### Abstract

We use CryptHOL to formalise commitment schemes and Sigma-protocols. Both are widely used fundamental two party cryptographic primitives. Security for commitment schemes is considered using game-based definitions whereas the security of Sigma-protocols is considered using both the game-based and simulation-based security paradigms. In this work, we first define security for both primitives and then prove secure multiple case studies: the Schnorr, Chaum-Pedersen and Okamoto Sigma-protocols as well as a construction that allows for compound (AND and OR statements) Sigma-protocols and the Pedersen and Rivest commitment schemes. We also prove that commitment schemes can be constructed from Sigma-protocols. We formalise this proof at an abstract level, only assuming the existence of a Sigma-protocol; consequently, the instantiations of this result for the concrete Sigma-protocols we consider come for free.

### License

BSD License

### Topics

- Computer science/Security/Cryptography

### Session Sigma_Commit_Crypto

- Commitment_Schemes
- Cyclic_Group_Ext
- Discrete_Log
- Number_Theory_Aux
- Uniform_Sampling
- Pedersen
- Rivest
- Sigma_Protocols
- Schnorr_Sigma_Commit
- Chaum_Pedersen_Sigma_Commit
- Okamoto_Sigma_Commit
- Xor
- Sigma_AND
- Sigma_OR

### Depends on

- CryptHOL

### Used by

- Constructive Cryptography in HOL: the Communication Modeling Aspect

### Auto-related entries

- The Stern-Brocot Tree

## Cite

×

Sigma_Commit_Crypto-AFP

    @article{Sigma_Commit_Crypto-AFP,
      author  = {David Butler and Andreas Lochbihler},
      title   = {Sigma Protocols and Commitment Schemes},
      journal = {Archive of Formal Proofs},
      month   = {October},
      year    = {2019},
      note    = {\url{https://isa-afp.org/entries/Sigma_Commit_Crypto.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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