Abstract
Very recently a topic in number theory and algebraic geometry — elliptic curves (more precisely, the theory of elliptic curves defined over finite fields) — has found application in cryptography. The basic reason for this is that elliptic curves over finite fields provide an inexhaustible supply of finite abelian groups which, even when large, are amenable to computation because of their rich structure. Before (§ IV.3) we worked with the multiplicative groups of fields. In many ways elliptic curves are natural analogs of these groups; but they have the advantage that one has more flexibility in choosing an elliptic curve than in choosing a finite field.
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
W. Fulton, Algebraic Curves‚ Benjamin, 1969.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms‚ Springer- Verlag, 1984.
N. Koblitz, “Why study equations over finite fields?,” Mathematics Magazine, Vol. 55 (1982), 144–149.
S. Lang, Elliptic Curves: Diophantine Analysis, Springer-Verlag, 1978.
J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.
References
R. Gupta and M. R. Murty, “Primitive points on elliptic curves,” Compositio Mathematics vol. 58 (1986), 13–44.
N. Koblitz, “Elliptic curve cryptosystems,” Mathematics of Computation‚ vol. 48 (1987).
N. Koblitz, “Primality of the number of points on an elliptic curve over a finite field,” to appear.
H. W. Lenstra, Jr., “Elliptic curves and number-theoretic algorithms,” Report 86–19, Mathematisch Instituut, Universiteit van Amsterdam, 1986.
V. Miller, “Use of elliptic curves in cryptography,” Abstracts for Crypto 85‚ 1985.
A. M. Odlyzko, “ Discrete logarithms in finite fields and their cryptographic significance,” Advances in Cryptology, Proceedings of Eurocrypt 84, Springer, 1985, 224–314.
R. Schoof, “Elliptic curves over finite fields and the computation of square roots mod p,” Mathematics of Computation, vol. 44 (1985), 483–494.
References
H. W. Lenstra, Jr., “Factoring integers with elliptic curves,” Report 86–18, Mathematisch Instituut, Universiteit van Amsterdam, 1986.
J. M. Pollard, “Theorems on factorization and primality testing,” Proceedings Cambridge Phil. Soc., vol. 76 (1974), 521–528.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Koblitz, N. (1987). Elliptic Curves. In: A Course in Number Theory and Cryptography. Graduate Texts in Mathematics, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0310-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4684-0310-7_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-0312-1
Online ISBN: 978-1-4684-0310-7
eBook Packages: Springer Book Archive