On Quantum Distinguishers for Type-3 Generalized Feistel Network Based on Separability

In this work, we derive a method for constructing quantum distinguishers for GFNs (Generalized Feistel-like schemes with invertible inner functions and XORs), where for simplicity 4 branches are considered. The construction technique is demonstrated on Type-₃ GFN, where some other cyclically inequivalent GFNs are considered as examples. Introducing the property of separability, we observe that finding a suitable partition of input blocks implies that some branches can be represented as a sum of functions with almost disjoint variables, which simplifies the application of Simon’s algorithm. However, higher number of rounds in most of the cases have branches which do not satisfy the previous property, and in order to derive a quantum distinguisher for these branches, we employ Simon’s and Grover’s algorithm in combination with a suitable system of equations given in terms of input blocks and inner functions involved in the round function. As a result, we are able to construct a 5-round quantum distinguisher for Type-₃ GFNs using only a quantum encryption oracle with query complexity 2^N/4 • Ö(N/4), where N size of the input block.