Towards Quantum One-Time Memories from Stateless Hardware

Anne Broadbent1, Sevag Gharibian2,3, and Hong-Sheng Zhou3

1Department of Mathematics and Statistics, University of Ottawa, Ontario, Canada
2Department of Computer Science, Paderborn University, Germany
3Department of Computer Science, Virginia Commonwealth University, Virginia, USA

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Abstract

A central tenet of theoretical cryptography is the study of the minimal assumptions required to implement a given cryptographic primitive. One such primitive is the one-time memory (OTM), introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008], which is a classical functionality modeled after a non-interactive 1-out-of-2 oblivious transfer, and which is complete for one-time classical and quantum programs. It is known that secure OTMs do not exist in the standard model in both the classical and quantum settings. Here, we propose a scheme for using quantum information, together with the assumption of stateless ($i.e.$, reusable) hardware tokens, to build statistically secure OTMs. Via the semidefinite programming-based quantum games framework of Gutoski and Watrous [STOC 2007], we prove security for a malicious receiver making at most 0.114$n$ adaptive queries to the token (for $n$ the key size), in the quantum universal composability framework, but leave open the question of security against a polynomial amount of queries. Compared to alternative schemes derived from the literature on quantum money, our scheme is technologically simple since it is of the "prepare-and-measure" type. We also give two impossibility results showing certain assumptions in our scheme cannot be relaxed.

In theoretical cryptography, many desirable primitives are provably impossible to implement securely. This raises a central question: What are the minimal assumptions one needs to add to the computational model in order to securely implement such primitives? In this work, the primitive we study is known as a "one-time memory (OTM)", introduced by Goldwasser, Kalai, and Rothblum [CRYPTO 2008]. Briefly, an OTM hides two secret bits, so that any user of the OTM can extract precisely one of the bits; the other bit is then permanently lost. Unfortunately, secure OTMs provably do not exist in both the classical and quantum computational models. Thus, here propose the addition of a minimal assumption to the quantum computational model – that of "state-less" (i.e., reusable) hardware tokens. With this assumption in hand, we give an OTM construction which we rigorously prove is (information theoretically) secure against a malicious user making at most 0.114n (adaptive) queries to the hardware token (for n the secret key size). This security is shown in the "quantum universal composability framework", meaning our protocol composes "nicely" and securely with other primitives in this framewoek. Compared to alternative schemes, our protocol is technologically simple. We leave open the question of security against a polynomial number of queries.

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