Decoding chaotic cryptography without access to the superkey

Some chaotic systems can be synchronized by sending only a part of the state space information. This property is used to create keys for cryptography using the unsent state spaces. This idea was first used in connection with the Lorenz equation. It has been assumed for that equation that access to the unsent information is impossible without knowing the three parameters of the equation. This is why the values of these parameters are collectively known as the “superkey”. The exhaustive search for this key from the existing data is time consuming and can easily be countered by changing the key. We show in this paper how the superkey can be found in a very rapid manner from the synchronizing signal. We achieve this by first transforming the Lorenz equation to a canonical form. Then we use our recently developed method to find highly accurate derivatives from data. Next we transform a nonlinear equation for the superkey to a linear form by embedding it in four dimensions. The final equations are solved by using the generalized inverse.