Part III
Generic Fully Homomorphic Encryption

This chapter explains the generic techniques of homomorphic computation adopted by various FHE schemes such as TFHE, CKKS, BGV, and BFV,

As we learned from §B-4, ${\mathsf{\text{𝖦𝖫𝖶𝖤}}}_{S,\sigma }(\mathrm{\Delta }M)=(A_0,A_1,\cdots ,A_{k-1},B)\in R_{⟨n,q⟩}^{k+1}$, where $R_{⟨n,q⟩}={\mathbb{Z}}_q[x]∕(x^n+1)$, and $B$ is computed as $B=\underset{i=0}{\overset{k-1}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}(A_i\cdot S_i)+\mathrm{\Delta }\cdot M+E$. Each $A_i$ is an $(n-1)$-degree polynomial as a public key, whose each coefficient is uniformly randomly sampled from $R_{⟨n,q⟩}$. $E$ is an $(n-1)$-degree polynomial as a noise, whose each coefficient is sampled from $R_{⟨n,q⟩}$ based on the Gaussian distribution ${\chi }_{\sigma }$. $S$ is a list of $k$ $(n-1)$-degree polynomials as a secret key, such that $S=(S_0,S_1,\cdots S_{k-1})\in R_{⟨n,q⟩}^k$, and each polynomial $S_i$’s each coefficient is a randomly sampled binary number in ${\mathbb{Z}}_2$ (i.e., $\{0,1\}$).

Based on this GLWE setup, this section will explain the following 5 homomorphic operations: ciphertext-to-ciphertext addition, ciphertext-to-plaintext addition, ciphertext-to-plaintext multiplication, ciphertext-to-ciphertext multiplication, and key switching.