Exploiting Small-Norm Polynomial Multiplication with Physical Attacks: Application to CRYSTALS-Dilithium

We present a set of physical attacks against CRYSTALS-Dilithium that accumulate noisy knowledge on secret keys over multiple signatures, finally leading to a full recovery attack. The methodology is composed of two steps. The first step consists of observing or inserting a bias in the posterior distribution of sensitive variables. The second step of an information processing phase which is based on belief propagation, which allows effectively exploiting that bias. The proposed concrete attacks rely on side-channel information, injection of fault attacks, or a combination of the two. Interestingly, the adversary benefits from the knowledge on the released signature, but is not dependent on it. We show that the combination of a physical attack with the binary knowledge of acceptance or rejection of a signature also leads to exploitable information on the secret key. Finally, we demonstrate that this approach is also effective against shuffled implementations of CRYSTALS-Dilithium