Paper 2019/1185

Formalising $\mathrm{\Sigma }$-Protocols and Commitment Schemes using CryptHOL

David Butler, Andreas Lochbihler, David Aspinall, and Adria Gascon

Abstract

Machine-checked proofs of security are important to increase the rigour of provable security. In this work we present a formalised theory of two fundamental two party cryptographic primitives: $\mathrm{\Sigma }$-protocols and Commitment Schemes. $\mathrm{\Sigma }$-protocols allow a prover to convince a verifier that they possess some knowledge without leaking information about the knowledge. Commitment schemes allow a committer to commit to a message and keep it secret until revealing it at a later time. We use CryptHOL to formalise both primitives and prove secure multiple examples namely; the Schnorr, Chaum-Pedersen and Okamoto $$-protocols as well as a construction that allows for compound (AND and OR) $$-protocols and the Pedersen and Rivest commitment schemes. A highlight of the work is a formalisation of the construction of commitment schemes from $$-protocols. We formalise this proof at an abstract level using the modularity available in Isabelle/HOL and CryptHOL. This way, the proofs of the instantiations come for free.