Let $\rho$ be a probability measure on $\mathrm{SL}_d(\mathbb{Z})$ and consider the random walk defined by $\rho$ on the torus $\mathbb{T}^d = \mathbb{R}^d / \mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved in [BFLM11] that under an assumption on the group generated by the support of $\rho$ the random walk starting at any irrational point equidistributes in the torus. In this article, we study the central limit theorem and the law of the iterated logarithm for this walk starting at some point having good diophantine properties.