Probability Calculation for Hamming Weight Distributions

This document presents the probability calculation for distinguishing Hamming weight distributions. Given two distinct Hamming weight scenarios (HW = 0 and HW = 1), we derive the probability that an observed value \(x\) originates from the HW = 0 distribution. Utilizing Bayes' theorem, we calculate the conditional probability and provide the final expression for \(P(HW = 0 \mid x)\). This result is fundamental for evaluating the likelihood of an observed value associated with a specific Hamming weight, particularly useful in statistical analysis of distributions in cryptographic contexts.