Light affine logic is a variant of linear logic with a polynomial cut-elimination procedure. We study the extensional expressive power of light affine logic with respect to a general notion of encoding of functions in the setting of the Curry–Howard correspondence. We consider light affine logic with both fixpoints of formulae and second-order quantifiers, and analyse the properties of polytime soundness and polytime completeness for various fragments of this system. In particular, we show that the implicative propositional fragment is not polytime complete if we place some reasonable conditions on the encodings. Following previous work, we show that second order leads to polytime unsoundness. We then introduce simple constraints on second-order quantification and fixpoints, and prove that the fragments obtained are polytime sound and complete.