Ciphertext-to-Ciphertext Addition

BGV’s ciphertext-to-ciphertext addition scheme is exactly the same as BFV’s scheme (Summary D-2.4 in §D-2.4).

$⟨$ Summary D-4.3 $⟩$ BGV Ciphertext-to-Ciphertext Addition

${𝖱𝖫𝖶𝖤}_{S, σ} (M^{⟨ 1 ⟩} + Δ E^{⟨ 1 ⟩}) + {𝖱𝖫𝖶𝖤}_{S, σ} (M^{⟨ 2 ⟩} + Δ E^{⟨ 2 ⟩})$

$= (A^{⟨ 1 ⟩}, B^{⟨ 1 ⟩}) + (A^{⟨ 2 ⟩}, B^{⟨ 2 ⟩})$

$= (A^{⟨ 1 ⟩} + A^{⟨ 2 ⟩}, B^{⟨ 1 ⟩} + B^{⟨ 2 ⟩})$

$= {𝖱𝖫𝖶𝖤}_{S, σ} ((M^{⟨ 1 ⟩} + M^{⟨ 2 ⟩}) + Δ E^{⟨ 1 ⟩} + Δ E^{⟨ 2 ⟩})$