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Motivated by the rise of quantum computers, existing public-key cryptosystems are expected to be replaced by post-quantum schemes in the next decade in billions of devices. To facilitate the transition, NIST is running a standardization process which is currently in its final Round. Only three digital signature schemes are left in the competition, among which Dilithium and Falcon are the ones based on lattices. Classical fault attacks on signature schemes make use of pairs of faulty and correct signatures to recover the secret key which only works on deterministic schemes. To counter such attacks, Dilithium offers a randomized version which makes each signature unique, even when signing identical messages. In this work, we introduce a novel Signature Correction Attack which not only applies to the deterministic version but also to the randomized version of Dilithium and is effective even on constant-time implementations using AVX2 instructions. The Signature Correction Attack exploits the mathematical structure of Dilithium to recover the secret key bits by using faulty signatures and the public-key. It can work for any fault mechanism which can induce single bit-flips. For demonstration, we are using Rowhammer induced faults. Thus, our attack does not require any physical access or special privileges, and hence could be also implemented on shared cloud servers. We perform a thorough classical and quantum security analysis of Dilithium and successfully recover 1,851 bits out of 3,072 bits of secret key $s_1$ for security level 2. The lattice strength against quantum attackers is reduced from $2^{128}$ to $2^{81}$ while the strength against classical attackers is reduced from $2^{141}$ to $2^{89}$. Hence, the Signature Correction Attack may be employed to achieve a practical attack on Dilithium (security level 2) as proposed in Round 3 of the NIST post-quantum standardization process.