This paper presents a set of physical profiled attacks against CRYSTALS-Dilithium that accumulate noisy knowledge on secret keys over multiple signatures, ultimately leading to full key recovery. The methodology involves two steps: first, observing or inserting bias in the posterior distribution of sensitive variables; second, using belief propagation to exploit this bias. The attacks leverage side-channel information, induced faults, or both. The adversary benefits most from this knowledge when targeting released signatures, though they are not strictly necessary. The combination of physical attacks with binary knowledge of signature acceptance or rejection also provides exploitable information on the secret key. The approach is effective against shuffled implementations of CRYSTALS-Dilithium. CRYSTALS-Dilithium is a lattice-based post-quantum cryptographic scheme selected by NIST. It has been the target of various attacks, including side-channel and fault attacks. The paper focuses on CRYSTALS-Dilithium and demonstrates attacks that recover secret key polynomials using noisy information from side-channel and fault attacks. The attacks are based on the Belief Propagation (BP) algorithm, which is used in Soft Analytical Side-Channel Attacks (SASCA). The paper shows that the number of signatures needed to recover a secret key polynomial depends on the noise level and the type of attack. The paper introduces a generic attack framework that can be applied to various scenarios, including physical attacks with accepted signatures, physical attacks without accepted signatures, and physical attacks with shuffled computations. The framework is evaluated through simulated experiments, showing that the number of traces needed to recover a secret key polynomial varies depending on the noise level and the type of attack. The paper also discusses the impact of these attacks on the security of CRYSTALS-Dilithium and suggests that masking is recommended for all sensitive variables to mitigate the risks. The results show that the proposed attacks are effective against CRYSTALS-Dilithium, highlighting the importance of robust countermeasures against physical attacks.This paper presents a set of physical profiled attacks against CRYSTALS-Dilithium that accumulate noisy knowledge on secret keys over multiple signatures, ultimately leading to full key recovery. The methodology involves two steps: first, observing or inserting bias in the posterior distribution of sensitive variables; second, using belief propagation to exploit this bias. The attacks leverage side-channel information, induced faults, or both. The adversary benefits most from this knowledge when targeting released signatures, though they are not strictly necessary. The combination of physical attacks with binary knowledge of signature acceptance or rejection also provides exploitable information on the secret key. The approach is effective against shuffled implementations of CRYSTALS-Dilithium. CRYSTALS-Dilithium is a lattice-based post-quantum cryptographic scheme selected by NIST. It has been the target of various attacks, including side-channel and fault attacks. The paper focuses on CRYSTALS-Dilithium and demonstrates attacks that recover secret key polynomials using noisy information from side-channel and fault attacks. The attacks are based on the Belief Propagation (BP) algorithm, which is used in Soft Analytical Side-Channel Attacks (SASCA). The paper shows that the number of signatures needed to recover a secret key polynomial depends on the noise level and the type of attack. The paper introduces a generic attack framework that can be applied to various scenarios, including physical attacks with accepted signatures, physical attacks without accepted signatures, and physical attacks with shuffled computations. The framework is evaluated through simulated experiments, showing that the number of traces needed to recover a secret key polynomial varies depending on the noise level and the type of attack. The paper also discusses the impact of these attacks on the security of CRYSTALS-Dilithium and suggests that masking is recommended for all sensitive variables to mitigate the risks. The results show that the proposed attacks are effective against CRYSTALS-Dilithium, highlighting the importance of robust countermeasures against physical attacks.