In [17] there is proposed a NICE-Schnorr-type signature scheme. In this scheme one uses the group Ker(φ _Cl ⁻¹ ) instead of IF _p ^* . Thus instead of modular arithmetic one would need to apply standard ideal arithmetic (multiply and reduce) using algorithms from [5] for example. Because every group operation needs the application of the Extended Euclidean Algorithm the implementation would be very inefficient. Especially the signing process, which would typically be performed on a smartcard with limited computational power would be too slow to allow practical application. In this work we will introduce an entirely new arithmetic for elements in Ker(φ _Cl ⁻¹ ), which uses the generator and ring-equivalence for exponentiation. Thus the signer essentially performs the exponentiation in ( , which turns out to be about twenty times as fast as conventional ideal arithmetic. Furthermore in [17] it is shown, how one can further speed up this exponentiation by application of the Chinese Remainder Theorem for . With this arithmetic the signature generation is about forty times as fast as with conventional ideal arithmetic and more than twice as fast as in the original Schnorr scheme [26].