BGV’s ModDrop works similarly to that of CKKS’s ModDrop (Summary D-3.7 in §D-3.7). Remember that CKKS’s ciphertext decryption relation is as follows:
# where represents a modulo reduction by
BGV’s ModDrop operation decreases its modulus from is performed by updating the ciphertext to a new one: , . After the ModDrop, the ciphertext’s modulus decreases from , yet its decryption relation still holds the same as follows:
# where is an integer (the -th prime element of )
# where is an integer
# since
As shown above, decrypts to the same , a plaintext with a scaled error. However, the noise budget (i.e., allowed threshold of the noise) decreases because the ciphertext modulus-to-noise ratio decreases.
CKKS’s ModDrop is summarized as follows:
Summary D-4.5 BGV’s ModDrop
Given a BGV ciphertext with the -th multiplicative level , a ModDrop operation is as follows:
After this, the ciphertext’s multiplicative level decreases by 1, the noise’s scaling factor and the plaintext are unaffected, and the noise budget (i.e., allowed noise threshold) decreases.