Homomorphic Ciphertext-to-Ciphertext Addition

TFHE’s ciphertext-to-ciphertext addition uses LWE’s ciphertext-to-ciphertext addition scheme, which is equivalent to GLWE’s ciphertext-to-ciphertext addition scheme (§C-1) with

n = 1

$⟨$ Summary D-1.2 $⟩$ TFHE Ciphertext-to-Ciphertext Addition

${𝖫𝖶𝖤}_{\vec{s}, σ} (Δ m^{⟨ 1 ⟩} + e^{⟨ 1 ⟩}) + {𝖫𝖶𝖤}_{\vec{s}, σ} (Δ m^{⟨ 2 ⟩} + e^{⟨ 2 ⟩})$

$= ({\vec{a}}^{⟨ 1 ⟩}, b^{⟨ 1 ⟩}) + ({\vec{a}}^{⟨ 2 ⟩}, b^{⟨ 2 ⟩})$

$= ({\vec{a}}^{⟨ 1 ⟩} + {\vec{a}}^{⟨ 2 ⟩}, b^{⟨ 1 ⟩} + b^{⟨ 2 ⟩})$

$= {𝖫𝖶𝖤}_{\vec{s}, σ} (Δ (m^{⟨ 1 ⟩} + m^{⟨ 2 ⟩}) + e^{⟨ 1 + 2 ⟩})$

D-1.2 Homomorphic Ciphertext-to-Ciphertext Addition