**Computational Indistinguishability between Quantum States and Its Cryptographic Application**

*Akinori Kawachi and Takeshi Koshiba and Harumichi Nishimura and Tomoyuki Yamakami*

**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 QSCDff 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 three cryptographic properties: (i) QSCDff has the trapdoor property; (ii) the average-case hardness of QSCDff coincides with its worst-case hardness; and (iii) QSCDff is computationally at least as hard in the worst case as the graph automorphism problem. These cryptographic properties enable us to construct a quantum public-key cryptosystem, 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 relying on the cryptographic properties of QSCDcyc.

**Category / Keywords: **foundations / quantum cryptography, computational indistinguishability, trapdoor property, worst-case/average-case equivalence, graph automorphism problem, quantum public-key cryptosystem

**Date: **received 14 Apr 2006, last revised 17 May 2006

**Contact author: **kawachi at is titech ac jp

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **References are added.

**Version: **20060517:094438 (All versions of this report)

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