Determine whether the distribution of approximate recomposition is correct or not (i.e. whether extra N factor exists)

[Janmajayamall]

I think the noise added with approximate decomposition must be

$$2^{2\gamma} / 12 var(m)$$

instead of

$$N 2^{2\gamma} / 12 var(m)$$

where $$m$$ is message to multiply incoming value with.

I am yet explain myself why.