  The next bit test was introduced by Blum and Micali and proved by Yao to be a
universal test for cryptographic pseudorandom generators. On the other hand, no
universal test for the cryptographic one-wayness of functions (or permutations)
is known, though the existence of cryptographic pseudorandom generators is
equivalent to that of cryptographic one-way functions. In the quantum
computation model, Kashefi, Nishimura and Vedral gave a sufficient condition of
(cryptographic) quantum one-way permutations and conjectured that the condition
would be necessary. In this paper, we affirmatively settle their conjecture and
complete a necessary and sufficient for quantum one-way permutations. The
necessary and sufficient condition can be regarded as a universal test for
quantum one-way permutations, since the condition is described as a collection
of stepwise tests similar to the next bit test for pseudorandom generators.
