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.

本文件阐述了用于区分汉明重量（Hamming weight）分布的概率计算方法。给定两种不同的汉明重量场景（即汉明重量为0与汉明重量为1），我们推导得出观测值$x$源自汉明重量为0的分布的概率。借助贝叶斯定理，我们对该条件概率进行计算，并给出$P( ext{HW}=0 mid x)$的最终表达式。该结果对于评估与特定汉明重量相关的观测值的似然性具有基础性意义，尤其适用于密码学场景下的分布统计分析。