  The Feistel scheme is an important structure in the block ciphers. The
security of the Feistel scheme is related to distinguishability with a random
permutation. In this paper, efficient quantum algorithms for distinguishing
classical 3,4-round and unbalanced Feistel scheme with contracting functions
from random permutation are proposed. Our algorithms realize an exponential
speed-up over classical algorithms for these problems. Furthermore, the method
presented in this paper can also be used to consider unbalanced Feistel schemes
with expanding functions.
