**Date:** Thu, 5 Jan 1995 14:11:40 EST
**Reply-To:** Francois Morain <morain@polytechnique.fr>
**Sender:** Number Theory List <NMBRTHRY@NDSUVM1.BITNET>
**From:** Francois Morain <morain@polytechnique.fr>
**Subject:** #E(GF(2^601))
To celebrate 1995, we are happy to announce the following record.

Let K = GF(2^601)=GF(2)[T]/(T^601+T^7+T^4+T^3+T^2+T+1) and put

a6 = T^16+T^14+T^13+T^9+T^8+T^7+T^6+T^5+T^4+T^3.

Let E be the curve E: Y^2+XY=X^3+a6 . Then
#E(K)=2^601+1-37775742763172180654637698179922762979897172920800\
67701458146624068364548898667013349009665.

The computations were done on several DEC alpha's and the total time was
60 days.

To check our computations, we factored the number N* of points of
the twisted curve E* Y^2+XY=X^3+T^395*X^2+a6 which is

N* = 2*7^4*2180141*26138069*45525212446330489231*P_144

The curve E* is cyclic. For example, the point with abscissa T+T^2 has
exactly order N*.

For the sake of comparison, here are the timings for the fields we
used in the recent past:

GF(2^n) Time

300 9 days
400 29 days
500 12 days
601 60 days

The reason of the drop for 500 is due to the simplification of the
computations required by Couveignes's algorithm, thus making it
possible to reach higher fields. The paper containing all the details
is currently being written and the first version will be available
shortly.

R. Lercier and F. Morain

LIX, Laboratoire d'Informatique de l'Ecole Polytechnique
Ecole Polytechnique, 91128 Palaiseau Cedex, France