I. INTRODUCTION Strong coupling constant α$_{s}$ is one of the fundamental parameters in the Standard Model. Its precise determination from various sources provides a crucial test of Quantum Chromo- dynamics (QCD). Experimentally, it is obtained through high energy particle scatterings involving quarks, such as e $^{+}$e − → hadrons (see [1, 2] for a summary), for which perturbative calculation of QCD is possible. Among other measurements, the hadronic decay rate of tau lepton [3] provides one of the most precise determinations of α$_{s}$ . The tau-lepton hadronic decay rate is written in terms of a vacuum polarization function (VPF) of weak currents. Since the perturbative QCD calculation cannot be directly applied for physical time-like momentum transfer (or the virtual W boson mass) q $^{2}$, one considers a total decay rate that involves an integral over q $^{2}$. The decay rate is then calculated using three-loop perturbative expansion and the Operator Product Expansion (OPE) to parametrize corrections at low energies. The final error in α$_{s}$ contains those from the perturbative expansion and uncertainties of condensates appearing in OPE, both of which are non-negligible because the relevant energy scale is of order of 1 GeV, i.e. lower than the tau lepton mass m$_{τ}$ . There is also an assumption of the quark-hadron duality, which is not trivially satisfied. Theoretically, the perturbative calculation and the extraction of α$_{s}$ become more trans- parent if the experimental data for VPF are available at space-like momenta. Although there is no such direct measurement, lattice QCD is able to provide a non-perturbative calculation of VPF at space-like momenta. Since the calculation is based on the first-principles of QCD, for which the input parameters are low-lying hadron spectrum, it gives another method to extract α$_{s}$ . Using lattice QCD, one can calculate VPF of vector ( V ) and axial-vector ( A ) channels from two-point functions of those currents. After a Fourier transformation in four space-time dimensions, VPF can be obtained as a function of space-like momentum squared Q 2 ranging from zero to the order of lattice cutoff squared. Our recent lattice study with two flavors of dynamical overlap fermions demonstrated that such data can indeed be used to extract α$_{s}$ combining with a perturbative calculation in the continuum scheme [4]. The result is consistent with other lattice calculations in two-flavor QCD [5, 6]. One of the main advantages of our method is that the calculation can be done on existing 2