I'm sorry Márcio but I don't any rigorous approach. I'll ask Murilo Baptista, who is in the room next to mine and depending on the response, I put here.

Do you know that synchronisation is required for a key or message is to be extracted? Bidirectional sync works better. What do you mean rigorous?

Alexander, in this paper we do a brief discussion abou your issue:

http://www.sciencedirect.com/science/article/pii/S0375960113000728

Hi,

i am not sure if this is exactly what you mean, but i once impleneted a simple program which does something similar. you can have a look at the references below.

1) l .M Pecora and T .L Caroll, Fundementals of Synchronization of chaotic systems, concepts and applications, Chaos (1997)

2) http://www.rle.mit.edu/dspg/documents/SynchroofLorenz.pdf

I mean a mathematical proof that any chaos-based cryptography is "secure" in any sense (e.g. semantic security) . I am starting to study formal Cryptography now and I can figure out how chaos-based schemes can be compared with the conventional ones.

i see. i have not dealt with cryptography, but i the shceme i dealt with, if one loses track of the synchronizing chaotic signal, then the system diverges exponentially from the correct one. Which is why they are called chaotic. So if someone is trying to crack the signal and makes a small diversion from the driving chaotic signal, he/she will get a solution which diverges fast from the correct one. Ofcourse one can get as sophisticated as he wants in choosing a chaotic ODE and and a synchronization scheme ;)

Ask professor Safwan El Assad from the university of Nantes,

[email protected]

He is specialist of chaos based cryptography and has several patents on this field.

check his publications as for example

http://infonomics-society.org/IJICR/A%20New%20Chaotic%20Encryption%20Algorithm%20to%20Enhance%20the%20Security%20of%20ZigBee%20and%20Wi_Fi%20networks.pdf

Thanks René Lozi