The key to blame: Gradual typing meets cryptography

We connect three ways to achieve relational parametricity: universal types, runtime type generation, and cryptographic sealing. We study a polymorphic blame calculus, λB, inspired by that of Ahmed, Findler, Siek, and Wadler (2011), that ties universal types to runtime type generation; and a cryptographic lambda calculus, λK, inspired by that of Pierce and Sumii (2000), that relies on cryptographic sealing. Our λB calculus avoids the ‘topsy turvy’ aspects of Ahmed et al., who evaluate terms one would expect to be values, and leave as values terms one would expect to be evaluated. We present translations from λB to λK and back that we show to be simulations. We extract from λB the subset λG that corresponds to the polymorphic lambda calculus λF of Girard (1972) and Reynolds (1974); λG is also a subset of the system G studied by Neis, Dreyer, and Rossberg (2009). We present translations from λF to λG and back that we show to be fully abstract. Further, we shed light on the embedding given by Pierce and Sumii of λF into λK, describing how it is related to the composition of our translations from λF to λG and λB to λK, and that the conversions and casts of λB relate to the C and G components of their embedding.