  We study abelianization of QCD in the maximally abelian (MA) gauge. In the MA
gauge, the off-diagonal gluon amplitude is strongly suppressed, and then the
off-diagonal gluon phase shows strong randomness, which leads to a large
off-diagonal gluon mass. Using lattice QCD, we find a large effective
off-diagonal gluon mass in the MA gauge: $M_{\rm off} \simeq 1.2 {\rm GeV}$ in
SU(2) QCD, $M_{\rm off} \simeq 1.1 {\rm GeV}$ in SU(3) QCD. Due to the large
off-diagonal gluon mass in the MA gauge, infrared QCD is well abelianized like
nonabelian Higgs theories. We investigate the inter-monopole potential and the
dual gluon field $B_\mu$ in the MA gauge, and find longitudinal magnetic
screening with $m_B \simeq$ 0.5 GeV in the infrared region, which indicates the
dual Higgs mechanism by monopole condensation. We propose a gauge invariant
description of the MA projection by introducing the ``gluonic Higgs scalar
field''.
