Homomorphic Key Switching

D-1.5 Homomorphic Key Switching

- Reference: TFHE Deep Dive - Part III - Key switching and leveled multiplications [9]

TFHE’s key switching scheme changes an LWE ciphertext’s secret key from $\vec{s}$ to ${\vec{s}}_{^{'}}$ , where the two key vectors may or may not have the same dimensions. This scheme is essentially LWE’s key switching scheme. Specifically, this is equivalent to the alternative GLWE version’s (§B-4.4) key switching scheme (§C-5) with $n = 1$ as follows:

$⟨$ Summary D-1.5 $⟩$ TFHE Key Switching

Given ${𝖫𝖶𝖤}_{\vec{s}, σ} (Δ m + e) = (\vec{a}, b)$ ,

${𝖫𝖶𝖤}_{{\vec{s}}_{^{'}}, σ} (Δ m + e^{'}) = (0, b) - ⟨ {𝖣𝖾𝖼𝗈𝗆𝗉}^{β, l} (\vec{a}), {𝖫𝖾𝗏}_{{\vec{s}}_{^{'}}, σ}^{β, l} (\vec{s}) ⟩$

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