A Critique on Average-Case Noise Analysis in RLWE-Based Homomorphic Encryption

Homomorphic encryption schemes based on the Ring-Learning-with-Errors problem require accurate ciphertext noise analysis to ensure correctness and security. However, ring multiplications during homomorphic computations make the noise in the result ciphertexts difficult to characterize. Existing average-case noise analyses derive a bound on the noise by either assuming it follows a Gaussian distribution, or giving empirical formulae, with strong independence assumption and the Central Limit Theorem extensively applied. In this work, we question the validity of these methods, by showing that the noise exhibits a heavy-tailed distribution via exact calculation of its variance and kurtosis, for both independent and dependent noises. The heavy-tailedness suggests the failing probability of bounds derived from these methods may not be negligible, and we experimentally demonstrate several cases where the noise growth is underestimated.