This paper presents a public-key cryptosystem based on a subclass of the well-known satisfiability problem from propositional logic, namely the doubly-balanced 3-sat problem. We first describe the construction of an instance of our system starting from such a 3-sat formula. Then we discuss security issues: this is achieved on the one hand by exploring best methods to date for solving this particular problem, and on the other hand by studying (systems of multivariate) polynomial equation solving algorithms in this particular setting. The result of our investigations is that both types of method fail to break our instances. We end the paper with some complexity considerations and implementation results.