### Monday March 24, 2003

**Speaker**: Eric Bone

**Title**: The Elliptic Curve Discrete Logarithm Problem in
Cryptography

**Abstract**: In a multiplicative group *G*, we can state the Discrete Logarithm Problem (DLP)
as follows: If *b* is in *G*, and *y* is in the subgroup generated by *b*, find any
*x*
in **Z** such that
*y* = *b*^{x}.
Until 1985, cryptographic uses of the DLP were limited to the multiplicative groups
of finite fields.
Then V. Miller and N. Koblitz
independently suggested that a finite subgroup of the points on an elliptic curve *E*
over a finite field could also be used.
From these suggestions, cryptosystems have been constructed using the
integer multiplication maps on
*E*. In collaboration with F. Diamond, I investigate the construction of generalized
cryptosystems using the entire endomorphism ring when End(*E*) =**Z**[*f*],
where *f*
is the *p*-th power Frobenius map.