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International Journal of Research in Engineering and Science (IJRES)
ISSN (Online): 2320-9364, ISSN (Print): 2320-9356
www.ijres.org Volume 1 Issue 1 ǁ May. 2013 ǁ PP.17-20
www.ijres.org 17 | Page
An Identity Based Encryption Scheme based on Pell’s
Equation With Jacobi Symbol
1
Kondala Rao M, 2
K V S S R S S Sarma, 3
P S Avadhani
1
Department of Computer Science and Systems Engineering, Andhra University, India,
2
Department of Computer Information Science, University of Hyderabad, Hyderabad, India,
3
Department of Computer Science and Systems Engineering Andhra University, Visakhapatnam, India,
Abstract:- In this paper, we propose a novel public key cryptosystem in which the public key of a
subscriber can be chosen to be a publicly known value, such as his identity. Identity based encryption
scheme developed is a public key cryptosystem and it is based on Pell’s equation. This paper
demonstrates algorithms for key generation, encryption and decryption based on Pell’s equation with
Jacobi symbol.
Keywords:- pells equation, Legendre symbol, Jacobi symbol, permanent private secret.
I. INTRODUCTION
In an offline public key system, in order to send encrypted data it is necessary to know the public key
of the recipient. This usually necessitates the holding of directories of public keys. In an identity based system a
user’s public key is a function of his identity (for example his email address), thus avoiding the need for a
separate public key directory. The possibility of such a system was first mentioned by Shamir[4]. But it has
proved difficult to find implementations that are both practical and secure, although recently an implementation
based on elliptic curves has been proposed[3]. This paper describes an identity based cryptosystem which uses
Pell’s equation.
The system has an authority which generates a universally available public M. This modulus is a
product of two primes p and q held privately by the authority. Also, the system will make use of a universally
available secure hash function. Then, if user Alice wishes to register in order to be able to receive encrypted data
she presents her identity (e.g. email address) to the authority. In return she will be given a private key with
properties. A user Bob who wishes to send encrypted data to Alice will be able to do this knowing only Alice’s
public identity and the universal system parameters. There is no need for a public key directory.
There were some methods developed in which administer verifies the information, such as RSA. In
recent years, because of the development of public key cryptography functions, a new method of verification is
accordingly being created. When comparing RSA with Pell’s Equation, Proposed scheme is more secure and
faster than RSA Public Key Cryptography to achieve the same level of security as Pell’s Equation.
II. PROPOSED SCHEME
1.1 Pell’s equation based encryption system
Our scheme is based on Pell’s equation, and to begin with, we briefly introduced this scheme. We can
divide Pell’s schemes into four parts, Pell’s equation, Key Management, Encryption and Decryption. This
scheme also needs a trusted third party can decide the contents of the published information, whereas others can
only download without permission to modify it.
1.2 Pell’s Equation
In number theory, for any constant integer D the equation x2
−D y2
≡ 1 is called the Pell’s equation[1].
This has many applications in various branches of science. The set of all pairs(x, y) describes cyclic group Gp
over the Pell’s equation x2
−D y2
≡ 1 (mod P), where P is an odd prime.
The properties are also found in the group GN over the Pell’s equation x2
−D y2
≡ 1 (mod N), where N is a
product of two primes. This group GN then developed to be a public key crypto scheme based on Pell’s
equations over the ring ZN
*
.
From the group GN, we find a group isomorphism mapping f : GN → ZN
*
such that a solution (x, y) of
the Pell’s equation x2
− D y2
≡ 1 (mod N), can easily be transformed to unique element u in ZN
*
. This implies
that the plain texts/cipher texts in the in the group GN can easily transformed to the corresponding plain
texts/cipher texts in the RSA scheme[2].
An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol
www.ijres.org 18 | Page
Let p be an odd prime and D be a non zero quadratic residue element modulo p, which we denote if
with Fp. Gp the set of solutions (x, y)  Fp  Fp to the Pell equation
x
2
− D y
2
 1 (mod P).
We then define an addition operation “  “ on Gp as follows.
If two pairs ( 1x , 1y ), ( 2x , 2y )  Gp, then the third pair ( 3x , 3y ) can be computed as
( 3x , 3y )  ( 1x , 1y )  ( 2x , 2y )
 ( 1x 2x + D 1y 2y , 1x 2y + 2x 1y ) (mod P).
It is easy to verify that the Gp together with the operation “  “ is an abelian group with the identity
element by (1, 0) and the inverse of the element (x, y) by (x,-y). Further it can be proved that <Gp,  > is a
cyclic group of order p-1. Now we want to prove that the group Gp is isomorphic to Fp*, Where Fp*, is all non-
zero elements of Fp denotes a multiplicative group of Fp.
Theorem1:
Gp together with the operation “  “ is a cyclic group of order p-1. Now we want to prove that the group Gp is
isomorphic to Fp
*
, where Fp
*
denotes a multiplicative group of Fp.
Theorem2:
Two groups Gp and Fp
*
are isomorphic Now we define another operation “  ” as follows :
( , ) ( , ) ( , ) ......... ( , )i x y x y x y x y     i times over Gp
.
If ( ix , y i ) = i  ( ix , y i ) , we expand the above expression and have
ix =
0 k i
kiseven
 
