Efficient Algorithms for GCD and Cubic Residuosity in the Ring of Eisenstein Integers

  • Ivan Bjerre Damgård
  • Gudmund Skovbjerg Frandsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2751)


We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers, Z[ζ], i.e. the integers extended with ζ, a complex primitive third root of unity. The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms. Our algorithms take time O(n 2) for n bit input. This is an improvement from the known results based on the Euclidean algorithm, and taking time O(n · M(n)), where M(n) denotes the complexity of multiplying n bit integers. The new algorithms have applications in practical primality tests and the implementation of cryptographic protocols.


Primary Number Euclidean Algorithm Modular Exponentiation Loop Invariant Binary Algorithm 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bach, E., Shallit, J.: Algorithmic number theory. Foundations of Computing Series, vol. 1. MIT Press, Cambridge (1996); Efficient algorithmszbMATHGoogle Scholar
  2. 2.
    Damgård, I.B., Frandsen, G.S.: An extended quadratic Frobenius primality test with average and worst case error estimates. Research Series RS-03-9, BRICS, Department of Computer Science, University of Aarhus, Extended abstract in these proceedings (February 2003)Google Scholar
  3. 3.
    Ireland, K., Rosen, M.: A classical introduction to modern number theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, New York (1990)zbMATHGoogle Scholar
  4. 4.
    Lemmermeyer, F.: The Euclidean algorithm in algebraic number fields. Exposition. Math. 13(5), 385–416 (1995)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Lemmermeyer, F.: Reciprocity laws. Springer Monographs in Mathematics. Springer, Berlin (2000); From Euler to EisensteinGoogle Scholar
  6. 6.
    Lenstra Jr., H.W.: Euclidean number fields. I. Math. Intelligencer 2(1), 6–15 (1979)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Eikenberry, S.M., Sorenson, J.P.: Efficient algorithms for computing the Jacobi symbol. J. Symbolic Comput. 26(4), 509–523 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Scheidler, R., Williams, H.C.: A public-key cryptosystem utilizing cyclotomic fields. Des. Codes Cryptogr. 6(2), 117–131 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schönhage, A.: Schnelle Berechnung von Kettenbruchentwicklungen. Acta Informat. 1, 139–144 (1971)zbMATHCrossRefGoogle Scholar
  10. 10.
    Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing (Arch. Elektron. Rechnen) 7, 281–292 (1971)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Shallit, J., Sorenson, J.: A binary algorithm for the jacobi symbol. ACM SIGSAM Bull 27(1), 4–11 (1993)CrossRefGoogle Scholar
  12. 12.
    Stein, J.: Computationals problems associated with Racah algebra. J. Comput. Phys. 1, 397–405 (1967)zbMATHCrossRefGoogle Scholar
  13. 13.
    Weilert, A.: (1 + i)-ary GCD computation in Z[i] is an analogue to the binary GCD algorithm. J. Symbolic Comput. 30(5), 605–617 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Weilert, A.: Asymptotically fast GCD computation in Z[i]. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 595–613. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Weilert, A.: Fast computation of the biquadratic residue symbol. J. Number Theory 96(1), 133–151 (2002)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Williams, H.C.: An M3 public-key encryption scheme. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 358–368. Springer, Heidelberg (1986)Google Scholar
  17. 17.
    Williams, H.C., Holte, R.: Computation of the solution of x 3 + Dy 3 = 1. Math. Comp. 31(139), 778–785 (1977)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ivan Bjerre Damgård
    • 1
  • Gudmund Skovbjerg Frandsen
    • 1
  1. 1.BRICS, Department of Computer ScienceUniversity of AarhusAarhus CDenmark

Personalised recommendations