Key-Reuse Vulnerability of Phase-Keyed Fourier-Curve Modulation: Relation Leakage and Key-Refresh Cost on Coded Links

The security of keyed modulation is often argued from the key-space size and the error rate of a key-less receiver. This evidence fails when the key is reused and the waveform is harmonically coupled. For a phase-keyed Fourier-curve constellation, whose $k$ tones share one data parameter, integer relations among the harmonic indices yield data-cancelling mixed moments of the received tones that expose key characters. A modular relation lattice characterizes the exposed characters; for consecutive harmonics, third-order moments recover the relative phases and a fourth-order moment completes the key up to cyclic relabeling whenever its coefficient is nonzero, as in all evaluated settings. A non-data-aided relation-moment estimator turns this leakage into an attack that never enumerates the key space. On a regular $(3,6)$ LDPC-coded link, one key per 168-symbol codeword leaves the eavesdropper a block error rate below $0.04$ at the middle noise level, and the attack meets a predeclared $0.1$ compromise criterion in eleven of twelve operating points. Tangent artificial noise and a harmonic set without relations below order four raise her measured error rate at intermediate reuse lengths but do not remove the one-codeword vulnerability. For a grid of $2^{128}$ protocol keys at the middle noise level, equal-length refresh schedules that keep a $95\mathrm{\%}$ lower confidence bound of her block error rate above $0.9$ consume at least $1.52$ fresh key bits per information bit, $1.52$ times the entropy rate of a one-time pad on the data.