Breaking the power-of-two barrier: noise estimation for BGV in NTT-friendly rings

Abstract

The Brakerski–Gentry–Vaikuntanathan (BGV) scheme is a Fully Homomorphic Encryption (FHE) cryptosystem based on the Ring Learning With Error (RLWE) problem. Ciphertexts in this scheme contain an error term that grows with operations and causes decryption failure when it surpasses a certain threshold. Consequently, the parameters of BGV need to be estimated carefully, with a trade-off between security and error margin. The ciphertext space of BGV is the ring \(\mathcal {R}_q=\mathbb {Z}_q[x]/(\Phi _m(x))\), where usually the degree n of the cyclotomic polynomial \(\Phi _m(x)\) is chosen as a power of two for efficiency reasons. However, the jump between two consecutive powers-of-two polynomials also causes a jump in security, resulting in parameters that are much bigger than what is needed. In this work, we explore the non-power-of-two instantiations of BGV. Although our theoretical research encompasses results applicable to any cyclotomic ring, the focus of our investigation is the case of \({m=2^s\cdot 3^t}\) where \(s,t\ge 1\), i.e., cyclotomic polynomials with degree \({n=\phi (m)=2^s\cdot 3^{t-1}}\). We provide a thorough analysis of the noise growth in this new setting using the canonical norm and compare our results with the power-of-two case considering practical aspects like NTT algorithms. We find that in many instances, the parameter estimation process yields better results for the non-power-of-two setting.