Optimization of Homomorphic Comparison Algorithm on RNS-CKKS Scheme

Abstract

The sign function can be adopted to implement the comparison operation, max function, and rectified linear unit (ReLU) function in the Cheon-Kim-Kim-Song (CKKS) scheme; hence, several studies have been conducted to efficiently evaluate the sign function in the CKKS scheme. Recently, Lee et al. (IEEE Trans. Depend. Sec. Comp.) proposed a practically optimal approximation method for the sign function in the CKKS scheme using a composition of minimax approximate polynomials. In addition, Lee et al. proposed a polynomial-time algorithm that finds the degrees of component polynomials that minimize the number of non-scalar multiplications. However, homomorphic comparison/max/ReLU functions using Lee et al.'s approximation method have not been successfully implemented in the residue number system variant CKKS (RNS-CKKS) scheme. In addition, the degrees of component polynomials found by Lee et al.'s algorithm are not optimized for the RNS-CKKS scheme because the algorithm does not consider that the running time of non-scalar multiplication depends significantly on the ciphertext level in the RNS-CKKS scheme. In this study, we propose a fast algorithm for the inverse minimax approximation error, which is a subroutine required to find the optimal set of degrees of component polynomials. The proposed algorithm facilitates determining the optimal set of degrees of component polynomials with higher degrees than in the previous study. In addition, we propose a method to find the degrees of component polynomials optimized for the RNS-CKKS scheme using the proposed algorithm for the inverse minimax approximation error. We successfully implement the homomorphic comparison, max function, and ReLU function algorithms on the RNS-CKKS scheme with a low comparison failure rate ( $< 2^{-15}$ ), and provide various parameter sets according to the precision parameter $\alpha $. We reduce the depth consumption of the homomorphic comparison, max function, and ReLU function algorithms by one depth for several values of $\alpha $. In addition, the numerical analysis demonstrates that the homomorphic comparison, max function, and ReLU function algorithms using the degrees of component polynomials found by the proposed algorithm reduce the running time by 6%, 7%, and 6% on average, respectively, compared with those using the degrees of component polynomials found by Lee et al.'s algorithm.