 ( i
k )D
/2k  
x i-k
yk
;
iy =
0 k i
kisodd
 
 ( i
k )D
/2k  
x i-k
yk
According to the definition of the mapping f, we have
f (( ix , y i ))  ix - a iy

0 k i
kiseven
 
 ( i
k )D
/2k  
x i-k
yk
- a
0 k i
kisodd
 
 ( i
k )D
/2k  
x i-k
yk

0 k i
kiseven
 
 ( i
k )D
/2k  
x i-k
yk
+
0 k i
kisodd
 
 ( i
k )D
/2k  
x i-k
(–a y)k
 (x-ay)i
Because Gp is a cyclic group of order p-1, we have that if k  1 (mod P-1), then
(x,y) = k  (x, y), for all (x, y)  Gp
Let N be a product of two large primes p and q. Zn
*
denotes a multiplicative group of ZN. From the above
theorem, it is easy to develop the following theorem.
Theorem3:
The mapping f : GN  ZN
*
satisfying f ((1,0))  ( 1 mod N ), f ((x, y))  x - ay (mod N), where (x, y) 
GN and a2
 D (mod N), is a group isomorphism . Its inverse mapping f -1
(1)  (1,0) (mod N), f -1
(u)  ((u
+ u-1
)/2, (u-1
- u)/2a (mod N), where u  ZN
*
.
Theorem 4:
If ( iX , y i ) = i  (x, y), over GN , we have ix - a iy ≡ (x - ay)i
( mod N).
Theorem 5:
If k ≡ 1 mod (l .c .m (p-1, q-1)), we have (x, y) = k  (x, y), for all (x, y)  GN.
An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol
www.ijres.org 19 | Page
Legendre symbol
Let a be an integer and p > 2 a prime, Define the Legendre Symbol (a/p)=0,1,-1 as follows
a
p
=
0 if p divides a
1 if a is QR mod p
−1 if a NQR mod p
Jacobi Symbol
If p is a positive odd integer with prime factorization
P =
1
r
i
 pi
ai
.
The Jacobi symbol ( n / p) is defined for all integers n by the equation
( n / p) =
1
r
i
 (n / pi)ai
where (n / pi) is the legendre symbol.
1.3 Key Management
In this step, the Bob chooses the Secrete Key Generation[5], and computes and publishes some
auxiliary information on the bulletin. The Bob will perform the following steps:
1.3.1 Key Generation:
Assume that the Bob to send messages E (cipher text) and Alice case receive the messages E and M.
1. The Bob chooses two primes p and q (p  q)
2. The Bob compute
p
 = p mod 4 and
q
 =q mod 4 where
p
 ,
q
  {1,-1}
3. The Bob find non square integer D > 0 such that Legender symbols
(D/p) = -
p
 and (D/p) = -
q

