Functional bootstrapping is a core technique in Fully Homomorphic Encryption (FHE). For large plaintext, to evaluate a general function homomorphically over a ciphertext, in the FHEW/TFHE approach, since the function in look-up table form is encoded in the coefficients of a test polynomial, the degree of the polynomial must be high enough to hold the entire table. This increases the bootstrapping time complexity and memory cost, as the size of bootstrapping keys and keyswitching keys need to be large accordingly. In this paper, we propose to encode the look-up table of any function in a polynomial vector, whose coefficients can hold more data. The corresponding representation of the additive group Zq used in the RGSW-based bootstrapping is the group of monic monomial permutation matrices, which integrates the permutation matrix representation used by Alperin-Sheriff and Peikert in 2014, and the monic monomial representation used in the FHEW/TFHE scheme. We make comprehensive investigation of the new representation, and propose a new bootstrapping algorithm based on it. The new algorithm has the prominent benefit of small bootstrapping key size and small key-switching key size, which leads to polynomial factor improvement in key size, in addition to constant factor improvement in run-time cost.

翻译：功能引导是全同态加密（FHE）中的核心技术。在FHEW/TFHE方案中，为在大明文域上同态评估任意函数，查找表形式的函数需编码于测试多项式的系数中，导致多项式度数必须足够高以容纳整个表。这增加了引导时间复杂度和存储开销，因为引导密钥与密钥转换密钥的规模需随之增大。本文提出将任意函数的查找表编码于多项式向量中，其系数可承载更多数据。基于RGSW引导方案中加法群Zq的表示相应采用首一单项置换矩阵群，该表示整合了Alperin-Sheriff与Peikert于2014年提出的置换矩阵表示法，以及FHEW/TFHE方案中使用的首一单项表示法。我们对新表示法进行了全面研究，并据此提出新型引导算法。该算法的显著优势在于引导密钥和密钥转换密钥尺寸较小，可使密钥规模实现多项式因子级改进，同时运行时间成本实现常数因子级提升。