# Improving efficiency of an “encryption” algorithm based on a chaotic map

I've implemented an "encryption" algorithm that uses the deterministic chaos of an extended logistic map:

$$x_{n+1}= r_n \cdot x_n \cdot (1-x_n)$$

$$y_{n+1} = \begin{cases} \text{x_n+y_n} &\text{if x_n+y_n<1}\\ \text{x_n+y_n-1} &\text{if x_n+y_n\ge 1}\\ \end{cases}$$

$$r_{n+1} = 3.57 + \left(0.23\cdot\sqrt{x^2+y^2}\right)$$

The data is first chunked into $$256\cdot64$$ bits which I visualise as "squares". The initial values of $$x,y,r$$ are determined per-chunk by sampling bytes in a SHA-256 hashed key-data pair mixed with a hash of the index of the data chunk. I then map $$x,y$$ values from the range of the unit square to specific bits in the chunk and flip those bits iteratively, $$(256\cdot64)^2$$ times per chunk.

I am now looking to increase the performance of this algorithm, and I had an idea to apply some kind of chaotic "surface" to each chunk, sensitive to initial values. I would then only need to compute the value of this surface once per bit with a single function to spread the chaos over each chunk.

Having browsed Wikipedia's list of chaotic maps, I couldn't really seem to find anything that fit what I had in mind. Are there any known functions that might yield a good result?

Edit: After suggestions that chaos theory isn't possible to make secure, I wanted to find out whether other algorithms exist that use chaos theory. Apparently there is a large quantity of literature on this topic, especially focused on image encryption. Could someone explain the main sources of insecurity in a chaos theory based encryption algorithm?

• Comments are not for extended discussion; this conversation has been moved to chat. – SEJPM Dec 5 '18 at 10:31
• – Paul Uszak Dec 6 '18 at 4:28
• @PaulUszak I don't think I came across that one, though yes I did come across similar papers – Daniel Castle Dec 6 '18 at 4:40