[Paper Review] Quantum oblivious transfer and bit commitment protocols based on two non-orthogonal states coding

Li Yang|arXiv (Cornell University)|Jun 25, 2013

Quantum Information and Cryptography2 references3 citations

TL;DR

This paper proposes quantum oblivious transfer (R-OT and OT₁²) and bit commitment protocols based on non-orthogonal quantum states, using two non-orthogonal states with a fixed angle (π/4) to enable secure information transfer. Although the protocols are unconditionally secure against individual attacks, they are not secure against entanglement-based attacks as per the no-go theorem, but are physically secure due to practical limitations in quantum memory and matrix size.

ABSTRACT

Oblivious transfer protocols (R-OT and OT$_{1}^{2}$) are presented based on non-orthogonal states transmission, and the bit commitment protocols on the top of OT$_{1}^{2}$ are constructed. Although these OT protocols are all unconditional secure, the bit commitment protocols based on OT protocols are not secure against attack similar to that presented by no-go theorem.

Motivation & Objective

• To design quantum oblivious transfer protocols based on non-orthogonal quantum states for unconditional security in key distribution.
• To construct bit commitment protocols using the OT₁² protocol as a foundation, aiming for practical security despite theoretical limitations.
• To analyze the physical feasibility of quantum bit commitment under realistic constraints such as finite quantum memory and matrix size.
• To demonstrate that protocols can be physically secure even if not unconditionally secure under the no-go theorem.
• To explore the trade-off between theoretical security and practical implementability in quantum cryptography.

Proposed method

• Uses two non-orthogonal quantum states |Ψ₀⟩ and |Ψ₁⟩ with ⟨Ψ₀|Ψ₁⟩ = cos(π/4) = √2/2 to encode bits, enabling distinguishability with optimal probability 1 - √2/2.
• Employs a measurement basis choice strategy: Bob randomly selects between B₀ = {|Ψ₀⟩, |Ψ₀⟩⊥} and B₁ = {|Ψ₁⟩, |Ψ₁⟩⊥}, accepting only when measuring |Ψₓ⟩⊥ to infer rₓ⊕1.
• Constructs OT₁² via Crépeau’s framework using R-OT, where Bob selects k conclusive bits from his results and commits to a random basis choice (X,Y) to enable Alice to encrypt messages.
• Applies XOR-based encryption: c₀ = b₀ ⊕ s₀, c₁ = b₁ ⊕ s₁, where s₀ and s₁ are XORs of selected bit strings, enabling Bob to recover only one message.
• Proposes a bit commitment protocol using Bell states |Φ⁻⟩, where Alice modifies qubits via Ry(−π/2) rotation based on her bit string, and Bob verifies via projective measurement.
• Introduces a simplified protocol using single-qubit states with angle encoding, where Alice rotates states by π/4 based on her bits, and Bob verifies using random basis measurements.

Experimental results

Research questions

• RQ1 Can oblivious transfer be securely implemented using only two non-orthogonal quantum states, without requiring entanglement?
• RQ2 To what extent can bit commitment protocols be made physically secure despite the no-go theorem forbidding unconditional security?
• RQ3 What is the impact of practical limitations—such as finite quantum memory and matrix size—on the feasibility of entanglement-based attacks?
• RQ4 How does the choice of measurement basis and state encoding affect the success probability and security of quantum oblivious transfer?
• RQ5 Can a bit commitment protocol be constructed that is secure in practice even if not unconditionally secure in theory?

Key findings

• The R-OT protocol achieves a conclusive bit success probability of 1/4 per qubit using a simple measurement basis strategy, with honest Bob obtaining approximately n/4 conclusive bits.
• The OT₁² protocol is unconditionally secure when the probability of honest Bob obtaining more than k conclusive bits (p₁) approaches 1 exponentially, and malicious Bob obtaining 2k or more bits (p₂) approaches 0 exponentially, as n → ∞.
• The bit commitment protocol based on Bell states |Φ⁻⟩ is not unconditionally secure due to entanglement-based attacks, but becomes physically secure due to the infeasibility of storing or manipulating large-scale unitary matrices (e.g., 2²⁰⁰ × 2²⁰⁰ for 2k=200).
• The simplified protocol using single-qubit states with angle encoding avoids quantum memory requirements if Bob measures during the commit phase, but remains vulnerable to qubit omission attacks unless detectors are perfect.
• The physical security of the protocols stems from the astronomical size of required unitary transformations (e.g., 2²⁰⁰ matrix entries) and limited quantum memory lifetime, making attacks infeasible in practice.
• Even though the protocols do not satisfy the full conditions of unconditional security under the no-go theorem, they are physically secure due to practical constraints on quantum operations and storage.