Quantitative Information Leakage and Secret Reconstruction Analysis of Photon Number Splitting Attacks in (3,5)-Threshold Quantum-Assisted Secret Sharing

Threshold secret sharing schemes form the cryptographic backbone of distributed key management, multiparty computation, and decentralized custody systems. While classical (k, n)- threshold schemes—most notably Shamir's Secret Sharing (SSS)—provide unconditional information-theoretic security at rest, their practical deployment depends entirely on the security of the channels used to distribute shares from the dealer to participants. Quantum-Assisted Secret Sharing (QASS) addresses classical transport vulnerabilities to quantum-capable adversaries by using Quantum Key Distribution (QKD) links to establish unconditionally secure one-time pad (OTP) channels for share transmission. However, practical QKD implementations rely on attenuated laser sources emitting Weak Coherent Pulses (WCP) rather than ideal single-photon sources. This hardware imperfection exposes the share distribution phase to Photon-Number-Splitting (PNS) attacks. In this paper, we present a rigorous mathematical model and quantitative information leakage analysis of PNS attacks against a \( (3,5) \) -threshold QASS architecture. We derive the exact bit-leakage rate as a function of the mean photon number \( μ \) and fibre channel length L, evaluate the structural breakdown of the polynomial interpolation threshold under multi-channel eavesdropping, and quantify the exponential collapse in the adversary's secret reconstruction search space. Finally, we prove that integrating a three-intensity vacuum +weak decoy-state protocol restores full information-theoretic security of the (3,5) threshold across metropolitan and long-haul optical networks.