```Date: Mon, 22 Apr 1996 09:16:35 EDT Reply-To: Francois Morain Sender: Number Theory List From: Francois Morain Subject: #E_X(GF(2^1301)) We have established a new (as far as we know) record in elliptic curve point counting over GF(2^n). This new record follows records we have already done before, mainly, n = 701 (cf. [LM95]) and n = 1009 (cf. [LM96]). Let K = GF(2^1301)=GF(2)[T]/(T^1301+T^11+T^10+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)=2^1301+1-t where t = -12520204660458550994342381347414364580981597602551541937475251045091\ 682467090516587665003925037001223752821603011654792837240825761437453\ 092925430162137172566628268880767876046805121177132601520639 The method used is the Schoof-Elkies-Atkin algorithm incorporating Lercier's algorithm [L96] for computing isogenies in small characteristic. The computation was done on several DEC alpha's. The time needed on a single DEC alpha would have been 206 days among which 7 days were spent computing isogenies. This is smaller than what we had for GF(2^1009) (243 days for the whole computation, among which 155 days for computing isogenies with the algorithm described in [C94]). All l up to 673 had to be used. Papers on this topic can be found in http://lix.polytechnique.fr/~morain/ and http://lix.polytechnique.fr/~lercier/ R. Lercier and F. Morain Bibliography ------------ [C94] Jean-Marc Couveignes, Quelques calculs en th\'eorie des nombres, Universit\'e de Bordeaux I, 1994. [LM95] R. Lercier and F. Morain, Counting the number of points of on elliptic curves over finite fields: strategies and performances, Proc. Eurocrypt '95, Lecture Notes in Computer Science 921, Springer, 1995, pp. 79-94. [LM96] R. Lercier and F. Morain, Counting points on elliptic curves over GF(p^n) using Couveigne's algorithm, Research Report LIX/RR/95/09, Ecole Polytechnique-LIX, September 1995 [L96] R. Lercier, Computing isogenies in F(2^n), H. Cohen(Ed.), Proc. ANTS II, Lecture Notes in Computer Science, Springer-Verlag, to appear. ```