Some applications of linear programming to Dilithium

Dilithium is a signature algorithm, considered post-quantum, and recently standardized under the name ML-DSA by NIST. Due to its security and performance, it is recommended in most use cases.

During this presentation, I will outline the main ideas behind two studies, conducted in collaboration with Andersson Calle-Vierra, Benoît Cogliati, and Louis Goubin, which provide a better understanding of Dilithium's practical security.

I show that by introducing an error during the generation of the Dilithium signature, it is possible to force the signer to provide “incorrect” signatures that reveal information about the secret key in the form of inequalities on its coefficients. We then reformulate the problem of recovering the secret key from incorrect signatures into a linear programming problem.

Among the data in the MLWE instance (A,t), which is at the heart of Dilithium's construction, the least significant part of t—denoted by t0—is not included in the public key and is formally considered part of the secret key. Knowing t_0 has no impact on Dilithium's black box security, but it does allow for much more effective side-channel attacks. I explain the main ideas for constructing an attack that finds t0 from Dilithium signatures, again by twisting linear programming tools.

Both of these results use linear programming techniques that are somewhat deviated from their traditional uses, which is quite unusual for cryptanalysis of public-key algorithms.