Secure state estimation is the problem of estimating the state of a dynamical system from a set of noisy and adversarially corrupted measurements. Intrinsically a combinatorial problem, secure state estimation has been traditionally addressed either by brute force search, suffering from scalability issues, or via convex relaxations, using algorithms that can terminate in polynomial time but are not necessarily sound. In this paper, we present a novel algorithm that uses a satisfiability modulo theory approach to harness the complexity of secure state estimation. We leverage results from formal methods over real numbers to provide guarantees on the soundness and completeness of our algorithm. Moreover, we discuss its scalability properties, by providing upper bounds on the runtime performance. Numerical simulations support our arguments by showing an order of magnitude decrease in execution time with respect to alternative techniques. Finally, the effectiveness of the proposed algorithm is demonstrated by applying it to the problem of controlling an unmanned ground vehicle.