  We present the results of our ALMA Cycle 4 high-spatial-resolution
(0.04-0.07") observations, at HCN J=3-2 and HCO+ J=3-2 lines, of the nucleus of
NGC 1068, the nearby prototypical type 2 active galactic nucleus (AGN). Our
previous ALMA observations identified the compact emission of these lines at
the putative location of the torus around a mass-accreting supermassive black
hole. We now report that we have detected the rotation of this compact
emission, with the eastern and western sides being redshifted and blueshifted,
respectively. Unlike the previously reported CO J=6-5 emission, both the
morphological and dynamical alignments of the HCN J=3-2 and HCO+ J=3-2 emission
are roughly aligned along the east-west direction (i.e., the expected torus
direction), suggesting that these molecular lines are better probes of a
rotating dense molecular gas component in the torus. The western part of the
torus exhibits larger velocity dispersion and stronger emission in the HCN
J=3-2 and HCO+ J=3-2 lines than the eastern part, revealing a highly
inhomogeneous molecular torus. The dense molecular gas in the torus and that of
the host galaxy at 0.5-2.0" from the AGN along the torus direction are found to
be counter-rotating, suggesting an external process happened in the past at the
NGC 1068 nucleus.
