Asymptotic results for the number of Wagner's solutions to a generalised birthday problem

Journal article, 2015

We study two functionals of a random matrix A with independent elements uniformly distributed over the cyclic group of integers {0, 1, . . ., M-1} modulo M. One of them, V0(A) with mean μ, gives the total number of solutions for a generalised birthday problem, and the other, W(A) with mean λ, gives the number of solutions detected by Wagner's tree based algorithm.We establish two limit theorems. Theorem 2.1 describes an asymptotical behaviour of the ratio λ/μ as M→∞. Theorem 2.2 gives bounds for the total variation distance between Poisson distribution and distributions of V0 and W.