CKKS multiplication turns a two-part ciphertext into three parts and doubles the error

CKKS – Addition and Multiplication

In this tutorial installment, I implement ciphertext addition and multiplication for the CKKS homomorphic encryption scheme using a 64-bit modulus. Addition is simple: add the polynomial pairs, and the error grows linearly. Multiplication, however, produces a three-tuple ciphertext whose decryption involves the square of the secret key, causing error to grow quadratically and the scaling factor to square. I show the code and tests, and explain why rescaling and relinearization are needed next.