Can combining two secure block ciphers be insecure? Let's say we have two secure block ciphers FOO and BAR. Will a new block cipher built by combining them into a single new block cipher result in any insecurity?
$$\begin{align}
\operatorname{BAZ}_K(x)&amp;=\operatorname{FOO}_K(\operatorname{BAR}_K(x))\\
\operatorname{BAZ}^{-1}_K(x)&amp;=\operatorname{BAR}^{-1}_K(\operatorname{FOO}^{-1}_K(x))
\end{align}
$$
It depends on what you mean by &quot;a secure block cipher&quot;, but under a strong enough definition (which is thought to hold in practice), the answer is &quot;no&quot; for boring reasons. If we had some block cipher $F_k$ such that $F_{k}^{-1}$ was also a secure block cipher, then the composition:
$$(F_k\circ F_k^{-1})(x) = x$$
Would trivially not be a secure block cipher.
Therefore, if we want to justify this (contrived) counterexample, we would need $F_k$ being a secure block cipher to imply that $F_k^{-1}$ is as well.
This does not hold for the &quot;standard&quot; notion of security of block ciphers (distinguishing the PRP from random), but does hold if you also allow the adversary to query the &quot;inverse&quot; function.
Most block ciphers in practice are thought to satisfy this stronger definition, but this kind of insecure construction (composing a block cipher with its inverse) is a &quot;dumb&quot; attack on your idea, and relies on the two block ciphers ($F_k$ and $F_k^{-1}$) sharing a great deal of structure mathematically.