Date:         Fri, 4 Nov 1994 17:21:11 EST
Reply-To:     Francois Morain <>
Sender:       Number Theory List <NMBRTHRY@NDSUVM1.BITNET>
From:         Francois Morain <>
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