Security of permutation cipher
How secure is this cipher?


At first glance, not very.  It would appear to be vulnerable to a ciphertext-only attack, for example, the attacker can recover the plaintext given a ciphertext of about 10k (actually, he probably can deal with less), even assuming that all the attacker initially knows is that the plaintext is "ASCII English", and he has no initial knowledge of the $(\sigma_1, \sigma_2, ..., \sigma_n)$ permutations.

Since you're learning, I won't spell out the details.  Here is how I started with:


"ASCII English" is approximately 20% spaces; that is, 20% of the plaintext is the exact same byte value.
Within a single generation (where "generation" means "within 255 consecutive times we shift only $\sigma_1$), $\sigma_2(\sigma_3(...\sigma_n(P))...)$ will be a fixed permutation; hence the result of that will be the same value 20% of the time.


How can we use that to attempt to reconstruct $\sigma_1$ (up to a circular shift)?  How can we correlate data from different generations (hint: it's easier if we just assume that the $\sigma_2(\sigma_3(...\sigma_n(P))...)$ permutations from different generations are independent)?  Once we've reconstructed $\sigma_1$, how can we immediately recover the plaintext?