Homomorphic Key Switching

D-2.8 Homomorphic Key Switching

BFV’s key switching scheme changes an RLWE ciphertext’s secret key from $S$ to $S^{'}$ . This scheme is essentially RLWE’s key switching scheme with the sign of the $A \cdot S$ term flipped in the encryption and decryption formula. Specifically, this is equivalent to the alternative GLWE version’s (§B-4.4) key switching scheme (§C-5) with $k = 1$ as follows:

$⟨$ Summary D-2.8 $⟩$ BFV’s Key Switching

${𝖱𝖫𝖶𝖤}_{S^{'}, σ} (Δ M) = (0, B) + ⟨ {𝖣𝖾𝖼𝗈𝗆𝗉}^{β, l} (A), {𝖱𝖫𝖾𝗏}_{S^{'}, σ}^{β, l} (S) ⟩$

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