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Abstract. We describe improvements to the performance of a key agreement protocol based in the infrastructure of a real quadratic field through investigating fast methods for exponentiating ideals. We present adaptations of non-adjacent form and signed base-3 exponentiation and compare these to the binary method. To adapt these methods, we introduce new algorithms for squaring, cubing, and dividing w-near (f, p) representations of ideals in the infrastructure. Numerical results from an implementation of the key agreement protocol using our new algorithms and all three exponentiation methods are presented, demonstrating that nonadjacent form exponentiation improves the speed of key establishment for most of the currently recommended security levels.