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I was wondering if we could construct a symmetric encryption scheme by assuming that the secret key itself in GGH is public and the shared "key" is the error vector $e$.

To encrypt we would take the lattice basis $B$ and our shared secret $e \in \mathbb{R}^n$ and do:

$c = m*B + e$

and to decrypt:

$m = (c-e) * B^{-1}$

Without the correct $e$ this should not decrypt correctly. But how could I attack such an encryption?

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That wouldn't appear to work. In a known plaintext attack, that is, where the attacker knows both $m$ and $c$, he can compute:

$$e = c - m * B$$

recovering the secret key.

Alternatively, in a ciphertext-only scenario, the attacker could take two ciphertexts $c, c'$ and compute

$$(c - c') * B^{-1} = m - m'$$

resulting in the difference between the two plaintexts.

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