**Date:** Fri, 4 Nov 1994 17:21:11 EST
**Reply-To:** Francois Morain <morain@polytechnique.fr>
**Sender:** Number Theory List <NMBRTHRY@NDSUVM1.BITNET>
**From:** Francois Morain <morain@polytechnique.fr>
**Subject:** #E(GF(2^400))
Let K = GF(2^400)=GF(2)[T]/(T^400+T^5+T^3+T^2+1) and put

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

and let E be the curve

E: Y^2+XY=X^3+a6

Then
#E(K)=2^400+1-831311719593373938759692923178171920626211278774179820465793.

The computation was done on several DEC alpha's and the total time was
roughly 29 days.

To check our computations, we factorized the number N* of points of
the twisted curve
E* Y^2+XY=X^3+T^395*X^2+a6
which is
N* = 2*3*5*13*54693143363*180002576549*534301174626341637071*
643348834482194230236923008115651453*
1956543037156666733352226362140503925063.

As this number is square free, the curve E* is cyclic. Indeed the point
with abscissa T^3 has exactly order N*.

This new record was made possible by the improvements of our
on going implementation of Couveignes's algorithm.

R. Lercier and F. Morain

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