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Is there any way to share an encrypted file with a group? In the other words, can I encrypt a file with a public key so that each person in a group decrypt it with his/her private key?

Is there any way that I have a public key and some different private key for it?

I want to find a way to encrypt a file with a public key, and then produce some private keys from this public key, so that whenever I wanted to share that file with some person, I gave him/her a private key from this set to decrypt my file.

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Yes, you encrypt the file with a symmetric key, then encrypt that symmetric key with each of the recipients public keys. gpg can do this by adding multiple --recipient options.

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This exists. It is called Broadcast Encryption http://en.wikipedia.org/wiki/Broadcast_encryption . Latest research even allows for Traitor tracing http://en.wikipedia.org/wiki/Traitor_tracing , meaning that even if two people give a part of their secret keys to form a "pirate decryptor", there is an algorithm which will find one of the users that colluded. It is used in practice for pay tv for example.

A trivial an non-efficient way to do Broadcast Encryption is as @John Meacham said to encrypt your file under some symmetric key and to send an encryption of the symmetric key to each user.

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