- Cocks IBE scheme
-
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.
Contents
Protocol
Setup
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
.
Extract
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.
Encrypt
To encrypt a bit (coded as
/
)
for
, the user
- chooses random
with
,
- chooses random
with
, different from
,
- computes
and
and
- sends
to the user.
Decrypt
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.
Correctness
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
):
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
, 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).
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 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.
References
- ^ Clifford Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues, Proceedings of the 8th IMA International Conference on Cryptography and Coding, 2001
Categories: - a public RSA-modulus
Wikimedia Foundation. 2010.