This paper studies the impact of side initial state information on the detectability of data deception attacks against cyber-physical systems. We assume the attack detector has access to a linear function of the initial system state that cannot be altered by an attacker. First, we provide a necessary and sufficient condition for an attack to be undetectable by any dynamic attack detector under each specific side information pattern. Second, we characterize attacks that can be sustained for arbitrarily long periods without being detected. Third, we define the zero state inducing attack, the only type of attack that remains dynamically undetectable regardless of the side initial state information available to the attack detector. Finally, we design a dynamic attack detector that detects detectable attacks.