When Simple Permutations Mix Poorly - Limited Independence does not Imply Pseudorandomness

Over the past two decades, several works have used (almost) $k$-wise independence as a proxy for pseudorandomness in block ciphers, since it guarantees resistance against broad classes of statistical attacks. For example, even the case $k = 2$ already implies security against differential and linear cryptanalysis.

Hoory, Magen, Myers, and Rackoff (ICALP ’04; TCS ’05) formulated an appealing conjecture: if the sequential composition of $T$ independent local randomized permutations is (close to) four-wise independent, then it should also be a pseudorandom permutation. Here, "local" means that each output bit depends on only a constant number of input bits. This conjecture offers a potential strong justification for analyses of block ciphers that establish (almost) $k$-wise independence of this type of constructions.

In this work, we disprove the conjecture in full generality by presenting an explicit local randomized permutation whose sequential composition is four-wise independent, but not a pseudorandom permutation. Our counterexample in fact extends to $k$-wise independence for any constant $k$.