24 2. PRELIMINARIES
• Key Generation: e key generation algorithm takes as input an access tree T , the master
key MK, and the public key PK to produce a secret key SK, such that SK can decrypt E iff
T matches .
• Decryption: e decryption algorithm takes as input the ciphertext E, the decryption key
SK and the public key PK and decrypts the ciphertext iff T matches , otherwise it pro-
duces ?.
2.2.5 PROXY RE-ENCRYPTION
Proxy re-encryption allows third parties called proxies to re-encrypt a ciphertext generated using
a secret key so that it can be decrypted using a different public key. Proxy re-encryption (PRE)
was first presented in [120]. is scheme known as the BBS scheme is a very simple PRE scheme
based on Elgamal encryption.
e concept of symmetric key proxy re-encryption (SRE) was explored by Syalim et al.
in [119]. e symmetric proxy re-encryption scheme of [119] makes use of an All Or Nothing
Transform (AONT) along with a symmetric cipher. AONT has the properties that it’s output
is pseudo-random, and the transformation of output back to input requires all the output blocks
be in their correct positions. ese properties are used along with a weak permutation cipher to
develop a secure symmetric proxy re-encryption scheme. For algorithmic details, the reader can
refer to [119]. Proxy re-encryption is used in Chapter 7 for ensuring the confidentiality of code
transferred from the base station to sensor nodes.
2.3 OTHER MATHEMATICAL PRIMITIVES
2.3.1 BILINEAR MAPS
A bilinear map is a function that takes as input elements from two spaces and outputs an element
of a third space. Let G
1
and G
T
be cyclic groups of the prime order q. Typically, G
1
is an elliptic
curve group and G
T
is a finite field. We, therefore, denote G
1
using additive notation and G
T
using multiplicative notation. Let P and Q be two generators of G
1
. A bilinear map then is an
injunctive function e W G
1
G
1
! G
T
, which has the following properties.
• Bilinearity: 8P; Q 2 G
1
; 8a; b 2 Z
q
, such that we have e.aP; bQ/ D e.P; Q/
ab
.
• Non-degeneracy: If P ¤ 0, then e.P; P / ¤ 1.
• Computability: ere exists an efficient algorithm for computing e.P; Q/; 8P; Q 2 G
1
.
Bilinear maps would be used in Chapter 6 for creating an access control scheme based on KP-
ABE that works on aggregate data.
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