. 26.
    Shanks D.: Five number-theoretic algorithms. In: Proceedings of the 2nd Manitoba Conference on Numberical Mathathematics, Manitoba, Canada, pp. 51–70 (1972).Google Scholar
  27. 27.
    Sutherland A.V.: Structure computation and discrete logarithms in finite abelian \(p\)-groups. Math. Comp. 80, 477–500 (2011).Google Scholar
  28. 28.
    Tonelli A.: Bemerkung über die auflösung quadratischer congruenzen, Göttinger Nachrichten, pp. 344–346 (1891).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Gook Hwa Cho
    • 1
  • Namhun Koo
    • 1
  • Eunhye Ha
    • 1
  • Soonhak Kwon
    • 1
  1. 1.Department of MathematicsSungkyunkwan UniversitySuwonSouth Korea

Personalised recommendations