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**11 - 13**of**13**### Computational Distinguishability between Quantum States: Random Coset States vs. Maximally Mixed States over the Symmetric Groups

, 2004

"... We introduce a new underlying problem for computational cryptographic schemes secure against quantum adversaries. The problem is a distinction problem between quantum states which is a natural generalization of distinction problems between probability distributions, which are commonly used in comput ..."

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We introduce a new underlying problem for computational cryptographic schemes secure against quantum adversaries. The problem is a distinction problem between quantum states which is a natural generalization of distinction problems between probability distributions, which are commonly used in computational cryptography. Specifically speaking, our problem QSCDff is defined as a quantum state computational distinguishability problem between random coset states with a hidden permutation and a maximally mixed state (uniform distribution) over the symmetric group. A similar problem to ours appears in the context of the hidden subgroup problem on the symmetric group in the research of quantum computation and is regarded as a hard problem. In this paper, we show that (i) QSCDff has the trapdoor property; (ii) the average-case complexity of QSCDff completely coincides with its worst-case complexity; (iii) the computational complexity of QSCDff is lower-bounded by the worst-case hardness of the graph automorphism problem. These properties enable us to construct cryptographic systems. Actually, we show a cryptographic application based on the hardness of QSCDff. Keywords: 1

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, 2001

"... We investigate definitions of and protocols for multi-party quantum computing in the scenario where the secret data are quantum systems. We work in the quantum information-theoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of veri ..."

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We investigate definitions of and protocols for multi-party quantum computing in the scenario where the secret data are quantum systems. We work in the quantum information-theoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to establish that any multi-party quantum computation can be securely performed as long as the number of dishonest players is less than n/6. This thesis is based on joint work with Claude Crépeau and Daniel Gottesman.

### Progress in Quantum Computational Cryptography

"... Abstract: Shor’s algorithms for the integer factorization and the discrete logarithm problems can be regarded as a negative effect of the quantum mechanism on publickey cryptography. From the computational point of view, his algorithms illustrate that quantum computation could be more powerful. It i ..."

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Abstract: Shor’s algorithms for the integer factorization and the discrete logarithm problems can be regarded as a negative effect of the quantum mechanism on publickey cryptography. From the computational point of view, his algorithms illustrate that quantum computation could be more powerful. It is natural to consider that the power of quantum computation could be exploited to withstand even quantum adversaries. Over the last decade, quantum cryptography has been discussed and developed even from the computational complexity-theoretic point of view. In this paper, we will survey what has been studied in quantum computational cryptography. Key Words: computational cryptography, quantum computing, quantum cryptography Category: E.3, F.1.1