In this paper, we propose a novel strategy for physical layer authentications based on the challenge-response concept for a transmitting drone (Alice). In a preliminary training phase, Alice moves over several positions, and Bob (either a drone or a ground device) estimates the Alice-Bob channel gains. Then Alice transmits its message from different random positions (challenge) and Bob, upon receiving the messages, authenticates the sender via a log-likelihood test on the estimated channel gains (response). In turn, the intruder Trudy selects random positions on which she transmits messages on behalf of Alice to Bob. In this paper, we design the probability mass distribution of Alice's challenge positions and the Trudy response positions by modeling the problem as a zero-sum game between Bob and Trudy, where the payoff of Trudy is the missed detection probability. Moreover, we propose three different approaches that minimize the energy spent by Alice without sacrificing security, which differ in computational complexity and resulting energy consumption. Finally, we test the proposed technique via numerical simulations, which include a realistic model of both Alice-Bob and Trudy-Bob fading channels, affected by shadowing.