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Are there any security concerns with using RSA in “CBC mode”?

Specifically: if I use RSA encryption as my block cipher operation, and apply the standard CBC mode operations including a random IV, will the resulting cipher text provide the same level of theoretical security as the underlying RSA problem?

I am only considering RSA on its own. No padding scheme. I know traditionally RSA is insecure without proper padding, but in CBC mode I don’t see why the padding would be necessary (and it complicates the issue by introducing non-deterministic elements)

I also understand that CTR, CFB, and other block cipher modes that use the block encryption function for both encryption and decryption would fail horribly with RSA. Only concerned with CBC

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This is a truly mad idea, so I applaud you for that. But it's seriously insecure. My interpretation of "RSA-CBC" would work like this:

$$ \begin{array}{l} \textsf{RSA-CBC}\Bigl( (N,e), m_1 \| m_2 \| \ldots \|m_\ell \Bigr): \\ \quad c_0 \gets \mathbb{Z}_N \\ \quad \mbox{for $i=1$ to $\ell$:} \\ \quad\quad c_i := (c_{i-1} + m_i)^e \bmod N \\ \quad \mbox{return } c_0 \| c_1 \| \ldots \| m_\ell \end{array}$$

Here each $m_i$ and each $c_i$ is a $\mathbb{Z}_N$-element. RSA-CBC chooses a random "IV" (element of $\mathbb{Z}_N$), then encrypts each plaintext block by adding the previous ciphertext block and then applying the RSA function.

So what's wrong with it? Suppose I see an encryption of some unknown plaintext. If I have a guess for $m_i$, then I can check whether my guess is correct via $c_i \overset?= (c_{i-1} + m_i)^e \bmod N$. I can indeed perform this check because the RSA exponent $e$ is public.

More generally, CBC doesn't work with public-key operations. Anyone can repeat the steps done during CBC encryption, if the block cipher is replaced by a public-key operation that anyone can perform.

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  • $\begingroup$ Beautiful. I knew there was something simple I was missing. $\endgroup$ Feb 10, 2022 at 4:21
  • $\begingroup$ Well, one can design this with secure PKCS and OAEP padding, too. $$c_i := (pad(c_{i-1} + m_i))^e \bmod N$$ (the reduced size of padding is omitted) $\endgroup$
    – kelalaka
    Feb 10, 2022 at 14:23
  • $\begingroup$ @kelalaka, $c_{i-1}+m_i$ is already a "full size" element of $\mathbb{Z}_N$, so I'm not sure what padding you could apply while still keeping the result in $\mathbb{Z}_N$. $\endgroup$
    – Mikero
    Feb 10, 2022 at 15:04
  • $\begingroup$ @Mikero one has to reduce the size of input of this scheme like 11-byte for PKCS#1 v1.5 padding ( see this for OAEP ) as I said (may be not clear, sorry for that). Then encryptions are secure as long as PKCS#1 v1.5 padding or OEAP is secure. Anyway, one has to use ECIES... $\endgroup$
    – kelalaka
    Feb 10, 2022 at 15:29
  • $\begingroup$ OK, then in that case the "CBC chaining" adds nothing. You might as well just do RSA-OAEP in "ECB" mode ;) $\endgroup$
    – Mikero
    Feb 10, 2022 at 16:18

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