## Abstract

We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is "secure" against any polynomial-time quantum adversary. Our problem, QSCD _{ff}, is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonly-used distinction problem between two probability distributions in computational cryptography. As our major contribution, we show that QSCD _{ff} has three properties of cryptographic interest: (i) QSCD _{ff} has a trapdoor; (ii) the average-case hardness of QSCD _{ff} coincides with its worst-case hardness; and (iii) QSCD _{ff} is computationally at least as hard as the graph automorphism problem in the worst case. These cryptographic properties enable us to construct a quantum public-key cryptosys-tem which is likely to withstand any chosen plaintext attack of a polynomial-time quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies on similar cryptographic properties of QSCDcyc.

Original language | English |
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Pages (from-to) | 528-555 |

Number of pages | 28 |

Journal | Journal of Cryptology |

Volume | 25 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Jul |

Externally published | Yes |

## Keywords

- Computational indistinguishability
- Graph automorphism problem
- Quantum cryptography
- Quantum publickey cryptosystem
- Trapdoor
- Worst-case/average-case equivalence

## ASJC Scopus subject areas

- Software
- Computer Science Applications
- Applied Mathematics