Secure and Efficient Two-party Quantum Scalar Product Protocol With Application to Privacy-preserving Matrix Multiplication

TL;DR

The paper tackles the inefficiency of existing quantum two-party scalar product protocols by introducing a Fourier Entangled state-based S2QSPP that achieves unconditional security under malicious models. It combines Entanglement Bondage with modular and quantum Fourier techniques to deliver polynomial-time scalar-product calculations and validates feasibility via IBM Qiskit simulations. Building on this core protocol, the authors present a privacy-preserving two-party matrix multiplication (P2MM) scheme by reducing matrix multiplication to multiple S2QSPP runs, preserving input privacy. The work demonstrates both theoretical security guarantees (via leakage bounds and zero-knowledge tests) and practical performance advantages over prior approaches, with potential impact on secure quantum computation and privacy-preserving linear algebra.

Abstract

Secure two-party scalar product (S2SP) is a promising research area within secure multiparty computation (SMC), which can solve a range of SMC problems, such as intrusion detection, data analysis, and geometric computations. However, existing quantum S2SP protocols are not efficient enough, and the complexity is usually close to exponential level. In this paper, a novel secure two-party quantum scalar product (S2QSP) protocol based on Fourier entangled states is proposed to achieve higher efficiency. Firstly, the definition of unconditional security under malicious models is given. And then, an honesty verification method called Entanglement Bondage is proposed, which is used in conjunction with the modular summation gate to resist malicious attacks. The property of Fourier entangled states is used to calculate the scalar product with polynomial complexity. The unconditional security of our protocol is proved, which guarantees the privacy of all parties. In addition, we design a privacy-preserving quantum matrix multiplication protocol based on S2QSP protocol. By transforming matrix multiplication into a series of scalar product processes, the product of two private matrices is calculated without revealing any privacy. Finally, we show our protocol's feasibility in IBM Qiskit simulator.

Figures (4)

• Figure 1: The quantum circuit of Operation stage.
• Figure 2: The circuits of quantum gates. a), b), c), d), e), f) are circuits of $\mathcal{QFT}$, $\mathcal{ROT}$, $\mathcal{SUM}$, $\mathcal{MUL}$, $\mathcal{BSUM}$, $\mathcal{XOR}$, respectively.
• Figure 3: The total circuit of our protocol.
• Figure 4: a), b), c), d) are the results $M_i$ for $i=1,2,3,4$ of the first experiment, respectively. Similarly, e), f), g), h) are the results of the second experiment.

Theorems & Definitions (15)

• Definition 1: Fourier Entangled (FE) State
• Definition 2: Malicious adversary model
• Definition 3: Leakage Degree
• Definition 4: Negligibility
• Definition 5: Unconditional Security under the Malicious Model
• Definition 6: Secure Two-party Scalar Product (S2SP)
• Theorem 1
• Lemma 1: Security of Phase-information
• Lemma 2: Security of Bit-information
• Lemma 3: Security under Forgery Attack