Binary Classifier Wire-Resistance Attack on KLJN: Bayes-Optimal Limit at a Narrow Resistor Gap

Abstract

Machine learning is shown not to increase Eve's successful bit-guessing probability in the binary-classifier-based wire-resistance attack on the Kirchhoff Law-Johnson Noise (KLJN) key exchange when the resistor gap is narrow. Using time-domain simulations of a non-ideal KLJN loop with $R_L = 9$ k$Ω$, $R_H = 10$ k$Ω$ and wire resistances from 0 to 3200 $Ω$, we generate 3000 secure bits per wire resistance and evaluate the mean-square noise voltages at both ends over 100-sample bit exchange periods (BEPs). This probability $p$ is evaluated for the sign rule, the nearest-line rule, logistic regression, support vector machine (SVM), $k$-nearest neighbors, random forest and gradient boosting. The sign rule gives $p = 0.711$ at 50 $Ω$ and 0.682 at 3200 $Ω$. The nearest-line rule, logistic regression and SVM stay close to it and the other classifiers perform worse. For Gaussian noise, the likelihood ratio between the HL and LH cases depends only on the difference of the two mean-square voltages, so the sign rule is Bayes-optimal. The resulting $p_{\max}$ reproduces the measured values and predicts that about 900 to 1200 samples per bit are needed for $p = 0.95$. This difference carries about 0.19 bit per secure bit, 1.5 times the hard-decision value.