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Backdoor : A secret or undocumented means of getting into a computer system, or software that uses such a means to penetrate a system. Some software has a backdoor placed by the programmer to allow them to gain access to troubleshoot or change the program. Software that is classified as a "backdoor" is designed to exploit a vulnerability in a system, and open it to future access by an attacker.
Trojan: Any program with a hidden intent. Trojans are one of the leading causes of breaking into machines. If you pull down a program from a chat room, new group, or even from unsolicited e-mail, then the program is likely trojaned with some subversive purpose. The word Trojan can be used as a verb: To trojan a program is to add subversive functionality to an existing program. For example, a trojaned login program might be programmed to accept a certain password for any user's account that the hacker can use to log back into the system at any time. Rootkits often contain a suite of such trojaned programs.