We present the first efficient compact e-cash scheme that does not rely on a random oracle in its security proof. To this end we construct efficient GS proofs for
the following two problems: (1) the *Euclidean* distance problem, and (2) the *Euclidean* intersection problem.

The Euclidean distance problem is a fundamental problem in computational geometry. It is known that the Euclidean distance problem is NP-hard [@GareyJohnson79]. The Euclidean