Fixing and Mechanizing the Security Proof of
         Fiat-Shamir with Aborts and Dilithium

The proofs (see proofs/) are split over two directories. The utils directory contains utility files that are not specific to the security proof of Dilithium. This includes various extensions to the EasyCrypt libraries (e.g., the yet-to-be-merged new matrix theory) and the games defining assumptions like MLWE and SelfTargetMSIS.

The security directory contains the specification of Dilithium together with the mechanized security proof. The key files are:

• Dilithium.ec: This is the top-level file, containing the specification of Dilithium, the security theorem for Dilithium (i.e., Theorem 4 from the paper), and a commented desciption of all the constants appearing in the security bound.

• SimplifiedScheme.ec : This file contains the simpified scheme (i.e., the one where the public key is (A,t)) and the proof of CMA security. This includes the proof of NMA security, the verification of the HVZK simulator, and the instantiation of the CMA-to-NMA reduction.

• FSa_CMAtoKOA.eca: This contains the proof of the CMA to NMA reduction (i.e., Theorem 3 from the paper). The file iself contains all the arguments corresponding to the proof of Lemma 1.

• ReprogHybrid.eca: This file contains the core arguments for the CMA to NMA reduction: bounding the loss of the combined hop from Prog to Trans (i.e., essentially the proof of Theorem 3).

• DRing.eca: The abstract theory specifying the minimal structure (on Rq, highBits, etc.) required to carry out the security proof.

• ConcreteDRing.eca an instance of DRing, showing that the ring Rq from the specification of Dilithium has all the properties needed in the security proof.

Checking all the proofs with EasyCrypt (EC) requires a development version. This is mainly due to our use of the expected-cost logic which is not yet part of the main development branch. Moreover, EC requires certain versions of the SMT solvers Alt-Ergo, Z3, and CVC4 to be available. To facilitate independent checking, we provide scripts to set up a docker container with the entire toolchain.

The configuration of docker is highly OS specific. On Linux, this usually amounts to:

• installing docker and enabling the docker daemon
• adding the local user to the docker group
• rebooting (to let the group changes take effect)

If docker is configured correctly,

$ make -C proofs tests

should set up a fresh Debian-based container, install the complete EasyCrypt toolchain based on the deploy-expected-cost branch and run easycrypt on all files in the development. Note that, depending on your internet connection and hardware, the initial setup of the container can take a while.

If desired, make -C proofs shell provides a shell in the same context as the tests. Note: Manually checking the files in the security folder requires adding -R utils to the arguments of easycrypt.

Alternatively, EasyCrypt can be installed locally by following the instructions at https://github.com/EasyCrypt/easycrypt Here we add some details specific to our development version.

The only difference is that, when pinning easycrypt, the correct branch needs to be chosen:

$ opam pin -yn add easycrypt https://github.com/EasyCrypt/easycrypt.git#dilithium

As the installation above instructions suggests, one can also install EasyCrypt through nix. Our proofs require all three SMT solvers (alt-ergo, Z3, and CVC4) available to EasyCrypt. The installation process amounts to the following:

0. Install nix

1. Install CVC4 manually, as the corresponding nix package is currently broken.

2. Clone the EasyCrypt repository and switch to the dilithium branch

3. In the EasyCrypt folder, run

   $ NIXPKGS_ALLOW_UNFREE=1 nix-shell --arg withProvers true
   

   to start nix. The alt-ergo and Z3 provers should now be available as well.

4. make to create the ec.native which is the "main" executable for EasyCrypt.

5. ./ec.native why3config for EasyCrypt to find the three SMT solvers.

6. exit out of nix shell and we can now use EasyCrypt

7. Our EasyCrypt proofs can be checked using

   $ ECRJOBS=1 path/to/ec.native runtest tests.config utils security
   

   from this folder. Alternatively, you can also symlink ./ec.native to easycrypt and use it with proof general.