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Regarding LWE schemes where the encryption is performed this way:

for $m \in \mathbb{Z}_t$, compute $c = LWE_{\mathbf{s}}^{t/q}(m) = \{ \mathbf{a}, \mathbf{a \cdot s} + m\cdot q/t + e\} \in \mathbb{Z}_{q}^{n}$

(where $\mathbf{s}$ is the secret vector of length $n$, $\mathbf{a}$ is a random vector and $e$ is some noise generated by a discrete Gaussian sampler)

What happens if for a given plaintext modulus $t$, we use only a small subspace of $Z_t$ for our messages? Does it have any influence on the overall security of the scheme?

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    $\begingroup$ Notice that in the CPA-security game, an attacker could restrict the set of messages they are working with to any subset. Therefore, if the scheme is proved to be CPA-secure, your scenario is also safe. $\endgroup$ – Hilder Vítor Lima Pereira Mar 13 at 15:55
  • $\begingroup$ Oh yeah true I didn't think about this... Thank you! $\endgroup$ – Robin S. Mar 14 at 1:08
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No, as long as your plaintext space is a subset of $Z_t$, then there is not any negative impact on security.

However, if you use a smaller message space $Z_x\subset Z_t$, you should consider setting the plaintext space to $Z_x$ directly as this could improve the efficiency.

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  • $\begingroup$ I am actually doing this for the purpose of homomorphic encryption, where I need to use a tiny subset. But thank you anyway! $\endgroup$ – Robin S. Mar 13 at 10:10

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