The Brakerski-Gentry-Vaikuntanathan (BGV) scheme is one of the most significant fully homomorphic encryption (FHE) schemes. It belongs to a class of FHE schemes whose security is based on the presumed intractability of the Learning with Errors (LWE) problem and its ring variant (RLWE). Such schemes deal with a quantity, called noise, which increases each time a homomorphic operation is performed. Specifically, in order for the scheme to work properly, it is essential that the noise remains below a certain threshold throughout the process. For BGV, this threshold strictly depends on the ciphertext modulus, which is one of the initial parameters whose selection heavily affects both the efficiency and security of the scheme. For an optimal parameter choice, it is crucial to accurately estimate the noise growth, particularly that arising from multiplication, which is the most complex operation. In this work, we propose a novel average-case approach that precisely models noise evolution and guides the selection of initial parameters, improving efficiency while ensuring security. The key innovation of our method lies in accounting for the dependencies among ciphertext errors generated with the same key, and in providing general guidelines for accurate parameter selection that are library-independent.

翻译：Brakerski-Gentry-Vaikuntanathan (BGV) 方案是最重要的全同态加密 (FHE) 方案之一。它属于一类基于带误差学习 (LWE) 问题及其环变体 (RLWE) 的预设难解性来保证安全性的FHE方案。此类方案需要处理一个称为噪声的量，该噪声在每次执行同态操作时都会增加。具体而言，为确保方案正常运行，噪声在整个过程中必须始终低于特定阈值。对于BGV方案，该阈值严格依赖于密文模数——这是初始参数之一，其选择对方案的效率和安全性均有重大影响。为实现最优参数选择，必须精确估计噪声增长，特别是由最复杂的乘法操作所产生的噪声。本研究提出了一种新颖的平均情况分析方法，能够精确建模噪声演化过程，并指导初始参数的选择，在保证安全性的同时提升效率。该方法的核心创新在于：考虑了使用相同密钥生成的密文误差之间的依赖性，并提供了与具体实现库无关的精确参数选择通用指导原则。