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I have forgotten the password for an encrypted 7zip file (the filenames are not hidden), and I have copies of some of the unencrypted files. Is there a way I can use this to find out what the password is by using these copies?

It feels like there should be a way of using the knowledge of what some of the files look like unencrypted, to find out the password in a faster way than just trying to brute force it. However, I don't know anything about how 7zip encrypts files.

Is there any program that can do this? If not, and you know about encryption: Is it possible to write such a program?

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You're looking for a known-plaintext attack. Usually (e.g. you're up against a high-quality algorithm), this is hopeless.

For example, there are some known-plaintext attacks on AES which are slightly faster than brute-forcing the key. However, this is pointless because you'd still wait gazillions of years before you identified the key.

Older ZIP versions, however, are known for notoriously weak encryption. So it might be that the encryption algorithm used to encrypt your zip is vulnerable. You'd need to find out which algorithm was used exactly in order to get a meaningful answer.

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