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 O(q^2r/N), showing that our security bound is tight up to a factor of (log r)^3.
Proceedings Title: LNCS: Advances in Cryptology ASIACRYPT 2017
Conference Dates: December 3-7, 2017
Conference Location: Hong Kong, -1
Conference Title: The 23rd Annual International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2017
Pub Type: Conferences
Iterated random function, random function, pseudorandom function, bitcoin, password hashing, Patarin, H-coefficient technique, provable security