and g 2 may be the points of an elliptic curve defined on the ground field gf q or the extension field gf q k whereas g 3 can be a subgroup of gf q k i e the multiplicative group of the extension field in this case g 1 is a l torsion subgroup of the group g 2 and satisfies l p o for all p u03b5g 1 where o is the identity element l is a large prime and l cofactor number of points in g 2 additionally the value of k is the smallest integer for which l q k u2212 1 both g 1 and g 3 are of prime order l furthermore the described embodiments use a bilinear function e that when applied to elements of the groups g 1 u00d7 g 2 gives an element of g 3 as a result for example this bilinear function may be the tate pairing known to those skilled in the art and that can be expressed as wherein f is a generic function the bilinear function e is a map or pairing having the following properties e k p r e p r k e p k r 4 wherein p and q u03b5g 1 r and s u03b5g 2 further k p represents the scalar multiplication of point p by the integer k so that k p p p p k.