We provide a method for deciding the insecurity of cryptographic protocols in presence of the standard Dolev-Yao intruder (with a finite number of sessions) extended with so-called oracle rules, i.e., deduction rules that satisfy certain conditions. As an instance of this general framework, we obtain that protocol insecurity is in NP for an intruder that can exploit the properties of the exclusive or (XOR) operator. This operator is frequently used in cryptographic protocols but cannot be handled in most protocol models. An immediate consequence of our proof is that checking whether a message can be derived by an intruder (using XOR) is in PTIME. We also apply our framework to an intruder that exploits properties of certain encryption modes such as cipher block chaining (CBC).