B-4 GLWE Cryptosystem

The GLWE cryptosystem is a generalized form to encompass both the LWE and RLWE cryptosystems. The GLWE cryptosystem’s ciphertext is a tuple $({\{A_i\}}_{i=0}^{k-1},B)$, where $B=\underset{i=0}{\overset{k-1}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}(A_i\cdot S_i)+\mathrm{\Delta }\cdot M+E$. The public key ${\{A_i\}}_{i=0}^{k-1}$ and the secret key ${\{S_i\}}_{i=0}^{k-1}$ are a list of $k$ $(n-1)$-degree polynomials, each. The message $M$ and the noise $E$ are an $(n-1)$-degree polynomial, each. Like in LWE and GLWE, a new public key $A$ is created for each ciphertext, whereas the same secret key $S$ is used for all ciphertexts. In this section, we denote each ciphertext instance as $({\{A_i\}}_{i=0}^{k-1},B)$ instead of $({\{A_i\}}_{i=0}^{k-1,⟨j⟩},B^{⟨j⟩})$ for simplicity.