Genetic attack on neural cryptography

Different scaling properties for the complexity of bidirectional synchronization and unidirectional learning are essential for the security of neural cryptography. Incrementing the synaptic depth of the networks increases the synchronization time only polynomially, but the success of the geometric attack is reduced exponentially and it clearly fails in the limit of infinite synaptic depth. This method is improved by adding a genetic algorithm, which selects the fittest neural networks. The probability of a successful genetic attack is calculated for different model parameters using numerical simulations. The results show that scaling laws observed in the case of other attacks hold for the improved algorithm, too. The number of networks needed for an effective attack grows exponentially with increasing synaptic depth. In addition, finite-size effects caused by Hebbian and anti-Hebbian learning are analyzed. These learning rules converge to the random walk rule if the synaptic depth is small compared to the square root of the system size.

Figure 1

Tree parity machine with $K=3$ and $N=4$.

Figure 2

Probability $P_r$ of repulsive steps as a function of the generalization error $ϵ$. The inset shows the probability $P_g$ for a successful geometric correction.

Figure 3

Synchronization time $t_{\mathrm{sync}}$ as a function of $H$ for $K=3$, $N=1000$, random walk learning rule, and different values of $L$, averaged over 10 000 simulations.

Figure 4

Success probability $P_E$ of the genetic attack for $K=3$, $N=1000$, random walk learning rule, and $M=4096$. Symbols represent results obtained from 1000 simulations, and the lines show a fit with Eq. (17).

Figure 5

Parameters $\mu$ and $\beta$ as a function of the synaptic depth $L$. Symbols denote results of fitting simulation data for different $M$ with Eq. (17) and the lines were calculated using the model given in Eq. (19).

Figure 6

Offset $\delta$ as a function of the number of attackers $M$, for $K=3$, $N=1000$, and the random walk learning rule. Symbols and the line were obtained by a fit with Eq. (19).

Figure 7

Success probability of the genetic attack as a function of the synchronization time for $K=3$, $N=1000$, random walk learning rule, $M=4096$, and different values of $L$. The dashed line shows the envelope of this set of curves.

Figure 8

Success probability of the genetic attack for $K=3$, $L=7$, $N=1000$, random walk learning rule, $M=4096$, and $H=2.28$. These results were obtained by averaging over 100 simulations.

Figure 9

Length of the weight vectors in the steady state for $K=3$ and $N=1000$. Symbols denote results averaged over 1000 simulations and lines show the first-order approximation given in Eq. (29) and Eq. (31).

Figure 10

Parameter $\mu$ and $\beta$ as a function of $L$ for the genetic attack with $K=3$, $N=1000$, and $M=4096$. The symbols represent results from 1000 simulations and the lines show a fit using the model given in Eq. (19).

Figure 11

Synchronization time $t_{\mathrm{sync}}$ as a function of $H$ for $K=3$, $L=7$, averaged over 100 simulations.

Figure 12

Success probability $P_E$ of the geometric attack with $M=1$, the majority attack with $M=100$, and the genetic attack with $M=4096$, for $K=3$, $N=1000$, and the random walk learning rule. Symbols represent results averaged over 1000 simulations, in part (a) for random inputs and in part (b) for queries with $H=0.32L$. The lines were obtained by fitting with Eq. (33).