Well, the guard knows Y, and the enemy knows Y... but the guard's authentication procedure involves taking an X (not Y) supplied by the friendly spy, executing

Well, the guard knows Y, and the enemy knows Y... but the guard's authentication procedure involves taking an X (not Y) supplied by the friendly spy, executing the calculation, and asserting the result is equal to Y. Since the enemy does not know X, and X cannot be trivially derived from Y (one-way function), the scheme is secure.