Homomorphic Encryption (HE) draws a significant attention as a privacy-preserving way for cloud computing because it allows computation on encrypted messages called ciphertexts. Among numerous HE schemes proposed, HE for Arithmetic of Approximate Numbers (HEAAN) is rapidly gaining popularity across a wide range of applications because it supports messages that can tolerate approximate computation with no limit on the number of arithmetic operations applicable to the corresponding ciphertexts. A critical shortcoming of HE is the high computation complexity of ciphertext arithmetic; especially, HE multiplication (HE Mul) is more than 10,000 times slower than the corresponding multiplication between unencrypted messages. This leads to a large body of HE acceleration studies, including ones exploiting FPGAs; however, those did not conduct a rigorous analysis of computational complexity and data access patterns of HE Mul. Moreover, the proposals mostly focused on designs with small parameter sizes, making it difficult to accurately estimate their performance in conducting a series of complex arithmetic operations. In this paper, we first describe how HE Mul of HEAAN is performed in a manner friendly to computer architects. Then we conduct a disciplined analysis on its computational and memory access characteristics, through which we (1) extract parallelism in the key functions composing HE Mul and (2) demonstrate how to effectively map the parallelism to the popular parallel processing platforms, multicore CPUs and GPUs, by applying a series of optimization techniques such as transposing matrices and pinning data to threads. This leads to the performance improvement of HE Mul on a CPU and a GPU by 42.9x and 134.1x, respectively, over the single-thread reference HEAAN running on a CPU. The conducted analysis and optimization would set a new foundation for future HE acceleration research.