  In this paper, we investigate the problem of planning an optimal infinite
path for a single robot to achieve a linear temporal logic (LTL) task with
security guarantee. We assume that the external behavior of the robot,
specified by an output function, can be accessed by a passive intruder
(eavesdropper). The security constraint requires that the intruder should never
infer that the robot was started from a secret location. We provide a sound and
complete algorithmic procedure to solve this problem. Our approach is based on
the construction of the twin weighted transition systems (twin-WTS) that tracks
a pair of paths having the same observation. We show that the security-aware
path planning problem can be effectively solved based on graph search
techniques in the product of the twin-WTS and the B\"{u}chi automaton
representing the LTL formula. The complexity of the proposed planning algorithm
is polynomial in the size of the system model. Finally, we illustrate our
algorithm by a simple robot planning example.
