Homomorphic Multiple Precision Multiplication for CKKS and Reduced Modulus Consumption

Homomorphic Encryption (HE) schemes such as BGV, BFV, and CKKS consume some ciphertext modulus for each multiplication. Bootstrapping (BTS) restores the modulus and allows homomorphic computation to continue, but it is time-consuming and requires a significant amount of modulus. For these reasons, decreasing modulus consumption is crucial topic for BGV, BFV and CKKS, on which numerous studies have been conducted. We propose a novel method, called Mult(2), to perform ciphertext multiplication in the CKKS scheme with lower modulus consumption. Mult(2) relies an a new decomposition of a ciphertext into a pair of ciphertexts that homomorphically performs a weak form of Euclidean division. It multiplies two ciphertexts in decomposed formats with homomorphic double precision multiplication, and its result approximately decrypts to the same value as does the ordinary CKKS multiplication. Mult(2) can perform homomorphic multiplication by consuming almost half of the modulus. We extend it to Mult(t) for any t >= 2, which relies on the decomposition of a ciphertext into t components. All other CKKS operations can be equally performed on pair/tuple formats, leading to the double-CKKS (resp. tuple-CKKS) scheme enabling homomorphic double (resp. multiple) precision arithmetic. As a result, when the ciphertext modulus and dimension are fixed, the proposed algorithms enable the evaluation of deeper circuits without bootstrapping, or allow to reduce the number of bootstrappings required for the evaluation of the same circuits. Furthermore, they can be used to increase the precision without increasing the parameters. For example, Mult(2) enables 8 sequential multiplications with 100 bit scaling factor with a ciphertext modulus of only 680 bits, which is impossible with the ordinary CKKS multiplication algorithm.