  We report the first lattice QCD results of the scattering amplitudes of the
$K\pi$ system for $I = 1/2$ channel together with $I=3/2$ case. We investigate
all quark diagrams contributing to these iso-spin states, and find that the
scattering amplitudes are expressed as combinations of only three diagrams
after setting the masses of $u$-quark and $d$-quark to be the same.
  The lattice simulations are performed in the quenched approximation at $\beta
= 2.23$ on a 12$^3 \times 24$ lattice with an improved Iwasaki gauge action. We
employ a new dilution-type noise method to get accuracy of data with reasonable
CPU time. A simple method is proposed and applied to eliminate lattice artefact
due to the finite extent of lattice along the time direction.
  A clear difference in the quark mass dependence between $I=3/2$ and $I=1/2$
channels is observed. Although the chiral extrapolation is subtle, we assume
$E_{K\pi}^2 \propto m_{u,d}^2 $, and obtain the $S$-wave scattering lengths as
$a_0(I=3/2) m_{\pi} =-0.084^{+0.051}_{-0.064}$ and $a_0(I=1/2) m_{\pi} = -0.625
\pm0.012$. We show all necessary formulas which make the calculation possible.
  We argue that $\Lambda N$ is the most appropriate target of the L\"uscher's
formula for baryonic system because it has no $\pi$ exchange diagrams and has a
scattering length suitable for a lattice QCD simulation.
