[Paper Review] Large N Quantum Cryptography

Adrian Kent|ArXiv.org|Dec 8, 2002

Quantum Computing Algorithms and Architecture3 references3 citations

TL;DR

This paper proposes novel quantum protocols for bit string commitment and random number generation that achieve high security despite the impossibility of perfectly secure quantum bit commitment. By leveraging large N quantum states and quantum coding techniques, the protocols ensure that cheating by either party is severely limited, even when individual bit commitment is classically and quantumly insecure.

ABSTRACT

In quantum cryptography, the level of security attainable by a protocol which implements a particular task $N$ times bears no simple relation to the level of security attainable by a protocol implementing the task once. Useful partial security, and even near-perfect security in an appropriate sense, can be obtained for $N$ copies of a task which itself cannot be securely implemented. We illustrate this with protocols for quantum bit string commitment and quantum random number generation between mistrustful parties.

Motivation & Objective

• To demonstrate that quantum protocols can achieve useful security for committing multiple bits (N > 1) even when perfect security for a single bit is impossible.
• To address the gap between classical and quantum cryptographic equivalence, showing that quantum generalizations of bit commitment are not bound by the same no-go theorems.
• To design protocols that provide meaningful security against both cheating parties using quantum state encoding and measurement-based verification.
• To explore the feasibility of secure quantum random number generation between mistrustful parties using entangled states and a cut-and-choose strategy.

Proposed method

• Uses a set of non-orthogonal quantum states |ψ₀⟩ = |0⟩ and |ψ₁⟩ = sinθ|0⟩ + cosθ|1⟩ to encode bit strings, with θ as a security parameter.
• Employs a protocol where Alice sends N qubits in states corresponding to her committed bit string, and Bob verifies by measuring projection onto the claimed state.
• Applies quantum coding theory to construct large sets of low-overlap quantum states in Hⁿ, enabling encoding of O(exp(Cn)) classical bits into n qubits.
• Uses Holevo’s theorem to bound the accessible information B can extract before unveiling, ensuring partial security against B.
• Implements a cut-and-choose mechanism in the random number generation protocol, where B tests M−1 batches of Bell singlets before accepting the last batch for bit generation.
• Leverages the fact that any cheating strategy by B is limited to learning only ≈log M bits of the N-bit string, which is negligible when M ≪ 2^N.

Experimental results

Research questions

• RQ1 Can quantum protocols achieve meaningful security for committing multiple bits when perfect security for a single bit is impossible?
• RQ2 To what extent can quantum coding of classical information in high-dimensional Hilbert spaces enhance security in bit string commitment?
• RQ3 Can a protocol for quantum random number generation between mistrustful parties be secure against both individual cheating strategies and collective attacks?
• RQ4 How does the security of quantum protocols scale with N, the number of qubits or bits involved, in the absence of perfect security for N=1?
• RQ5 What is the trade-off between the number of bits committed and the accessible information an eavesdropper can gain, given quantum limits like Holevo’s bound?

Key findings

• The protocol achieves security against Alice such that her probability of successfully cheating on any one of r possible strings is bounded by 1 + (r−1)ε, where ε = sinθ, and this bound can be made arbitrarily small by choosing small θ.
• Security against Bob is ensured by Holevo’s theorem, which limits the accessible information about the N-bit string to at most log n bits, where n is the number of qubits.
• For large N, the protocol can encode O(exp(Cn)) classical bits into n qubits using sets of vectors with pairwise inner products less than sinθ, enabling scalable commitment of long strings.
• In the random number generation protocol, Bob’s ability to bias the outcome is limited to learning only ≈log M bits of information, which is negligible when M ≪ 2^N.
• The protocol remains secure even when individual bit commitment is impossible, showing that multi-copy protocols can achieve security not possible in the single-copy case.
• Explicit constructions of low-overlap quantum states exist for large N, with f(n) = O(exp(Cn)) codewords in Hⁿ, supporting the feasibility of the scheme.