Two-layer tree-connected feed-forward neural network model for neural cryptography

Neural synchronization by means of mutual learning provides an avenue to design public key exchange protocols, bringing about what is known as neural cryptography. Two identically structured neural networks learn from each other and reach full synchronization eventually. The full synchronization enables two networks to have the same weight, which can be used as a secret key for many subsequent cryptographic purposes. It is striking to observe that after the first decade of neural cryptography, the tree parity machine (TPM) network with hidden unit $K=3$ appears to be the sole network that is suitable for a neural protocol. No convincingly secure neural protocol is well designed by using other network structures despite considerable research efforts. With the goal of overcoming the limitations of a suitable network structure, in this paper we develop a two-layer tree-connected feed-forward neural network (TTFNN) model for a neural protocol. The TTFNN model captures the notion that two partners are capable of exchanging a vector with multiple bits in each time step. An in-depth study of the dynamic process of TTFNN-based protocols is then undertaken, based upon which a feasible condition is theoretically obtained to seek applicable protocols. Afterward, according to two analytically derived heuristic rules, a complete methodology for designing feasible TTFNN-based protocols is elaborated. A variety of feasible neural protocols are constructed, which exhibit the effectiveness and benefits of the proposed model. With another look from the perspective of application, TTFNN-based instances, which can outperform the conventional TPM-based protocol with respect to synchronization speed, are also experimentally confirmed.

Figure 1

A 3-3-2 TTFNN.

Figure 2

Dynamics of $p_{a,b}$ for TTFNN-based protocols with $L=3$: (a) At the beginning ($\rho =0$), $p_{a,b}$ is given by Eq. (15). (b) In the middle ($\rho =0.57$). (c) Toward full synchronization ($\rho =0.97$), $p_{a,b}$ is close to the distribution in Eq. (16).

Figure 3

$\langle \Delta {\rho }_{\mathrm{a}}\left(\rho \right)\rangle$ and $\langle \Delta {\rho }_{\mathrm{r}}\left(\rho \right)\rangle$ for TTFNN-based protocols, $L=10$. The linear fitting equation $\Gamma \left(\rho \right)$ is given by Eq. (25).

Figure 4

Two typical types of dynamics of $\langle \Delta \rho \left(\rho \right)\rangle$ for TTFNN-based protocols. The arrow lines display the moving directions on average.

Figure 5

The methodology for seeking additional feasible protocols.

Figure 6

The average synchronization time $\langle T_{\mathrm{sync}}\rangle$ as a function of $L$ for different protocols. Symbols denote averages over 1000 simulations with learning rule ($L_1$) using $N=100$.

Figure 7

$\langle \Delta {\rho }_{\mathrm{a}}\left(\rho \right)\rangle$ and $\langle \Delta {\rho }_{\mathrm{r}}\left(\rho \right)\rangle$ of (a) TPM ($K=3$), (b) TPCM ($K=3$), (c) TCPM ($K=3$), and (d) CI ($q=1,2$). $L=10$. The linear fitting equation $\Gamma \left(\rho \right)$ is given by Eq. (25).