February 6, 2006

The cardinality of the curve E: Y^2 = X^3+4589*X+91128 
modulo (p=10^2099+6243) is p+1-t with

t= 46309039515660266466504436829543902982037303111273
   73133329015939600620473192265145917348534609271301
   80463487786678542979959407234769815726981464645620
   45942259943796863867474309156215573353558251312947
   05279002886049129524152450854617338507270325443970
   45009251083206675470344648078432773977487544433481
   80799415881345112208263533866935960995501879593481
   23563494292428730241372116337183868845956990766720
   16762883130518896917848778957234951929827379632139
   23336753918213564996900545193436665475820821636114
   40351895261067767318730922734900512756008172744976
   01902169276283488975926314494300515154111433860489
   79698992682115877314851670619809465979774090915096
   58990522856709034389876336674924978610876297672927
   22366580514359521188254467575882285940677159779534
   48091024653607152243664910219594822529312596688627
   12674193865426844485114378638574361621622675608807
   03434187870680581930897304946865615024745778306695
   51482064698450271097057130249581451860620439840908
   82472930864017990614559843280193688352934271319965
   36691264709607426192741260260055570692444330408489

Here are the timings on an AMD 64 Processor 3400+ (2.4GHz), with our NTL
implementation, for some large numbers (excluding the time for
computing modular equations):

what  | 500dd | 1000dd | 1500dd | 1600dd | 1700dd | 2005dd | 2100dd
--------------------------------------------------------------------
X^p   |  6h   | 134h   |  35d   |   46d  |  60d   | 133d   | 121d
Total | 10h   | 180h   |  77d   |   63d  |  80d   | 195d   | 190d

	AEnge, PGaudry and FMorain