The Iterated Random Function Problem.

At CRYPTO 2015, Minaud and Seurin introduced and studied the iterated random permutation problem, which is to distinguish the r-th iterate of a random permutation from a random permutation. In this paper, we study the closely related iterated random function problem, and prove the first almost-tight bound in the adaptive setting. More specifically, we prove that the advantage to distinguish the r-th iterate of a random function from a random function using q queries is bounded by (O(q^2r(log r)^3/N)), where N is the size of the domain. In previous work, the best known bound was (O(q^2r^2/N)), obtained as a direct result of interpreting the iterated random function problem as a special case of CBC-MAC based on a random function. For the iterated random function problem, the best known attack has an advantage of (varOmega (q^2r/N)), showing that our security bound is tight up to a factor of ((log r)^3).