Since $p:\mathbb{R}\times\mathbb{R}\rightarrow S^{1}\times S^{1}$ given by $$(p\times p)(x,y)=(\cos2\pi x,\sin2\pi x,\cos2\pi y,\sin2\pi y)$$ is a covering map, letting $e=(0,0)$ and $b=p(e)=(1,0,1,0)$ , we have $p^{-1}(b)$ consisting of all integer points $\mathbb{Z}^{2}\subset\mathbb{R}^{2}$ and the bijective ($\mathbb{R}^{2}$ is simply connected, Theorem 54.4) correspondence $$\phi:\pi_{1}(S^{1}\times S^{1},b)\rightarrow\mathbb{Z}^{2}.$$ To show that $\phi$ is isomorphism, we consider two loops in $S^{1}\times S^{1}$ such that their liftings in $\mathbb{R}^{2}$ starting at $e$ end at some integer points $(m,n)$ and $(m',n')$ , and show that their product has a lifting starting at $e$ and ending at $(m+m',n+n')$ .