Construction of models of bounded arithmetic by restricted reduced powers

We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR 2(g) is weaker than R 2(g). In particular, if such a permutation can be defined by a term in the language of R 2, then strictR 1 2 is weaker than R 1 2.