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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 \textstyle n = pq, where \textstyle p,q,\,p \equiv q \equiv 3 \mod 4 are prime and kept secret,
  2. the message and the cipher space \textstyle \mathcal{M} = \left\{-1,1\right\}, \mathcal{C} = \mathbb{Z}_n and
  3. a secure public hash function \textstyle f: \left\{0,1\right\}^* \rightarrow \mathbb{Z}_n.


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

  1. derives \textstyle a with \textstyle \left(\frac{a}{n}\right) = 1 by a determistic process from \textstyle ID (e.g. multiple application of \textstyle f),
  2. computes \textstyle r = a^{\frac{n+5-p-q}{8}} \mod n (which fulfils either \textstyle r^2 = a \mod n or \textstyle r^2 = -a \mod n, see below) and
  3. transmits \textstyle r to the user.


To encrypt a bit (coded as \textstyle 1/\textstyle -1) \textstyle m \in \mathcal{M} for \textstyle ID, the user

  1. chooses random \textstyle t with \textstyle m = \left(\frac{t}{n}\right),
  2. computes \textstyle c_1 = t + at^{-1} \mod n and c2 = tat − 1 and
  3. sends \textstyle s=(c_1, c_2) to the user.


To decrypt a ciphertext s = (c1,c2) for user ID, he

  1. computes α = c1 + 2r if r2 = a or α = c2 + 2r otherwise, and
  2. computes m = \left(\frac{\alpha}{n}\right).

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 \textstyle p \equiv q \equiv 3 \mod n (i.e. \left(\frac{-1}{p}\right) = \left(\frac{-1}{q}\right) = -1) and \textstyle \left(\frac{a}{n}\right) \Rightarrow \left(\frac{a}{p}\right) = \left(\frac{a}{q}\right), either \textstyle a or \textstyle -a is a quadratic residue modulo \textstyle n.

Therefore, \textstyle r is a square root of \textstyle a or \textstyle -a:

 \begin{align} r^2 &= \left(a^{\frac{n+5-p-q}{8}}\right)^2 \ &= \left(a^{\frac{n+5-p-q - \Phi\left(n\right)}{8}}\right)^2 \ &= \left(a^{\frac{n+5-p-q - (p-1)(q-1)}{8}}\right)^2 \ &= \left(a^{\frac{n+5-p-q - n+p+q-1}{8}}\right)^2 \ &= \left(a^{\frac{4}{8}}\right)^2 \ &= \pm a \ \end{align}

Moreover (for the case that \textstyle a is a quadratic residue, same idea holds for \textstyle -a):

 \begin{align} \left(\frac{s+2r}{n}\right) &= \left(\frac{t + at^{-1} +2r}{n}\right) = \left(\frac{t\left(1+at^{-2} +2rt^{-1}\right)}{n}\right) \ &= \left(\frac{t\left(1+r^2t^{-2} +2rt^{-1}\right)}{n}\right) = \left(\frac{t\left(1+rt^{-1}\right)^2}{n}\right) \ &= \left(\frac{t}{n}\right) \left(\frac{1+rt^{-1}}{n}\right)^2 = \left(\frac{t}{n} \left(\pm 1\right)\right)^2 = \left(\frac{t}{n}\right) \ \end{align}


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 \textstyle n, make the choice of \textstyle t uniform and random and moreover include some authenticity checks for \textstyle t (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.


  1. ^ 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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