  In this paper, we study applications of Bernstein-Vazirani algorithm and
present several new methods to attack block ciphers. Specifically, we first
present a quantum algorithm for finding the linear structures of a function.
Based on it, we propose new quantum distinguishers for the 3-round Feistel
scheme and a new quantum algorithm to recover partial key of the Even-Mansour
construction. Afterwards, by observing that the linear structures of a
encryption function are actually high probability differentials of it, we apply
our algorithm to differential analysis and impossible differential
cryptanalysis respectively. We also propose a new kind of differential
cryptanalysis, called quantum small probability differential cryptanalysis,
based on the fact that the linear structures found by our algorithm are also
the linear structure of each component function. To our knowledge, no similar
method was proposed before. The efficiency and success probability of all
attacks are analyzed rigorously. Since our algorithm treats the encryption
function as a whole, it avoid the disadvantage of traditional differential
cryptanalysis that it is difficult to extending the differential path.
