Improve security of transposition ciphers As we know, breaking a Transposition Cipher is quite trivial and can be easily done by either exploiting single letter frequencies (not $d$-gram frequencies though, given that $2 &lt; d &lt; L$), or by employing a simple CPA, thus deriving the snippets length $L$ and the transposition matrix (key).

If we imagine to encrypt the plaintext twice, employing two different keys, would this help improving security?

If we employ the same snippet length $L_1 = L_2$, this clearly won't improve security as an attacker could easily find the key $k_3 = k_1 \times k_2$ (where $\times$ is the matrix product), without even bothering about $k_1$ and $k_2$.

However, that's not the case if $L_1 \neq L_2$. So I guess there's some kind of improvement; can you show me if and how this is true?
First determine the permutation domain. If $L_2 &gt; L_1$, then the cipher is broken by $\lceil  log_2 (L_2) \rceil$ pairs of plaintext/ciphertect. The security improvement is small, as the attacker only needs to use $\lceil  log_2 (L_2) \rceil - \lceil  log_2 (L_1) \rceil$ additional pairs to find the key.