# LWE: does using only a small subspace of the plaintext space influence the security of the encryption scheme?

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?

• 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. – Hilder Vitor Lima Pereira Mar 13 '19 at 15:55
• Oh yeah true I didn't think about this... Thank you! – Binou Mar 14 '19 at 1:08

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.