Elliptic congruence function fields. (English) Zbl 0899.11055

Stein, Andreas

Cohen, Henri (ed.), Algorithmic number theory. Second international symposium, ANTS-II, Talence, France, May 18-23, 1996. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1122, 375-384 (1996).

The Diffie-Hellman key exchange protocol, usually set in the multiplicative group of a prime field, can be extended to real quadratic congruence function fields where the key space is the set of reduced principal ideals. While not a group structure, it is “almost” a group and possesses a so-called infrastructure. It can be shown that the discrete logarithm problem for elliptic congruence function fields is equivalent to that for elliptic curves over finite fields. This paper explains the main properties and arithmetic of reduced principal ideals in elliptic congruence function fields where the underlying structure of the key exchange protocol lies. For elliptic congruence function fields, it is “closer” to a group than that for the real quadratic congruence fields, although still not a group. In addition, some properties of reduced principal ideals, which have no analogies for real quadratic number fields are shown.
For the entire collection see [Zbl 0852.00023].

Reviewer: Ian F.Blake (Palo Alto)

MSC:

11R58	Arithmetic theory of algebraic function fields
11T71	Algebraic coding theory; cryptography (number-theoretic aspects)
94A60	Cryptography
11Y16	Number-theoretic algorithms; complexity
11G05	Elliptic curves over global fields
14H52	Elliptic curves

Keywords:

public key cryptography; key exchange; discrete logarithm problem in elliptic congruence function fields