Let $n, q, \sigma$ be the polynomial degree($x^n+1$), coefficient modulo, and the standard derivation, respectively. I often see some parameters such as enter image description here

For RLWE, we can use the CRT to decompose the $\text{RLWE}_{q}$ to some $\text{RLWE}_{q_i}$ for $1\leq i\leq l$, where $q = q_1 q_2\cdots q_l$, then when we consider the security of RLWE, we should take $\log q$ or $\log q_i$ to be considered?


1 Answer 1


You should consider $\log q$.

Considering that all the other parameters are fixed, the smaller $q$ is, the higher is the security. Even the table in your question is showing this (the table probably supposes that $\sigma$ is a small fixed value).

The hardness of the LWE and RLWE problems increases as the ratio $q / || \text{noise}||$ decreases, i.e., larger noise for the same $q$ makes the (R)LWE harder. You can think about the two extreme cases: if the noise is zero, then you can find the secret $s$ with Gaussian elimination; if the noise is as big as $q$, then it is essentially impossible to find $s$ because everything will be (very close to) uniform.

So if you take your (R)LWE samples $(a, b)$ defined mod $q$ and reduce mod a smaller $q'$ (that divides $q$), your new samples have the same noise terms, but with respect to a smaller modulus, so the ratio $q' / || \text{noise}||$ is smaller and the (R)LWE instance that you get is harder.


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