# Ring LWE: How can the secret be chosen from the “uniform distribution”?

The key generation algorithm for Ring-LWE is as follows. Have a ring $$R_p =Z_p[x]/(x^n + 1)$$. Then pick a uniformly random $$a$$ from $$R_p$$. Pick $$s$$ from an appropriate distribution. Pick $$e$$ from the error distribution (typically a constant-size Gaussian). Compute $$b = as + e$$. Reveal $$(a, b)$$ as the public key and $$s$$ is the secret key.

I've been reading The Homomorphic Encryption Standard. It says that $$s$$ can be picked as a uniform random element from $$R_p$$. How is this possible? For instance if the (admittedly insecure) choice $$n=1$$ was picked, then for any values of $$a$$, $$b$$ and $$e$$ there is some value $$s$$ which satisfies the equation. More seriously, the LPR scheme, for instance, assumes that $$(re - se_1 + e_2)$$ is small in order for decryption to be correct. How can it be small if $$s$$ is chosen uniformly at random from $$R_p$$?