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How could quantum cryptography help us achieve what are not achievable inclassical cryptography? In this work we consider the following problem, whichwe call succinct RSPV for classical functions (SRC). Suppose $f$ is a functiondescribed by a polynomial time classical Turing machine, which is public; theclient would like to sample a random $x$ as the function input and use aprotocol to send $f(x)$ to the server. What's more, (1) when the server ismalicious, what it knows in the passing space should be no more than $f(x)$;(2) the communication should be succinct (that is, independent to the runningtime of evaluating $f$). Solving this problem in classical cryptography seemsto require strong cryptographic assumptions.  We show that, perhaps surprisingly, it's possible to solve this problem withquantum techniques under much weaker assumptions. By allowing for quantumcommunication and computations, we give a protocol for this problem assumingonly collapsing hash functions [Unr16]. Our work conveys an interesting messagethat quantum cryptography could outperform classical cryptography in a new typeof problems, that is, to reduce communications in meaningful primitives withoutusing heavy classical cryptographic assumptions.

A Quantum Approach for Reducing Communications in Classical Cryptographic Primitives