Homomorphic encryption (HE) allows computations to be directly carried out on ciphertexts and enables privacy-preserving cloud computing. The computations on the coefficients of the polynomials involved in HE are always followed by modular reduction, and the overall complexity of ciphertext multiplication can be reduced by utilizing the quotient. Our previous design considers the cases that the dividend is an integer multiple of the modulus and the modulus is in the format of $2^w-2^u\pm1$, where $u<w/2$. In this paper, the division is generalized for larger $u$ and dividend not an integer multiple of the modulus. An algorithm is proposed to compute the quotient and vigorous mathematical proofs are provided. Moreover, efficient hardware architecture is developed for implementing the proposed algorithm. Compared to alternative division approaches that utilize the inverse of the divisor, for $w=32$, the proposed design achieves at least 9% shorter latency and 79\% area reduction for 75% possible values of $u$.

翻译：同态加密（HE）允许直接在密文上执行计算，从而支持隐私保护的云计算。HE中涉及的多项式系数运算始终伴随模约简，且利用商数可降低密文乘法的整体复杂度。我们先前的研究考虑了被除数为模数的整数倍且模数格式为$2^w-2^u\pm1$（其中$u<w/2$）的情形。本文将该除法推广至更大的$u$值以及被除数非模数整数倍的情形。提出了一种用于计算商数的算法，并给出了严格的数学证明。此外，开发了高效的硬件架构来实现所提算法。与利用除数逆元的替代除法方法相比，当$w=32$时，所提设计在75%的$u$取值范围内实现了至少9%的延迟缩短和79%的面积缩减。