Neural cryptography with feedback

Neural cryptography is based on a competition between attractive and repulsive stochastic forces. A feedback mechanism is added to neural cryptography which increases the repulsive forces. Using numerical simulations and an analytic approach, the probability of a successful attack is calculated for different model parameters. Scaling laws are derived which show that feedback improves the security of the system. In addition, a network with feedback generates a pseudorandom bit sequence which can be used to encrypt and decrypt a secret message.

Figure 1

A tree parity machine with $K=3$ and $N=4$.

Figure 2

Probability $P_{\mathit{sync}}$ as a function of the fraction $a_w$ of initially known weights, calculated from 1000 simulations with $K=3$ and $N=100$.

Figure 3

The average overlap between student and generator as a function of the number of steps for $K=3$, $L=5$, and $N=100$.

Figure 4

Average synchronization time $t_{\mathit{sync}}$ and its standard deviation as a function of $L$, found from $\mathrm{10\hspace{0.17em}000}$ simulation runs with $K=3$ and $N=\mathrm{10\hspace{0.17em}000}$. The line $52L^2$ is a result of linear regression for $R=0$.

Figure 5

The synchronization time $t_{\mathit{sync}}$ and its standard deviation as a function of $L$, averaged over $\mathrm{10\hspace{0.17em}000}$ runs of the iterative equations for $K=3$.

Figure 6

The success probability $P_E$ as a function of $L$, averaged over $\mathrm{10\hspace{0.17em}000}$ simulations with $K=3$ and $N=1000$.

Figure 7

The coefficient $u$ as a function of the feedback parameter $R$, calculated from the results shown in Fig. 6.

Figure 8

The success probability $P_E$ as a function of $L$, found from $\mathrm{10\hspace{0.17em}000}$ runs of the iterative equations for $K=3$.

Figure 9

The coefficient $y$ as a function of the feedback parameter $R$, calculated from the results shown in Fig. 8.

Figure 10

The success probability $P_E$ as a function of the average synchronization time $t_{\mathit{sync}}$, calculated from the results shown in Figs. 5, 8.