Complexity Analysis of Encoding in CKKS-Fully Homomorphic Encryption Algorithm

In this paper, we analyze the complexity of encoding in CKKS algorithm. The encoding process in CKKS requires some steps such as Inversion of PI $\pi^{-1}$, Scaling, Discretization, and Inverse of Sigma $\sigma^{-1}$, and Rounding. However, the decoding requires Sigma function to convert the encoded message back to the plaintext. In this paper, we evaluate each function and determine the complexities. Experimental results show that the complexity of the encoding is ${O}(n^{3})$ and the complexity of the decoding is ${O}(n^{2})$. The work is useful for optimization and speeding up the CKKS computation on custom computing platforms such as GPUs and FPGAs.