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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

11T06 11Y16 68W40 

Notes

Acknowledgments

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).

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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

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