Noise of ciphertexts in LWE/RLWE based FHE Often times $[\langle \textbf{c}, \textbf{s} \rangle]_q$ is referred to as the noise associated to the ciphertext $\textbf{c}$, and that decryption is correct when the norm of the noise is $&lt; q/2$. But shouldn't $\langle \textbf{c}, \textbf{s} \rangle$ be the noise instead of $[\langle \textbf{c}, \textbf{s} \rangle]_q$, since $[\langle \textbf{c}, \textbf{s} \rangle]_q$ means the reduction mod $q$ already happened and so the norm of $[\langle \textbf{c}, \textbf{s} \rangle]_q$ is always $&lt; q/2$?
This is because of the decryption equation: If two ciphertexts differ ($\mathbb{Z}$-component-wise..., not homomorphically) in a multiple of $q$, they both decrypt to the same thing because of the modular reduction. Therefore, upon reception of a 'big' ciphertext, it makes sense to take modular reduction of this ciphertext and then look at its noise. Conversely, to cheat silly attackers one could add 'false noise' in this fashion (severely augmenting ciphertext size, however)