I've been given this key for an elliptic curve crypto -system:
A:=4569782456273849
B:=74578265973825694738
p:=164516845864567592349187678956932587156973824569837657473
So the EC group is G:=[A,B,p]
The two points from the group are g:=[48563875638, 59953686086232350377099851919349194342354278013502470154]
b:=[94505880155199340954270580844602804762931528214881553767, 125066642042178110526107700200751399553923146040555130047]
The tolerance parameter is t:=49.
The ciphertext is:
e:=[93639666479598242411823008371484237433314325218557988665, 52243374701368396864838008344430211295076041963702946225]
y:=[141014893828387250204699736658837464619673828728217488819, 54503942085017207420659567572818024668020348976893792817]
I need to decrypt the messsage and determine the private key "x" for which $b=g^x$.
The problem is I can't calculate the group order for G, so I can't use an HPS attack. I think that eliminates Pollard-Rho and Baby Step attacks as well.
Any ideas for a strategy?
I find with Sage that the order of the group is $2^2 * 3 * 179 * 769 * 99597802815676113473680829865485023921426165084723$
So you can use Pohlig–Hellman to reduce the discrete log a little bit but not so much.