International Association
for Cryptologic Research

The Module-Lattice-Based Digital Signature Standard (ML-DSA), formerly known as CRYSTALS-Dilithium, is a lattice-based post-quantum cryptographic scheme. In August 2024, the National Institute of Standards and Technology (NIST) officially standardized ML-DSA under FIPS 204. Dilithium generates one valid signature and multiple rejected signatures during the signing process. Most Side-Channel Attacks targeting Dilithium have focused solely on the valid signature, while neglecting the hints contained in rejected signatures. In this paper, we propose a method for recovering the private key by simultaneously leveraging side-channel leakages from both valid signatures and rejected signatures. This approach minimizes the number of signing attempts required for full key recovery. We construct a factor graph incorporating all relevant side-channel leakages and apply the Belief Propagation (BP) algorithm for private key recovery.

We conducted a proof-of-concept experiment on a Cortex M4 core chip, where the results demonstrate that utilizing rejected signatures reduces the required number of traces by at least $42\%$ for full key recovery. A minimum of a single trace can recover the private key with a success rate of $30\%$. Our findings highlight that protecting rejected signatures is crucial, as their leakage provides valuable side-channel information. We strongly recommend implementing countermeasures for rejected signatures during the signing process to mitigate potential threats.