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Cocks IBE scheme is an Identity based encryption system proposed by Clifford Cocks in 2001 [1]. The security of the scheme is based on the hardness of the quadratic residuosity problem.



The PKG chooses:

  1. a public RSA-modulus , where are prime and kept secret,
  2. the message and the cipher space and
  3. a secure public hash function .


When user wants to obtain his private key, he contacts the PKG through a secure channel. The PKG

  1. derives with by a determistic process from (e.g. multiple application of ),
  2. computes (which fulfils either or , see below) and
  3. transmits to the user.


To encrypt a bit (coded as /) for , the user

  1. chooses random with ,
  2. computes and and
  3. sends to the user.


To decrypt a ciphertext for user , he

  1. computes if or otherwise, and
  2. computes .

Note that here we are assuming that the encrypting entity does not know whether has the square root of or . 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 is the square of or ), 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.


  1. Clifford Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues, Proceedings of the 8th IMA International Conference on Cryptography and Coding, 2001