- Cocks IBE scheme
The PKG chooses:
- a public RSA-modulus , where are prime and kept secret,
- the message and the cipher space and
- a secure public hash function .
When user wants to obtain his private key, he contacts the PKG through a secure channel. The PKG
- derives with by a determistic process from (e.g. multiple application of ),
- computes (which fulfils either or , see below) and
- transmits to the user.
To encrypt a bit (coded as /) for , the user
- chooses random with ,
- chooses random with , different from ,
- computes and and
- sends to the user.
To decrypt a ciphertext s = (c1,c2) for user ID, he
- computes α = c1 + 2r if r2 = a or α = c2 + 2r otherwise, and
- computes .
Note that here we are assuming that the encrypting entity does not know whether ID has the square root r of a or − a. In this case we have to send a ciphertext for both cases. As soon as this information is known to the encrypting entity, only one element needs to be sent.
First note that since (i.e. ) and , either or is a quadratic residue modulo .
Therefore, is a square root of or :
Moreover (for the case that is a quadratic residue, same idea holds for ):
It can be shown that breaking the scheme is equivalent to solving the quadratic residuosity problem , which is suspected to be very hard. The common rules for choosing a RSA modulus hold: Use a secure , make the choice of uniform and random and moreover include some authenticity checks for (otherwise, an adaptive chosen ciphertext attack can be mounted by altering packets that transmit a single bit and using the oracle to observe the effect on the decrypted bit).
A major disadavantage of this scheme is that it can encrypt messages only bit per bit - therefore, it is only suitable for small data packets like a session key. To illustrate, consider a 128 bit key that is transmitted using a 1024 bit modulus. Then, one has to send 2 * 128 * 1024 bit = 32 KByte (when it is not known whether r is the square of a or − a), which is only acceptable for environments in which session keys change infrequently.
This scheme does not preserve key-privacy, i.e. a passive adversary can recover meaningful information about the identity of the recipient observing the ciphertext.
- ^ Clifford Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues, Proceedings of the 8th IMA International Conference on Cryptography and Coding, 2001
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