Formalising cryptography using CryptHOL.

Security proofs are now a cornerstone of modern cryptography. Provable security has greatly increased the level of rigour of the security statements, however proofs of these statements often present informal or incomplete arguments. In fact, many proofs are still considered to be unverifiable due to their complexity and length. Formal methods offers one way to establish far higher levels of rigour and confidence in proofs and tools have been developed to formally reason about cryptography and obtain machine-checked proof of security statements. In this thesis we use the CryptHOL framework, embedded inside Isabelle/HOL, to reason about cryptography. First we consider two fundamental cryptographic primitives: Σ-protocols and Commitment Schemes. Σ-protocols allow a Prover to convince a Verifier that they know a value $x$ without revealing anything beyond that the fact they know $x$. Commitment Schemes allow a Committer to commit to a chosen value while keeping it hidden, and be able to reveal the value at a later time. We first formalise abstract definitions for both primitives and then prove multiple case studies and general constructions secure. A highlight of this part of the work is our general proof of the construction of commitment schemes from Σ-protocols. This result means that within our framework for every Σ-protocol proven secure we obtain, for free, a new commitment scheme that is secure also. We also consider compound $\Sigma$-protocols that allow for the proof of AND and OR statements. As a result of our formalisation effort here we are able to highlight which of the different definitions of Σ-protocols from the literature is the correct one; in particular we show that the most widely used definition of Σ-protocols is not sufficient for the OR construction. To show our frameworks are usable we also formalise numerous other case studies of Σ-protocols and commitment schemes, namely: the Σ-protocols by Schnorr, Chaum-Pedersen, and Okamoto; and the commitment schemes by Rivest and Pedersen. Second, we consider Multi-Party Computation (MPC). MPC allows for multiple distrusting parties to jointly compute functions over their inputs while keeping their inputs private. We formalise frameworks to abstractly reason about two party security in both the semi-honest and malicious adversary models and then instantiate them for numerous case studies and examples. A particularly important two party MPC protocol is Oblivious Transfer} (OT) which, in its simplest form, allows the Receiver to choose one of two messages from the other party, the Sender; the Receiver learns nothing of the other message held by the sender and the Sender does not learn which message the Receiver chose. Due to OTs fundamental importance we choose to focus much of our formalisation here, a highlight of this section of our work is our general proof of security of a 1-out-of-2 OT (OT₂¹) protocol in the semi-honest model that relies on Extended Trapdoor Permutations (ETPs). We formalise the construction assuming only that an ETP exists meaning any instantiations for known ETPs only require one to prove that it is in fact an ETP --- the security results on the protocol come for free. We demonstrate this by showing how the RSA collection of functions meets the definition of an ETP, and thus show how the security results are obtained easily from the general proof. We also provide proofs of security for the Naor Pinkas (OT₂¹) protocol in the semi-honest model as well as a proof that shows security for the two party GMW protocol --- a protocol that allows for the secure computation of any boolean circuit. The malicious model is more complex as the adversary can behave arbitrarily. In this setting we again consider an OT₂¹ protocol and prove it secure with respect to our abstract definitions

Formalising Σ\Sigma-Protocols and Commitment Schemes using CryptHOL

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: $\Sigma$-protocols and Commitment Schemes. $\Sigma$-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 $\Sigma$-protocols as well as a construction that allows for compound (AND and OR) $\Sigma$-protocols and the Pedersen and Rivest commitment schemes. A highlight of the work is a formalisation of the construction of commitment schemes from $\Sigma$-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