Formalising $$\varSigma $$-Protocols and Commitment Schemes Using CryptHOL
D. Butler, Andreas Lochbihler, David Aspinall, Alberto Hijazo Gascón · Journal of Automated Reasoning · 2020
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: $$\varSigma $$ Σ -protocols and Commitment Schemes. $$\varSigma $$ Σ -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 (Lochbihler in Archive of formal proofs, 2017) to formalise both primitives and prove secure multiple examples namely; the Schnorr, Chaum-Pedersen and Okamoto $$\varSigma $$ Σ -protocols as well as a construction that allows for compound (AND and OR) $$\varSigma $$ Σ -protocols and the Pedersen and Rivest commitment schemes. A highlight of the work is a formalisation of the construction of commitment schemes from $$\varSigma $$ Σ -protocols (Damgard in Lecture notes, 2002). 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.