Impact of the Hamming weight of the difference of two random variables on the probability of its preservation after addition and subtraction

We study how does the Hamming weight of the difference between two values influence the probability of this difference preservation after modulo addition and subtraction. By the difference between two random variables we mean the operation XOR which is standard for cryptanalysis. We prove that if the most significant bit of the difference is equal to 0 (is equal to 1) then the probability of the difference preservation is equal to 2 −h (equal to 2 −( h −1) ), where h is the Hamming weight of the difference. The theoretical results are confirmed experimentally.