This paper presents a distributed (Bulk-Synchronous Parallel or bsp) algorithm to compute on-the-fly whether a structured model of a security protocol satisfies a ctl {Mathematical expression} formula. Using the structured nature of the security protocols allows us to design a simple method to distribute the state space under consideration in a need-driven fashion. Based on this distribution of the states, the algorithm for logical checking of a ltl formula can be simplified and optimised allowing, with few tricky modifications, the design of an efficient algorithm for ctl {Mathematical expression} checking. Some prototype implementations have been developed, allowing to run benchmarks to investigate the parallel behaviour of our algorithms.