Implement optimized bootstrapping algorithm from eprint.iacr.org/2020/1203

[Yanbin-Li-Oct]

Overview

This PR implements enhanced CKKS bootstrapping functionality based on the papers "Efficient Bootstrapping for Approximate Homomorphic Encryption with Non-Sparse Keys" (https://eprint.iacr.org/2020/1203.pdf) and "Better bootstrapping for approximate homomorphic encryption" (https://eprint.iacr.org/2019/688.pdf). The changes include new operator functions, improved bootstrapping algorithms, and support for non-sparse secret keys.

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Functional Enhancements

1. New Bootstrapping Implementation

• regular_bootstrapping_v2(): Implements efficient bootstrapping with non-sparse key support and improved precision
• mod_up_from_q0(): Fixes modulus raising from Q₀ to full Q, and adds optional key switching between dense and sparse keys
• coeff_to_slot_v2(), slot_to_coeff_v2(): Update BSGS (Baby-Step Giant-Step) algorithm for transformation matrices based on configurable BSGS ratio
• eval_mod(): Computes modular reduction using range-reduced Chebyshev approximation of cos(2^(-r)(x - 0.5)) with optimally-placed interpolation nodes, followed by r double-angle iterations. Implements errorless and depth-optimal polynomial evaluation.

2. New Utility Functions

• Complex arithmetic operations:
  • mult_i(), div_i(): Efficient imaginary unit operations without consuming modulus levels
• Enhanced plaintext operations:
  • add_plain_v2(): Adds complex constant to ciphertext using Gaussian integer addition. Employs NTL big integer library for overflow-safe
    scaling, computes accurate per-modulus RNS representation.
  • multiply_plain_v2(): Multiplies ciphertext by complex constant with automatic precision preservation for fractional parts (scales by prime modulus when needed), employs NTL library for overflow-safe Gaussian integer multiplication
  • scale_up(): Wrapper around multiply_plain_v2() for real-only scaling, commonly used in bootstrapping to adjust ciphertext scale factors

3. Key Generation Support

• generate_secret_key_v2(): Collision-free secret key generation with exact Hamming weight control using CPU-side Fisher-Yates shuffle algorithm
• Switch key generation: Support for dense-to-sparse and sparse-to-dense key conversions

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Testing and Examples

4. New Example Program

• example/bootstrapping/5_ckks_regular_bootstrapping_v2.cu: Demonstrates the new regular_bootstrapping_v2() function with non-sparse keys

Key Benefits

• Removes sparse key constraints for bootstrapping operations, complies with the most recent security recommendations
• Preserves more modulus levels, enabling deeper homomorphic computation circuits after refresh, and reducing the frequency of costly bootstrapping operations

Performance

• Runtime: About ~277 ms on NVIDIA GeForce RTX 3090 (poly_modulus_degree = 2^16)
• Precision: Achieves ~17-18 bits average precision (log₂ scale) after bootstrapping

[Alisah-Ozcan]

Hi, thanks for PR, I will review it and let you know.