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'''Cocks IBE scheme''' is an [[identity based encryption]] system proposed by [[Clifford Cocks]] in 2001.<ref>Clifford Cocks, [http://www.cesg.gov.uk/site/ast/idpkc/media/ciren.pdf An Identity Based Encryption Scheme Based on Quadratic Residues], ''Proceedings of the 8th IMA International Conference on Cryptography and Coding'', 2001</ref> The security of the scheme is based on the hardness of the [[quadratic residuosity problem]].
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==Protocol==
 
===Setup===
The PKG chooses:
# a public RSA-modulus <math>\textstyle n = pq</math>, where <math>\textstyle p,q,\,p \equiv q \equiv 3 \mod 4</math> are prime and kept secret,
# the message and the cipher space <math>\textstyle \mathcal{M} = \left\{-1,1\right\}, \mathcal{C} = \mathbb{Z}_n</math> and
# a secure public hash function <math>\textstyle f: \left\{0,1\right\}^* \rightarrow \mathbb{Z}_n</math>.
 
===Extract===
When user <math>\textstyle ID</math> wants to obtain his private key, he contacts the PKG through a secure channel. The PKG
# derives <math>\textstyle a</math> with <math>\textstyle \left(\frac{a}{n}\right) = 1</math> by a determistic process from <math>\textstyle ID</math> (e.g. multiple application of <math>\textstyle f</math>),
# computes <math>\textstyle r = a^{\frac{n+5-p-q}{8}} \mod n</math> (which fulfils either <math>\textstyle r^2 = a \mod n</math> or <math>\textstyle r^2 = -a \mod n</math>, see below) and
# transmits <math>\textstyle r</math> to the user.
 
===Encrypt===
To encrypt a bit (coded as <math>\textstyle 1</math>/<math>\textstyle -1</math>) <math>\textstyle m \in \mathcal{M}</math> for <math>\textstyle ID</math>, the user
# chooses random <math>\textstyle t_1</math> with <math>\textstyle m = \left(\frac{t_1}{n}\right)</math>,
# chooses random <math>\textstyle t_2</math> with <math>\textstyle m = \left(\frac{t_2}{n}\right)</math>, different from  <math>\textstyle t_1</math>,
# computes <math>\textstyle c_1 = t_1 + at_1^{-1} \mod n </math> and <math> c_2= t_2 - at_2^{-1}</math> and
# sends <math>\textstyle s=(c_1, c_2)</math> to the user.
 
===Decrypt===
To decrypt a ciphertext <math>s=(c_1, c_2)</math> for user <math>ID</math>, he
# computes <math>\alpha = c_1 + 2r</math> if <math> r^2=a </math> or <math>\alpha = c_2 + 2r</math> otherwise, and
# computes <math>m = \left(\frac{\alpha}{n}\right)</math>.
 
Note that here we are assuming that the encrypting entity does not know whether <math> ID</math> has the [[Quadratic_residue#Complexity_of_finding_square_roots|square root]] <math>r</math> of <math> a</math> or <math> -a</math>. 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.
 
===Correctness===
 
First note that since <math>\textstyle p \equiv q \equiv 3 \mod 4</math> (i.e. <math>\left(\frac{-1}{p}\right) = \left(\frac{-1}{q}\right) = -1</math>)  and <math>\textstyle \left(\frac{a}{n}\right) \Rightarrow \left(\frac{a}{p}\right) = \left(\frac{a}{q}\right)</math>, either <math>\textstyle a</math> or <math>\textstyle -a</math> is a [[quadratic residue]] modulo <math>\textstyle n</math>.
 
Therefore, <math>\textstyle r</math> is a square root of <math>\textstyle a</math> or <math>\textstyle -a</math>:
 
<math>
\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}
</math>
 
Moreover (for the case that <math>\textstyle a</math> is a quadratic residue, same idea holds for <math>\textstyle -a</math>):
 
<math>
\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}\right)\left(\pm 1\right)^2 = \left(\frac{t}{n}\right) \\
\end{align}
</math>
 
==Security==
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 <math>\textstyle n</math>, make the choice of <math>\textstyle t</math> uniform and random and moreover include some authenticity checks for <math>\textstyle t</math> (otherwise, an [[adaptive chosen ciphertext attack]] can be mounted by altering packets that transmit a single bit and using the [[Random oracle|oracle]] to observe the effect on the decrypted bit).
 
==Problems==
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 <math>r</math> is the square of <math>a</math> or <math>-a</math>), 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.
 
==References==
<references/>
 
[[Category:Identity-based cryptography]]

Latest revision as of 20:19, 8 January 2015

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