- Neal Koblitz
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
In this paper we discuss a source of finite abelian groups suitable for cryptosystems based on the presumed intractability of the discrete logarithm problem for these groups. They are the jacobians of hyperelliptic curves defined over finite fields. Special attention is given to curves defined over the field of two elements. Explicit formulas and examples are given, and the problem of finding groups of almost prime order is discussed.
- J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S. Wagstaff, Jr.,Factorization of b n±1, b=2, 3, 5, 6, 7, 10, 11, 12 up to High Powers, American Mathematical Society, Providence, RI, 1983.
- D. Cantor, Computing in the jacobian of a hyperelliptic curve,Math. Comp.,48 (1987), 95–101.
- W. Diffie and M. Hellman, New directions in cryptography,IEEE Trans. Inform. Theory,22 (1976), 644–654.
- T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithms,IEEE Trans. Inform. Theory,31 (1985), 469–472.
- W. Fulton,Algebraic Curves, Benjamin, New York, 1969.
- N. Koblitz,Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1984.
- N. Koblitz,A Course in Number Theory and Cryptography, Springer-Verlag, New York, 1987.
- N. Koblitz, Elliptic curve cryptosystems,Math. Comp. 48 (1987), 203–209.
- N. Koblitz, Primality of the number of points on an elliptic curve over a finite field,Pacific J. Math.,131 (1988), 157–165.
- S. Lang,Introduction to Algebraic Geometry, Interscience, New York, 1958.
- R. Lidl and H. Niederreiter,Finite Fields, Addison-Wesley, Reading, MA, 1983.
- V. Miller, Use of elliptic curves in cryptography,Advances in Cryptology-Crypto '85, Springer-Verlag, New York, 1986, pp. 417–426.
- A. M. Odlyzko, Discrete logarithms and their cryptographic significance,Advances in Cryptography: Proceedings of Eurocrypt 84, Springer-Verlag, New York, 1985, pp. 224–314.
- E. Seah and H. C. Williams, Some primes of the form (a n − 1)/(a − 1),Math. Comp.,33 (1979), 1337–1342.
- D. Shanks,Solved and Unsolved Problems in Number Theory, 3rd edn., Chelsea, New York, 1985.
- W. C. Waterhouse, Abelian varieties over finite fields,Ann. Sci. École Norm. Sup. (4),2 (1969), 521–560.
- Hyperelliptic cryptosystems
Journal of Cryptology
Volume 1, Issue 3 , pp 139-150
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Public key
- Discrete logarithm
- Hyperelliptic curve
- Industry Sectors
- Neal Koblitz (1)
- Author Affiliations
- 1. Department of Mathematics GN-50, University of Washington, 98195, Seattle, WA, U.S.A.