encryption/decryption with multiple keys - encryption

Question

Is it possible to encrypt data, such that it can be decrypted with several different keys?
Example:
I've encrypted data with key1, but I want to be able to decrypt with keys 2, 3, and 4.
Is this possible?

Answer 1

GnuPG does multi-key encryption in standard.
The following command will encrypt doc.txt using the public key for Alice and the public key for Bob. Alice can decrypt using her private key. Bob can also decrypt using his private key.
gpg --encrypt --recipient alice#example.com \
--recipient bob#example.com doc.txt
This feature is detailed in the user guide section entitled "Encrypting and decrypting documents"

Answer 2

Yes it's possible
Yes encryption for multiple recipients is possible. Also it seems logical when you think that you might want to be able to read what you've sent to someone and to do so you need to be in the recipients list.
Command line
Here is how to do it through gpg command line (as described in David Segonds' answer):
gpg --encrypt \
--recipient alice#example.com \
--recipient bob#example.com \
clear-message.txt
GUI client
Your GUI must provide a way to encrypt for several people
Mechanism
There is a question on Information Security, GPG File size with multiple recipients?, that explain the encryption mechanism:
GPG encrypts the file once with a symmetric key, then places a header
identifying the target keypair and an encrypted version of the
symmetric key.
[...] When encrypted to multiple recipients, this
header is placed multiple times providing a uniquely encrypted version
of the same symmetric key for each recipient.

Answer 3

GnuPG and PGP clients in general usually encrypt the actual data with a symmetric key called a "session key". The session key is then encrypted with each "recipient key" (i.e. the ones you specify with -r/--recipient). This is sometimes referred to as a hybrid cipher. Right now, I believe GnuPG by default uses an 256 bit session keys and AES to encrypt the plaintext data to that AES-256 session key, and your recipient keys are your RSA/DSA/ECDSA/etc. assymetric key in this case.
One reason for doing it this way is that symmetric cryptographic algorithms like AES are generally a lot faster than asymmetric ones like RSA. GnuPG thus only has to encrypt ~256 bits (the session key) with RSA, and can use AES to encrypt the data (as large as you want it to be!) with that session key. Intel machines even have a built in instruction, AES-NI, to do some steps of the algorithm in hardware, which makes GnuPG extra snappy at encrypting/decrypting data.
Another reason for doing it this way is that it allows PGP-encrypted documents to be encrypted to multiple parties without having to double the size of the document. Notice that when you specify multiple recipients for an encrypted document (e.g. gpg -ea -r Alice -r Bob -o ciphertext.asc), the encrypted document that gets stored (ciphertext.asc) is not 2x as large as if you had just encrypted it to Alice.
See also the --show-session-key parameter in the gpg man page to be able to decrypt just the session key, for example to allow a third party to decrypt a document that is encrypted to you without having to transfer to them your private key or the plaintext data.

Answer 4

Yes, it's possible. Google "multiparty encryption" for a start.
AFAIK, there are no drop 'em in and use 'em packages for it though.
-- MarkusQ
P.S. For a sketch of how it could be done, consider this. The encrypted message consists of:
the payload, encrypted with a one-time pad
the one time pad, encrypted with key1
the one time pad, encrypted with key2
...
the one time pad, encrypted with keyN
The recipient who hold key i just decrypts their copy of the pad with their key, and then decrypts the payload.
However, this is just a proof that it could be done and would suck as an actual implementation. If at all possible, you should avoid rolling your own encryption. If you don't understand why, you should definitely avoid rolling your own encryption.
-----Edit ------------
If I'm wrong and the Gnu tools do that, use them. But I can't seem to find any information on how to do it.

Answer 5

Multiple (more than two) key RSA is maybe like this - well i'm not a mathematician, so this algorithm is not necessarily secure, i just want to give an idea with it.
m=p*q*r; p,q,r are big prime numbers
fi(m)=(p-1)(q-1)(r-1)
d==(e1*e2*e3*...*ei)^(-1) (mod fi(m)); e1...ei are arbitrary numbers, d is calculated to fulfill the equation
y1==x^e1 (mod m)
y2==y1^e2 (mod m)
y3==y2^e3 (mod m)
...
x==yi^d (mod m)
This algorithm could be used for example to increase the speed of The Onion Router.