Designs, Codes and Cryptography

, Volume 56, Issue 1, pp 65–78 | Cite as

NTRU over rings beyond \({\mathbb{Z}}\)

  • Monica Nevins
  • Camelia KarimianPour
  • Ali Miri


The NTRU cryptosystem is constructed on the base ring \({\mathbb{Z}}\) . We give suitability conditions on rings to serve as alternate base rings. We present an example of an NTRU-like cryptosystem based on the Eisenstein integers \({\mathbb{Z}[\zeta_3]}\) , which has a denser lattice structure than \({\mathbb{Z}}\) for the same dimension, and which furthermore presents a more difficult lattice problem for lattice attacks, for the same level of decryption failure security.


Public-key cryptography Lattice NTRU 

Mathematics Subject Classification (2000)

Primary 11T71 Secondary 13G 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buchmann J.: Reducing lattice bases by means of approximations. Algorithmic number theory, Ithaca, NY, 1994. Lecture Notes in Computer Science, vol. 877, pp. 160–168. Springer, Berlin (1994).Google Scholar
  2. 2.
    Coglianese M., Goi B.-M.: MaTRU: a new NTRU-based cryptosystem. Indocrypt 2005. Lecture Notes in Computer Science, vol. 3797, pp 232–243. (2005).Google Scholar
  3. 3.
    Conway J.H., Sloane N.J.A.: Sphere packings, lattices and groups. In: Grundlehren der Mathematischen Wissenschaften, 3rd edn., vol. 290. Springer-Verlag, New York (1999).Google Scholar
  4. 4.
    Coppersmith D., Shamir A.: Lattice attacks on NTRU. Advances in cryptology—EUROCRYPT 1997. Lecture Notes in Computer Science, vol. 1233, pp. 52–61. Springer, Berlin (1997).Google Scholar
  5. 5.
    Gaborit P., Ohler J., Sole P.: CTRU, A polynomial analogue of NTRU. NTRU Technical Report #Inria RR-4621 (2006).Google Scholar
  6. 6.
    Hirschhorn P., Hoffstein J., Howgrave-Graham N., Whyte W.: Choosing NTRUEncrypt parameters in light of combined lattice reduction and MITM approaches. In: Proceedings of the 7th international conference on applied cryptography and network security, Paris-Rocquencourt, France. Lecture Notes In Computer Science, vol. 5536, pp. 437–455. (2009).Google Scholar
  7. 7.
    Hoffstein J., Pipher J., Silverman J.H.: NTRU, a ring-based public-key cryptosystem. Algorithmic number theory, Portland, OR, 1998. Lecture Notes in Computer Science, vol. 1423, pp. 267–288. Springer, Berlin (1996).Google Scholar
  8. 8.
    Hoffstein J., Pipher J., Silverman J.: An introduction to mathematical cryptography. In: Undergraduate Texts in Mathematics. Springer, New York (2008).Google Scholar
  9. 9.
    Howgrave-Graham N.: Computational mathematics inspired by RSA. PhD thesis, University of Bath (1998).Google Scholar
  10. 10.
    Howgrave-Graham N., Silverman J.H., Whyte W.: Choosing parameter sets for NTRUEncrypt with NAEP and SVES-3. Topics in cryptology—CT-RSA 2005. Lecture Notes in Computer Science, vol. 3376, pp. 118–135. Springer, Berlin (2005).Google Scholar
  11. 11.
    Kouzmenko R.: Generalizations of the NTRU cryptosystem. Diploma Project, École Polytechnique Fédérale de Lausanne, (2005–2006).Google Scholar
  12. 12.
    Lemmermeyer F.: The Euclidean algorithm in algebraic number fields. Exposition. Math. 13, 385–416 (1995)MATHMathSciNetGoogle Scholar
  13. 13.
    Masley J.M., Montgomery H.L.: Cyclotomic fields with unique factorization. J. Reine Angew. Math. 248–256 (1976).Google Scholar
  14. 14.
    Rhai T.-S.: A characterization of polynomial domains over a field. Am. Math. Mon. 69, 984–986 (1962)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.School of Information Technology and Engineering (SITE), and Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

Personalised recommendations