Paper 2024/1703 Original: eprint.iacr.org/2024/1703 ↗

Free-XOR Gate Bootstrapping

Abstract

The FHEW-like gate bootstrapping framework operates in a 2-bit plaintext space, where logic gates such as NAND, XOR, and AND are implemented by adding two ciphertexts and extracting the most significant bit. However, each gate operation requires bootstrapping with a primary cost of one blind rotation, which is expensive, when processing circuit operations for applications. We propose a novel Free-XOR gate bootstrapping framework based on a single-bit plaintext space, in which the XOR operation is realized by simply adding two ciphertexts, resulting in an almost free computational cost. To form a minimal complete set for logical operations, we design an algorithm for the AND gate within this framework. The AND gate cost of our Free-XOR gate bootstrapping involves two blind rotations. However, by utilizing a single-bit plaintext space to enhance noise tolerance and swapping some operations of the bootstrapping process, we can adopt a more compact parameter setting, which in turn accelerates the speed of blind rotation. We propose an instantiation of the NTRU-based AND gate operation, which requires two blind rotations. Despite the additional rotation, the overall computational cost is marginally lower than the state-of-the-art gate bootstrapping scheme LLW+ [TCHES24], which utilizes only a single blind rotation. In addition, our approach achieves a significant reduction in key size, reducing it to 3.3 times the size of LLW+ [TCHES24].