Deconvolution of a discrete uniform distribution

Let ξξ be a discrete random variable (r.v.) with uniform distribution on the support set {0,1,…,N}{0,1,…,N}. We study the problem of construction of non-degenerate independent r.v.’s ξ1ξ1 and ξ2ξ2 such that ξ=ξ1+ξ2ξ=ξ1+ξ2, if these r.v.’s exist. We describe a general form for the solutions to this problem, offer some analytic constructions and develop algorithms for computing the distributions of ξ1ξ1 and ξ2ξ2.