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Cocks IBE scheme

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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 deterministic 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. chooses random with , different from ,
  3. computes and and
  4. 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 disadvantage 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 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.


  1. ^ Clifford Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues Archived 2007-02-06 at the Wayback Machine, Proceedings of the 8th IMA International Conference on Cryptography and Coding, 2001