1. An Efficient Identity-based Cryptosystem for
End-to-end Mobile Security
IEEE Transactions on Wireless Communications, 2006
Jing-Shyang Hwu, Rong-Jaye Chen, Yi-Bing Lin
Presented by Jang Chol Soon

2. Contents Introduction Background
ID-based Encryption
Elliptic Curves
Divisor
Weil Pairing
Efficient Computation for Weil Pairing
Point Halving
Halve-and-Add Method for Weil Pairing
Performance Evaluation
Application System
Conclusions

3. Introduction Mobile security
Mobile operators have provided security protection including
authentication and encryption for circuit-switched voice services.
Wireless data services(e.g. mobile banking) are likely to be offered
by third parties(e.g. banks)
The third parties can’t trust the security mechanisms of mobile operators.
: their own solution for end-to-end security.
End-to-end security mechanisms in mobile services
: public-key cryptosystem
The main concern in public-key cryptosystem
: the authenticity of public key ⇒ “certificate”
The certificate is issued by a trusted third party consisting of the user name
and his public key.

4. Introduction ID-based cryptography In 1984, Shamir
The public key of a user can be derived from public information
that uniquely identifies the user. (e.g. , telephone number)
The first complete ID-based cryptosystem
· In 2001, Boneh and Franklin
· use a bilinear map(Weil pairing) over elliptic curves
Major advantages
· No certificate
· Users need not memorize extra public keys.
Drawback
· Overhead for the pairing computing

5. Background Background A. ID-based Encryption (scheme)
B. Elliptic Curves
C. Divisor
D. Weil pairing

6. Background A. ID-based Encryption (IBE) scheme
use a bilinear map called Weil pairing over elliptic curves.
bilinear map
· transform a pair of elements(P, Q) in group G1
· send the pair to an element in group G2 in a way that satisfies
some properties (bilinearity: It should be linear in each entry of the pair.)
Weil pairing on elliptic curves is selected as the bilinear map
· G1 : the elliptic curve group →
· G2 : the multiplicative group →
The decryption procedure yields the correct message
because of the bilinearity of the Weil pairing.

7. Background A. ID-based Encryption (IBE) scheme
The security level depends on the size of the finite field
because the scheme is constructed on an elliptic curve.
ex) an elliptic curve over 163-bit finite field = 1024-bit RSA
The most significant overhead is the computation of Weil pairing.
Sender
Receiver
PKG
Weil pairing
Elliptic curves

8. Background B. Elliptic Curves p : a prime larger than 3
: infinity point → the identity element
An elliptic curve over a finite field of size p noted by GE(p)
are
The group operation is written as addition
instead of multiplication.
λ : the slope of the line passing through P and Q

9. Background
C. Divisor
A useful device for keeping track of the zeros and poles of relational
functions
defined as a formal sum of points on elliptic curve group
: a non-zero integer that specifies the zero/pole property of point P
and its respective order.
A formula for adding two divisors in canonical form
· provide a method of finding a rational function f
· critical for computing Weil pairing

10. Background D. Weil Pairing Weil pairing e(P, Q) is defined as follows
The Weil pairing has the bilinearity property.
The first algorithm for e(P, Q) computation is Miller’s Algorithm.

11. one field multiplication
Efficient Computation for Weil Pairing
Point halving algorithm
proposed by Knudsen
Fast computation for scalar multiplication on elliptic curve
one field multiplication
Three operations

12. Halve-and-Add Method for Weil Pairing
Method for the evaluation of rational functions used in the Miller’s algorithm
To take advantage of point halving
· require 1 inversion, 3 multiplications,
1 squaring, and 1 square root computing
· advantage over the doubling

13. Performance Evaluation
By using halving, save
· 2n inversions
· 2n-3k multiplications
· n squaring at the cost of
solving n quadratic equation
· 2n square roots
· n trace computing

14. Performance Evaluation

15. Application System ID-based End-to-End Mobile Encryption System
typically based on Public-key cryptosystem
Traditional public-key cryptosystem
· The sender has to request the receiver’s public-key and verify its validity
before encrypting a message.
· When the receiver is off-line,
the sender can not communication with the receiver to request
the public-key
ID-based cryptosystem
· The sender can user the receiver’s ID(i.e., telephone number) as a public
key without any request and verification.
· Even if the receiver’s device is power-off,
the sender can still send an encrypted short message.

16. Bob’s phone number (public-key)
Application System
ID-based End-to-End Mobile Encryption System
Private Key Generator
(PKG)
ID=
(1)
SIM Card KR
Subscription
time
Alice
Cipher
Bob( )
SIM Card
SIM Card
(5) KR
GSM Network
ID-based
Decryption
ID-based
Decryption
(6)
Message
(2)
ID-based
Encryption
ID-based
Encryption
Message
(3)
Cipher
Bob’s phone number (public-key)
( )

17. Conclusion Conclusion
An efficient ID-based cryptography scheme for end-to-end mobile
security system
A fast method for computing the Weil pairing using point halving algorithm
: λ-representation in a normal basis
Contribution
to apply point halving algorithm to the ID-based scheme
an efficient approach to compute the rational function evaluation
algorithm