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arXiv:quant-ph/0405111 · New Journal of Physics

Security trade-offs in ancilla-free quantum bit commitment in the presence of superselection rules

David Peter Divincenzo, John A. Smolin, Barbara M. Terhal

Published Jul 13, 2004• 5 citations
arXivHTMLPDF

Abstract

Security trade-offs have previously been established for one-way bit commitment. We study this trade-off in two superselection settings. We show that for an [lsquo]abelian[rsquo] superselection rule (exemplified by particle conservation), the standard trade-off between sealing and binding properties still holds. For the non-abelian case (exemplified by angular momentum conservation), the security trade-off can be more subtle, which we illustrate by showing that if the bit commitment is forced to be ancilla-free, an asymptotically secure quantum bit commitment is possible.

DOI
10.1088/1367-2630/6/1/080

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Security trade-offs have previously been established for one-way bit commitment. We study this trade-off in two superselection settings. We show that for an [lsquo]abelian[rsquo] superselection rule (exemplified by particle conservation), the standard trade-off between sealing and binding properties still holds. For the non-abelian case (exemplified by angular momentum conservation), the security trade-off can be more subtle, which we illustrate by showing that if the bit commitment is forced to be ancilla-free, an asymptotically secure quantum bit commitment is possible.
We show that superselection rules do not enhance the information-theoretic security of quantum cryptographic protocols. Our analysis employs two quite different methods. The first method uses the concept of a reference system---in a world subject to a superselection rule, unrestricted operations can be simulated by parties who share access to a reference system with suitable properties. By this method, we prove that if an nnn-party protocol is secure in a world subject to a superselection rule, then the security is maintained even if the superselection rule is relaxed. However, the proof applies only to a limited class of superselection rules, those in which the superselection sectors are labeled by unitary irreducible representations of a compact symmetry group. The second method uses the concept of the format of a message sent between parties---by verifying the format, the recipient of a message can check whether the message could have been sent by a party who performed charge-conserving operations. By this method, we prove that protocols subject to general superselection rules (including those pertaining to non-Abelian anyons in two dimensions) are no more secure than protocols in the unrestricted world. However, the proof applies only to two-party protocols. Our results show in particular that, if no assumptions are made about the computational power of the cheater, then secure quantum bit commitment and strong quantum coin flipping with arbitrarily small bias are impossible in a world subject to superselection rules.
Currently, it is believed in the literature that unconditionally secure bit commitment is impossible in non-relativistic quantum cryptography while only a weaker notion of bit commitment with finite commitment time is achievable in relativistic quantum setting. Moreover, relativistic classical bit commitment protocols allow arbitrary long commitment time but such protocols are not practically feasible; either because of multiple rounds of communication that result in exponential increase in communication complexity or due to asymptotic nature of security argument. The impossibility of practically feasible standard bit commitment leaves an obvious skepticism on the completeness of Hilbert space quantum cryptography and its claims of unconditional security. Contrary to the previously proposed results, we demonstrate here that an information-theoretic standard bit commitment scheme can be devised by using rules of purely non-relativistic quantum mechanics only; neither additional resources from theory of relativity nor multiple rounds of communications are required. The proposed bit commitment scheme can be applied efficiently with existing quantum technologies without long term quantum memory; quantum entanglement is required only for time t+δ where t is the communication time between the committer and receiver while δ << t is the processing time at laboratory.
The relationship between the quantum bit commitment (QBC) and quantum seal (QS) is studied. It is elaborated that QBC and QS are not equivalent, but QS protocols satisfying a stronger unconditional security requirement can lead to an unconditionally secure QBC. In this sense, QS is strictly stronger than QBC in security requirements. Based on an earlier proposal on sealing a single bit, a feasible QBC protocol is also put forward, which is secure in practice even if the cheater has a strong quantum computational power.
Bit commitment is one of the cornerstones of both classical and quantum cryptography. In the classical world, security analysis is straightforward because both hiding and binding each admit a single natural interpretation. In the quantum world, however, security notions can be defined in different ways that are not necessarily equivalent. Over three decades, several works have introduced numerous binding notions whose precise statements and mu- tual implications remain scattered across the literature. This thesis clarifies that ambiguity. We study quantum binding notions within the framework of canonical purified quantum bit commitment (QBC) schemes, a class that captures both the theoretical generality and the practical mechanisms of recent constructions. In the statistical setting, we examine six binding notions: strong-binding, honest-binding, sum-binding, collapse-binding, extractor- binding, and arbitrary-binding. We prove that all these definitions except strong-binding are equivalent for the class of canonical purified schemes. In the computational setting, we establish that honest-binding, sum-binding, and collapse-binding are equivalent for the same class of schemes. Moreover, we introduce a quantitative framework based on the work of Ref. [26] to quantify the tradeoff between hiding and binding properties, and we use it to provide a geometric proof of the impossibility theorem of Mayers [21] and Lo & Chau [20], for canonical purified schemes. We next revisit the quantum notion of hiding, which is far less ambiguous than binding. Finally, we survey some technical tools that can be used to construct secure QBC schemes.
Bit commitment is an essential primitive in quantum cryptography. It is the basic building block of many cryptographic protocols in which it provides security over distrustful parties. This paper proposes a secured quantum two-bit commitment protocol for any two classical bits. The protocol contains two parties - a committer and a receiver - who share both quantum and classical communication channels. The main phases of this protocol are commitment and revealing phases where the concealing and binding conditions are proved in order to accept the commitment message. The proposed framework utilizes different security layers to hide specific quantum states in a superposition alongside trivial one qubit operations. The secured unitary transformations are applied by the two parties to meet the concealing and binding conditions. To verify the success of the proposed protocol, the measured output of the two parties are compared. The protocol operates under the assumption that the two parties might cheat and the possible existence of an eavesdropper.