Modify noise flooding for BGV/BFV threshold FHE decryption about openfhe-development HOT 1 CLOSED

The protocol for BFV threshold noise flooding is as follows:
Let our current RNS basis for decryption be Q=q_0 * q_1 * ... * q_k. Add two more 60-bit primes p_0 and p_1. The extended basis is QP = q_0 * p_1 * p_2 * ...q_1 * q_k.

As the input, we get c1 mod QP. We first generate a random uniform ring element b w.r.t. to mod Qprime = QP/q_0. Then we do exact RNS basis extension from Qprime to QP.

Then we do flooding c1 * s_i + b mod QP.

Each party sends the results, and the fuser adds them up mod QP. Finally, we do BFV rounding and output the decryption result.

Possible optimizations for future:

1. In BFV, right before BFV rounding before decryption, we modswitch from QP to q_0.
2. Modswitch to 3 moduli, i.e., QP = q_0 * p_0 * p_1 before generating/adding the random elements.
3. Do modswitch by each party from QP to q_0 before sending the partial decryption results to the fuser.
4. Change c1 * s_i to COEFFICIENT representation after the product. Do all other operations in COEFFICIENT representation.

For BGV threshold noise flooding, the protocol is exactly the same, except we use b' = tb for flooding instead of b.

from openfhe-development.