Abstract
We discuss the different possibilities to choose elliptic curves over different finite fields with respect to application for public key cryptosystems.
In 1985 it was proposed to use the multiplication on elliptic curves for the implementation of one way functions.
Supersingular curves E with #E(F q) = q + 1 elements were proposed at that time. New results due to A. Menezes, T. Okamoto and S. Vanstone show, that these curves are not well suited for that purpose. They can be attacked with a new division algorithm recently presented.
However, by using non-supersingular elliptic curves this attack can be avoided. We show how to construct suitable curves. Furthermore some aspects of a VLSI-implementation for such a cryptosystem are discussed.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
T. Beth, D. Gollmann; Algorithm Engineering for Public Key Algorithms; IEEE Journal on Selected Areas in Comm., Vol. 7, No. 4, 1989, pp 458–466.
T. Beth, W. Geiselmann, F. Schaefer; Arithmetics on Elliptic Curves; Algebraic and Combinatorial Coding Theory, 2nd Int. Workshop, Leningrad, 1990, pp 28–33.
T. Beth, F. Schaefer; Non Supersingular Elliptic Curves for Public Key Cryptosystems; to appear in Proc. of EUROCRYPT'91.
D. Coppersmith; Fast evaluation of logarithms in fields of characteristic two; IEEE Trans. Inform. Theory, IT 30, 1984, pp 587–594.
M. Deuring; Die Typen der Multiplikatorenringe elliptischer Funktionenkörper; Abh. Math. Sem. Hamburg, Bd. 14, 1941, pp 197–272.
W. Diffie, M. Hellman; New directions in cryptography; IEEE Trans. Inform. Theory, IT 22, 1976, pp 644–654.
T. ElGamal; A public key cryptosystem and a signature scheme based on discrete logarithms; IEEE Trans. Inform. Theory, IT 31, 1985, pp 469–472.
N. Koblitz; Elliptic Curve Cryptosystems; Mathematics of Computation, Vol. 48, No177, 1987, pp 203–209.
A. Menezes, S. A. Vanstone; The Implementation fo Elliptic Curve Cryptosystems; Advances in Cryptology-AUSCRYPT90, Springer LNCS 453, 1990, pp 2–13.
A. Menezes, T. Okamoto, S. A. Vanstone; Reducing Elliptic Curve Logarithms to Logarithms in a Finite Field; Proc. of the 22nd Annual ACM Symposium on the Theory of Comp., 80–89, 1991.
V. S. Miller; Use of Elliptic Curves in Cryptography; Advances in Cryptology: Proceedings of Crypto 85, Springer LNCS 218, 1986, pp 417–426.
P. Montgomery; Speeding the Pollard and elliptic curve methods of factorization; Math. Comp., Vol. 48, 1977, pp 243–264.
R. Schoof; Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p; Math. Of Comp., Vol. 44, No. 170, 1985, pp 483–494.
J. H. Silverman; The Arithmetic of Elliptic Curves; Springer-Verlag, New York, 1986.
J. T. Tate; The Arithmetic of Elliptic Curves; Inventiones Math. 23, Springer-Verlag, 1974, pp 179–206.
W. C. Waterhouse; Abelian Varieties over finite fields; Ann. Scient. Ec. Norm. Sup., 4th serie, 1969, pp 521–560.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beth, T., Schaefer, F. (1991). Arithmetic on non supersingular elliptic curves. In: Mattson, H.F., Mora, T., Rao, T.R.N. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1991. Lecture Notes in Computer Science, vol 539. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54522-0_97
Download citation
DOI: https://doi.org/10.1007/3-540-54522-0_97
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54522-4
Online ISBN: 978-3-540-38436-6
eBook Packages: Springer Book Archive