I am wondering whether we can use technique of gadget decomposition in decryption to replace degree 2 PRNG in your paper (I believe the PRNG is there to add smudging noise).

Let's say user provides $\beta^k s'$ where $\beta^k s'[i] = \beta^k s[i] + e_{i,k}$ for $k \in [0,l)$, where $i \in [0, n)$, $n$ being the LWE dimension. We can assume $e_{i, k}$ to be bounded by some bound $B$.

To decrypt LWE ciphertext
$$LWE_{s}(m) = (b, a) = (b = a \cdot s + e + \Delta m, a)$$

decompose vector $a$ in multiple limbs, $a_0, .., a_{l-1}$ s.t. $a[i] = \sum \beta^k a_j[i]$. Then calculate inner product,

$$b - a \cdot s + E = b - \sum_{k \in [0, l)} a_k \cdot \beta^k s'$$

where $E$ is bounded by $l\beta B$.

This is same as gadget decomposition technique in FHE and $E$ acts as smudging noise.