Implicit Computational Complexity of Subrecursive Definitions and Applications to Cryptographic Proofs

Abstract

We define a call-by-value variant of Godel’s system $$\textsf {T} $$ with references, and equip it with a linear dependent type and effect system, called $$\textsf {d}\ell \textsf {T} $$ , that can estimate the time complexity of programs, as a function of the size of their inputs. We prove that the type system is intentionally sound, in the sense that it over-approximates the complexity of executing the programs on a variant of the CEK abstract machine. Moreover, we define a sound and complete type inference algorithm which critically exploits the subrecursive nature of $$\textsf {d}\ell \textsf {T} $$ . Finally, we demonstrate the usefulness of $$\textsf {d}\ell \textsf {T} $$ for analyzing the complexity of cryptographic reductions by providing an upper bound for the constructed adversary of the Goldreich–Levin theorem.