There is a 128-bit AES key $m$, and it's encrypted by 1024-bit RSA modulus $n$ as $$c = (m\mathbin\| m\mathbin\|\ldots\mathbin\|m)^e \bmod n.$$

There are eight $m$ in $m\mathbin\| m\mathbin\|\ldots\mathbin\|m$.

Is there any way to find a message $m$?

The key is repeated 8 times, but I can't utilize this information to decrypt.

  • $\begingroup$ Is this a homework question? What is the source of this question? $\endgroup$ – kelalaka Apr 23 '20 at 7:58
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    $\begingroup$ If $e$ is small (such as $3$), then it is to possible to recover $m$ easily, but this information is missing. $\endgroup$ – user69015 Apr 23 '20 at 8:05

There is no proper padding and the message $m$ can be recovered easily if $e$ is small.

First, we can rewrite $(m\mathbin\| m\mathbin\|\ldots\mathbin\|m)$ as $$ (m + m2^{128} + \cdots + m2^{128\times 7}) = m(1 + 2^{128} + \cdots + 2^{128\times 7}), $$ and when we put it back in the equation, we have $$ c = m^e(1 + 2^{128} + \cdots + 2^{128\times 7})^e \mod n. $$ We can compute $$ c' = c\times (1 + 2^{128} + \cdots + 2^{128\times 7})^{-e} \mod n, $$ so we have the relation $c' \equiv m^e \bmod n$. In the case that $m^e < n$ (which happens for $e=3$, $5$ or $7$), then this is in fact an equality: $$ c' = m^e. $$ The value $m$ can be recovered by taking the $e$-nth root of $c'$.

  • $\begingroup$ Yes. That works for $e\in\{3,5,7\}$. $\endgroup$ – fgrieu Apr 23 '20 at 8:44
  • $\begingroup$ When you compute $c'$, it should be $c'=c\times(1+...)^{-e}$ to get $c' \equiv m^e \bmod n$, not $c'=c\times(1+...)^{-1}$. I can't edit a single character, and don't want to modify the rest of your answer ;-) $\endgroup$ – Faulst Apr 23 '20 at 12:38
  • $\begingroup$ @Faulst Thanks! $\endgroup$ – user69015 Apr 23 '20 at 12:52

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