Computational indistinguishability between quantum states and its cryptographic application

We introduce a problem of distinguishing between two quantum states as a new underlying problem to build a computational cryptographic scheme that is "secure" against quantum adversary. Our problem is a natural generalization of the distinguishability problem between two probability distributions, which are commonly used in computational cryptography. More precisely, our problem QSCD_ff is the computational distinguishability problem between two types of random coset states with a hidden permutation over the symmetric group. We show that (i) QSCD_ff has the trapdoor property; (ii) the average-case hardness of QSCD_ff coincides with its worst-case hardness; and (iii) QSCD_ff is at least as hard in the worst case as the graph automorphism problem. Moreover, we show that QSCD_ff cannot be efficiently solved by any quantum algorithm that naturally extends Shor's factorization algorithm. These cryptographic properties of QSCD_ff enable us to construct a public-key cryptosystem, which is likely to withstand any attack of a polynomial-time quantum adversary.