Universal projective hash families were introduced by Cramer and Shoup and
proved an invaluable cryptographic primitive for deriving provable secure
encryption schemes. So called "group systems" provide very natural designs
for UPHF which cryptographic robustness relies on the hardness of a mathematical 
problem that can be formulated as a group-theoretic language membership problem.
In an attempt to derive constructions from a broader scope (in particular, using
non abelian groups as a base) we introduce a new atomic unit, so called group 
action systems, which in a similar fashion allow the construction of 
cryptographically useful projective hash families.