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This is motivated by this question about the recovery of (corrupted) encrypted files, and in particular the statement (in the answer):

But If I know the algorithm and the key and I make a custom software to decrypt the contents of the file no matter what, I might get a decrypted JPEG that may be partially viewable/cluttered at some places.

In general, how things like full-disk encryption interact with randomly flipped bits is something I haven't thought about. There are two obvious solutions:

  • Wrap the full-disk encryption in an error-correcting code
  • Minimize the size of each "block" of data which is encrypted, and accept a full loss of data (to that single small block) if there is an error (although this wouldn't be full disk encryption I don't believe).

This question is asking about other potential solutions.


For a fixed key $k\in\mathcal{K}$ and encryption randomness $r\in\mathcal{R}$, one can view an encryption scheme $m\mapsto \mathsf{Enc}_k(m;r)$ as a code (in the sense of coding theory) with domain $\mathcal{M}$ and codomain $\mathcal{C}_{k, r}\subseteq \mathcal{X}$. The correctness of the encryption scheme states that the code is uniquely decodable, which one might write as the condition: $$\forall k\in\mathcal{K} : \mathsf{Dec}_k(\mathsf{Enc}_k(m; \mathcal{R})) = m$$ One could could define a code between $\mathcal{M}$ and $\mathcal{X}$ abstractly via the pair of maps $\mathsf{encode}$ and $\mathsf{decode}$. Then several coding-theoretic properties can be captured via conditions such as the above, like unique decodability: $$\forall m\in\mathcal{M} : \mathsf{decode}(\mathsf{encode}(m)) = m$$ A more interesting ability is that of being able to correct errors in some bounded set $\mathcal{E}$. This can often have a "geometric" flavor ($\mathcal{E}$ is a ball in some norm, so the errors are "small"), but I will not impose this requirement. The code $(\mathsf{encode}, \mathsf{decode}$) can correct $\mathcal{E}$ if: $$\forall m\in\mathcal{M} : \mathsf{decode}(\mathsf{encode}(m) + \mathcal{E}) = m$$

I'm curious about what can be said about the error-correction capabilities of IND-CPA encryption schemes. They clearly can have some, for the simple reason that if $(\mathsf{Enc}_k, \mathsf{Dec}_k)$ is IND-CPA, and $(\mathsf{encode}, \mathsf{decode})$ can correct $\mathcal{E}$ errors, then $(\mathsf{encode}\circ \mathsf{Enc}_k, \mathsf{Dec}_k\circ\mathsf{decode})$ is still IND-CPA (its ciphertexts are publicly computable from the initial scheme's ciphertexts), but can now correct $\mathcal{E}$ errors.

The question is therefore if any common schemes have any error-correction properties to speak of. Specifically, if I take some ciphertext $c$ and "flip" a single bit in it (or add any other potentially tiny error), can anything be said about the plaintext's relationship to the "true plaintext"? Do any IND-CPA secure schemes come with error-correction "baked in", or must they be explicitly combined with an error-correcting code as in the construction I mentioned above?

Note that I would be interested even if the schemes are no longer correct, but are "close to correct" in a certain sense. In particular, if $\forall k\in\mathcal{K}$: $$\mathsf{Dec}_k(\mathsf{Enc}_k(m;\mathcal{R}) + \mathcal{E}) \in m + \mathcal{E}'$$ Where $\mathcal{E}'$ is some set of allowable errors, and will hopefully be related to $\mathcal{E}$ (i.e. if the noise is promised to be small, the resulting errors are small as well).

Note that an obvious answer to the above (LWE encryption, which already has a natural coding-theoretic interpretation) appears to only partially work --- ciphertexts of the form $(a, b)$ are error-correcting "in the $b$ slot" naturally, but not "in the $a$ slot". One could then define the set of errors $\mathcal{E} = \{0\}^n\times \overline{\mathcal{E}}$ (where $a$ is $n$-dimensional), and $\overline{\mathcal{E}}$ is "small enough" in a precise sense. I find this slightly unsatisfying due to the "contrived" shape of the errors $\mathcal{E}$ (and that LWE-based encryption arguably requires choosing an error-correcting code already, so that it has error-correction properties "baked in" is unsurprising).

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Too long for a comment:

Let ${\cal M}=\{0,1\}^n,$ and ${\cal K}=\{0,1\}^k.$ Let the set $\cal E,$ have cardinality $2^{f},$ for simplicity (no Hamming sphere would have this value if you were only interested in stopping low weight errors, btw). You require that for every message $m\in {\cal M}$ and for every $e\in {\cal E}$ we have: $$ E^{-1}_k(E_k(m)+e)=m ,\quad \forall k\in {\cal K} $$ This has efficiency implications since you are no longer really encrypting messages $m$ one to one but encrypting equivalence classes of messages, where the equivalence class of message $m$ is $$ [m]=\{m+e: e \in {\cal E}\}. $$ In particular, your effective message space is now ${\cal M}^\ast=\{[m]: m \in {\cal E}\}$ which has size $2^{n-f}.$ In fact this setup is reminiscent of Gilbert-Sloane's setup for (keyed) authentication codes (MACs) with no secrecy.

In terms of security parameters, I'd venture that if you wanted $N$ bit security you might want to choose $n=N+f,$ and $k=N,$ in case you like the keylength=blocklength regime, balancing dictionary and brute force attacks. Presumably AES-256 might be used this way.

A decently designed block cipher, however, having been designed with avalanch properties, would be unlikely to have a set $\cal E$ which overlaps with a Hamming sphere centred at zero in any significant way.

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