Computational Number Theory and Curve Cryptography
Andreas Stein
Department of Mathematics
University of Illinois, Urbana-Champaign
Abstract:
In 1976 Diffie and Hellman introduced their well-known protocol for
exchanging a secret cryptographic key. Their scheme was based on
arithmetic in the multiplicative group of finite prime fields, but
can be extended to a more general setting of a finite group G.
The Diffie-Hellman technique and all of its extensions make use of
this idea, only the choice of G varies. Of course,
G here should
be selected such that the discrete logarithm problem (DLP) in this
structure is a hard problem. A very popular and efficient choice of
G is the group of points on an elliptic curve over a finite field.
Elliptic curve cryptosystems have the basic advantage that the
corresponding elliptic curve discrete logarithm problem (ECDLP)
appears to be significantly harder than the DLP in conventional
discrete logarithm systems.
In this talk, we discuss various aspects of computational number theory
and cryptography. We first compare elliptic curve cryptography
to RSA. In particular, we provide a survey of attacks to the elliptic
curve discrete logarithm problem by taking into account recent
developments. Then we present the state of the art of other curve
cryptosystems. In particular, we talk about an application of the
Weil descent methodology and its effects on elliptic curve cryptography.
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