Ciphertext-to-Plaintext Multiplication

BGV’s ciphertext-to-plaintext multiplication scheme is exactly the same as BFV’s scheme (Summary D-2.6 in §D-2.6).

$⟨$ Summary D-4.5 $⟩$ BGV Ciphertext-to-Plaintext Multiplication

${𝖱𝖫𝖶𝖤}_{S, σ} (M + Δ E) \cdot Λ$

$= (A, B) \cdot Λ$

$= (A \cdot Λ, B \cdot Λ)$

$= {𝖱𝖫𝖶𝖤}_{S, σ} ((M \cdot Λ) + Δ E \cdot Λ)$

Notice that BGV’s ciphertext-to-plaintext multiplication does not consume any multiplicative level.