# Is the error distribution in Learning with Errors (LWE), the discrete Gaussian distribution?

In $$\mathbb{Z}$$, the discrete Gaussian distribution is defined as $$D_{Z,s}(x) = \frac{\rho_s(x)}{\rho_s(\mathbb{Z})}, x\in \mathbb{Z}$$.

In LWE, $$(\overrightarrow{a}, b = \langle \overrightarrow{a}, \overrightarrow{s}\rangle + e)\in \mathbb{Z}_p^n\times \mathbb{Z}_p$$, the error $$e$$ is just sampled by rounding from continuous Gaussian, but in [GPV08], the author said that rounding is not the $$D_{\mathbb{Z},s}$$, their statistical distance is not negligible($$\Omega(1/s^3)$$).

The error in LWE is not discrete Gaussian? Or what's the relation between the error in LWE and the $$D_{Z,s}$$?

• Hello. Could you please tell us where you saw that "$e$ is just sampled by rounding from continuous Gaussian"? – Hilder Vitor Lima Pereira Apr 24 at 6:08
• In Seal, for BFV and CKKS, the error term $e(x)= e_0 + e_1x + \cdots + e_{N-1}x^{N-1}$, the coefficients $e_i$'s are just sampled by rounding from continuous Gaussian. – 2646jiaxing Apr 25 at 1:19
• I'm not sure that whether the LWE and RLWE are different, i.e, the RLWE error could be sampled by rounding, but the LWE could not? – 2646jiaxing Apr 25 at 1:20
• The answers to this should address your question: crypto.stackexchange.com/questions/88685/… – Chris Peikert May 19 at 0:54