We present a kind of groups suitable for cryptographic
applications: the tracezero subvariety. The construction is
based on Weil descent from curves of genus one or two over
extension fields F_{p}^{n}, n=3 (or 5).
On the Jacobian of the curve the group can be seen as a prime order subgroup,
however, considering the construction as Weil descent reveals that
the security is equivalent to that of groups based on
lowgenus hyperelliptic curves over prime fields.
The advantage is that the complexity to compute scalar multiples
is lower in most cases as one can make use of the Frobenius
endomorphism of the initial curve.
Thus the tracezero subvariety can be used efficiently in protocols
based on the discrete logarithm problem.
