Range extension for weak PRFs the good the bad and the ugly

We investigate a general class of (black-box) constructions for range extension of weak pseudorandom functions: a construction based on m independent functions F 1,...,F m is given by a set of strings over {1,...,m}*, where for example {〈2〉, 〈1,2〉} corresponds to the function X ↦[F 2(X),F 2(F 1(X))]. All efficient constructions for range expansion of weak pseudorandom functions that we are aware of are of this form. We completely classify such constructions as good, bad or ugly, where the good constructions are those whose security can be proven via a black-box reduction, the bad constructions are those whose insecurity can be proven via a black-box reduction, and the ugly constructions are those which are neither good nor bad. Our classification shows that the range expansion from [10] is optimal, in the sense that it achieves the best possible expansion (2 m  − 1 when using m keys). Along the way we show that for weak quasirandom functions (i.e. in the information theoretic setting), all constructions which are not bad – in particular all the ugly ones – are secure.