Abstract
In the paper an upper bound is established for certain exponential sums, analogous to Gaussian sums, defined on the points of an elliptic curve over a prime finite field. The bound is applied to prove the existence of group generators for the set of points on an elliptic curve over \(\mathbb{F}_{q}\) among certain sets of bounded size. We apply this estimate to obtain a deterministic O(q 1/2 + ε) algorithm for finding generators of the group in echelon form, and in particular to determine its group structure.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bach, E., Shallit, J.: Algorithmic Number Theory. MIT Press, Cambridge (1996)
Bombieri, E.: On exponential sums in finite fields. Amer. J. Math. 88, 71–105 (1966)
Chalk, J.H.H.: Polynomial congruences over incomplete residue systems modulo k. Proc. Kon. Ned. Acad. Wetensch. A92, 49–62 (1989)
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (1997)
Elkies, N.: Elliptic and modular curves over finite fields and related computational issues. Computational perspectives on number theory (Chicago, IL, 1995), Stud. Adv. Math., 7, 21–76. Amer. Math. Soc., Providence, RI (1998)
Gong, G., Bernson, T.A., Stinson, D.A.: Elliptic curve pseudorandom sequence generators. Research Report CORR-98-53, Faculty of Math., Univ. of Waterloo, 1–21 (1998)
Hallgren, S.: Linear congruential generators over elliptic curves. Preprint CS-94- 143, Dept. of Comp. Sci., Cornegie Mellon Univ, 1–10 (1994)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge Univ. Press, Cambridge (1997)
Menezes, A.J., Okamoto, T., Vanstone, S.A.: Reducing elliptic curve logarithms to logarithms in a finite field. Trans. IEEE Inform. Theory 39, 1639–1646 (1993)
Menezes, A.J.: Elliptic Curve Public Key Cryptosystems. Kluwer Acad. Publ., Boston (1993)
Schoof, R.J.: Elliptic curves over finite fields and the computation of square roots Mod p. Math. Comp. 44, 483–494 (1985)
Shoup, V.: Searching for primitive roots in finite fields. Math. Comp. 58, 369–380 (1992)
Shparlinski, I.E.: On primitive elements in finite fields and on elliptic curves. Matem. Sbornik 181, 1196–1206 (1990) (in Russian)
Shparlinski, I.E.: On Gaussian sums for finite fields and elliptic curves. In: Lobstein, A., Litsyn, S.N., Zémor, G., Cohen, G. (eds.) Algebraic Coding 1991. LNCS, vol. 573, pp. 5–15. Springer, Heidelberg (1992)
Shparlinski, I.E.: On finding primitive roots in finite fields. Theor. Comp. Sci. 157, 273–275 (1996)
Shparlinski, I.E.: Finite Fields: Theory and Computation. Kluwer Acad. Publ., North-Holland (1999)
Shparlinski, I.E.: On the Naor–Reingold pseudo-random function from elliptic curves. Appl. Algebra in Engin., Commun. and Computing (to appear)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Springer, Berlin (1995)
Vinogradov, I.M.: Elements of Number Theory. Dover Publ., NY (1954)
Weil, A.: Basic of Number Theory. Spinger, Heidelberg (1974)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kohel, D.R., Shparlinski, I.E. (2000). On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_24
Download citation
DOI: https://doi.org/10.1007/10722028_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67695-9
Online ISBN: 978-3-540-44994-2
eBook Packages: Springer Book Archive