Byzantine-repeater-aware mediated multi-party QKD in untrusted networks: a corrected-frame finite-key security framework via entropy accumulation

Mediated multi-party quantum key distribution (M-MQKD) is studied for repeater-assisted quantum networks with an untrusted repeater layer and end users restricted to single-qubit local operations and one-way quantum reception. Building on the 1D+star mediated construction, the repeater path is treated as an untrusted measurement layer and the security analysis adopts a corrected-frame GHZ model. Under authenticated timing windows, trusted user-side measurements, a calibrated Check observable, and an explicit public-reconciliation transcript, we establish a corrected-frame finite-key theorem yielding a conference-key length bound from two experimentally accessible statistics: Share-mode disagreement and Check-mode parity-violation. The virtual phase-error variable for privacy amplification is related to the corrected-frame Check parity observable by a stabilizer-complementarity argument with quantum side information, while the entropy term in entropy accumulation is replaced by a predeclared affine lower bound on $1- h_{2} (x)$ . Error-correction leakage is accounted for through a designated-leader pairwise reconciliation transcript, including syndrome traffic, verification tags, and auxiliary public messages. The protocol is hardened by announcement-consistency traps, salted commit-then-open commitments over modes and parameter-estimation outcomes, randomized reveal order, and authenticated timing windows. These mechanisms detect or exclude dishonest classical disclosures before parameter estimation and treat trap failures as an abort/detection statistic rather than an unproved entropy penalty. The framework gives an auditable finite-key extraction rule as a function of observed error rates, finite-sampling radii, reconciliation leakage, EAT finite-size penalties, trap thresholds, and commitment length. Numerical evaluation maps rate-security trade-offs and gives parameter-selection guidance, supporting resilient infrastructure for secure multi-party quantum communication.