How BGV's modulus switching works after multiplication?

[Caesar]

@gogo9th

BGV Modulus Switch

We find new polynomials A’A′A’ and B’B′B’ as follows:

• Among all possible A’A′A’ such that A’ \equiv A \bmod tA′≡AmodtA’ \equiv A \bmod t, choose the one whose each term’s coefficient is the closest integer to that of polynomial \dfrac{q_{l-1}}{q_l}\cdot Aql−1ql⋅A\dfrac{q_{l-1}}{q_l}\cdot A
• Among all possible B’B′B’ such that B’ \equiv B \bmod tB′≡BmodtB’ \equiv B \bmod t, choose the one whose each term’s coefficient is the closest integer to that of polynomial \dfrac{q_{l-1}}{q_l}\cdot Bql−1ql⋅B

Thank you for the detailed proof. I am sure this will be useful to the community.

I have not thoroughly reviewed your proof, but the premise (referenced here) seems logical to me and aligns with my previous response.