The CKKS Fully Homomorphic Encryption (FHE) scheme enables approximate arithmetic on encrypted complex numbers for a desired precision. Most implementations use RNS with carefully chosen parameters to balance precision, efficiency, and security. However, a key limitation in RNS-CKKS is the rigid coupling between the scale factor, which determines numerical precision, and the modulus, which ensures security. Since these parameters serve distinct roles—one governing arithmetic correctness and the other defining cryptographic structure—this dependency imposes design constraints, such as a lack of suitable NTT primes and limited precision flexibility, ultimately leading to inefficiencies.

The CKKS fully homomorphic encryption (FHE) scheme enables computations on vectors of approximate complex numbers. A moderate precision of ≈ 20 bits often suffices but, in many applications, a higher precision is required for functionality and/or security. Indeed, to obtain IND-CPA-D security [Li-Micciancio; Eurocrypt'21], secure threshold-FHE [Asharov et al; Eurocrypt'12] and circuit privacy [Gentry; STOC'09], all known approaches require a precision that supports noise flooding. This may lead to a precision of ≈ 80 bits, or more. High-precision CKKS is hard to achieve, notably because of bootstrapping. The main difficulty is modulus consumption: every homomorphic multiplication consumes some, out of an overall modulus budget. Unfortunately, in high precision, most known bootstrapping algorithms consume so much modulus that one needs to increase the parameters to increase the budget. The state-of-the-art approach, Meta-BTS [Bae et al; CCS'22], performs moderate-precision bootstrapping several times to enable high-precision bootstrapping, with similar modulus consumption as the base bootstrapping it builds upon. It however damages latency.

Bootstrapping is currently the only known method for constructing fully homomorphic encryptions. In the BFV scheme specifically, bootstrapping aims to reduce the error of a ciphertext while preserving the encrypted plaintext. The existing BFV bootstrapping methods follow the same pipeline, relying on the evaluation of a digit extraction polynomial to annihilate the error located in the least significant digits. However, due to its strong dependence on performance, bootstrapping could only utilize a limited form of plaintext modulus, such as a power of a small prime number.

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.

Bootstrapping, which enables the full homomorphic encryption scheme that can perform an infinite number of operations by restoring the modulus of the ciphertext with a small modulus, is an essential step in homomorphic encryption. However, bootstrapping is the most time and memory consuming of all homomorphic operations. As we increase the precision of bootstrapping, a large amount of computational resources is required. Specifically, for any of the previous bootstrap designs, the precision of bootstrapping is limited by rescaling precision.