Key reuse weakens phase-keyed Fourier-curve modulation security
Key-Reuse Vulnerability of Phase-Keyed Fourier-Curve Modulation: Relation Leakage and Key-Refresh Cost on Coded Links
Cryptography and Security
Summary
This paper shows that reusing keys in a certain type of secure signal modulation called phase-keyed Fourier-curve modulation can leak information about the key itself. The authors found that mathematical relationships among the signal’s harmonics expose parts of the key, allowing an attacker to recover it more easily than previously thought. Even with error-correcting codes and noise added, the attack can succeed if keys are used too often without refreshing. The paper quantifies the cost of refreshing keys frequently enough to maintain security, which turns out to be higher than a traditional one-time pad.
What this means in practice
- •For communications engineers: Design secure modulation schemes avoiding key reuse vulnerabilities to prevent key leakage in coded wireless links.
- •For cryptographic protocol developers: Calculate minimal key-refresh rates and entropy costs to maintain key secrecy in protocols using Fourier-based keyed modulation.
Authors
Bin Han, Muxia Sun, H. Vincent Poor, Hans D. Schotten
Abstract
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 $95\%$ 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.