4. The Bob compute n = p * q and m= (p+
p
 ) (q+
q
 ) / 4
5.
The Bob select a integer value for S such that the jacobi symbol
( ( S2
- D)/n) = -1. As there are closely to ( ) / 2n  such values of S.
6. The Bob select a integer value for e such that ( e, m ) =1. and makes {n, e, S, D} as public.
7. The Bob compute d * e  (m+1)/2 mod m for d and keeps as private key
8. D is Key for the Cipher Text.
1.4 Encryption:
The Bob changes the messages M to E.
1. The Bob get the M be a message to communicate / encrypt
2. The Bob compute j1 = (M2
-D)/n
3. The Bob compare If j1 = 1 go to step (4) else go to step (6)
4. The Bob compute x  (M2
+D) / ( M2
-D) (mod n )
And y  2M / (M2
-D) (mod n)
5. Go to step (8)
6. The Bob compare If j1 = -1 go to step (7) else go to stop().
7. The Bob compute x  ((M2
+D)(S2
+ D) + 4DMS) / (( M2
-D) (S2
- D) ) (mod n )
And y  (2S (M2
+D) + 2M (S2
+ D)) / (( M2
-D) (S2
- D) ) (mod n )
8. The Bob compute j2 = x (mod 2) where j2 {0, 1}
(nothing that x2
- Dy2
=1(mod n) for these values of x,
y and assume
that (y, n) =1)
9. The Bob Put Xl = x and Yl = y
10. The Bob compute (Xl+1, Xl) (mod n ) such that
if i  j
An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol
www.ijres.org 20 | Page
Xi+j = Xj + D Yi Yj and Yi+j = Xi Yj + Xj Yi
if i = j
X2i = Xi
2
+D Yi
2
or 2 Xi
2
– 1 and Y2i = 2 Xi Yi
11. The Bob compute E = DYXl (Xl+1 – x Xl)-1
mod n (here E is the cipher text) with 0<E<n
12. Send the {E, j1, j2 ) = cipher text.
1.5 Decryption
After receiving the ciphertext (E, j1,j2), the Alice checks that x2
−D y2
≡ 1
If yes, Alice can continue
1. The Alice compute
X2l = (E2
+ D) / (E2
- D) (mod n) and
i. Y2l = (2E / (E2
- D)) (mod n )
2. The Alice compute Xd (X2e)  X2de (x)(mod n) and
i. Xd+1 (X2e)  X2de+2e (x)(mod n)
ii. We have X2ed =  x (mod n) and j2  x (mod 2)
3. The Alice compute  and therefore determines x (mod n)
And find y   Y 2de   (X2de+2e - X2e Xd ) / DX2e mod n
we have t  x + y D mod n
Compute t1
such that
t1
= t if j1 = 1 else if j1 = -1
t1
= t (S- D ) / (S + D )
And t1
= (M + D ) / (M - D ) mod n
4. The Alice compute
M  (t1
+1) ( D ) / ( t1
-1) (mod n)
The Bob sends message (M) to key generation function that produces a secure private key (D). This
private key is then encrypted with public key cryptosystem using the Bob’s private key to form the result. Both
the message and the result are prepended and then transmitted. The Alice takes the message (M) and produces a
secure private key (D). The Alice also decrypts the result using the Bob’s public key. If the calculated secure
private key (D) matches the decrypted results, the result is accepted as valid. Because only the Bob knows the
private key and only the Bob could have produced a valid result.
III. CONCLUSION
In this paper, we have proposed an Identity based encryption scheme using Pell’s equation with
Legendre and Jacobi symbols. This scheme is a multi-use scheme, which means every Alice only needs to keep
one private secrete. The Alice just use the public information and their own private information to get the
information back. When then identity is changed, the Bob only needs to adjust the private key. And one of the
benefits of this scheme is that the Bob will not ask for some extra information from all the Alice while in the
information updating stage. We also show how to verify the information which comes from other Alice.
Furthermore, because this scheme is constructed on the pells equation, we attain a much higher level of security
comparing with the original public key problem in the same bits length.
REFERENCES
[1]. Chen C.Y., Chang C.C. Yang W.P.,"Fast RSA type cryptosystem based on Pell equation”, Proceeding
of International Conf. On Cryptology and Information Security Taiwan, Dec.1-5, 1996
[2]. C Cocks Split “Generation of RSA Parameters with multiple participants” proceeding of 6th
IMA
conference on Cryptography and Coding, Spring LNCS 1355.
[3]. A. Shamir, “Identity- Based Cryptosystems and Signature schemes advances in Cryptology” –
Proceeding of Crypto’84.
[4]. D. Boneh and M. Franklin, “Identity-based encryption from the Weil paring,” Advances in Cryptology-
Crypto’2001, Lecture Notes on Computer Science 2139, Springer-Verlag (2001), pp.213-229.
[5]. Michael J.JacobSon and Jr.Hugh C.Williams 2009 “Solving the Pell Equation”, pages 353-359.

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D0111720

  • 1. International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 1 Issue 1 ǁ May. 2013 ǁ PP.17-20 www.ijres.org 17 | Page An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol 1 Kondala Rao M, 2 K V S S R S S Sarma, 3 P S Avadhani 1 Department of Computer Science and Systems Engineering, Andhra University, India, 2 Department of Computer Information Science, University of Hyderabad, Hyderabad, India, 3 Department of Computer Science and Systems Engineering Andhra University, Visakhapatnam, India, Abstract:- In this paper, we propose a novel public key cryptosystem in which the public key of a subscriber can be chosen to be a publicly known value, such as his identity. Identity based encryption scheme developed is a public key cryptosystem and it is based on Pell’s equation. This paper demonstrates algorithms for key generation, encryption and decryption based on Pell’s equation with Jacobi symbol. Keywords:- pells equation, Legendre symbol, Jacobi symbol, permanent private secret. I. INTRODUCTION In an offline public key system, in order to send encrypted data it is necessary to know the public key of the recipient. This usually necessitates the holding of directories of public keys. In an identity based system a user’s public key is a function of his identity (for example his email address), thus avoiding the need for a separate public key directory. The possibility of such a system was first mentioned by Shamir[4]. But it has proved difficult to find implementations that are both practical and secure, although recently an implementation based on elliptic curves has been proposed[3]. This paper describes an identity based cryptosystem which uses Pell’s equation. The system has an authority which generates a universally available public M. This modulus is a product of two primes p and q held privately by the authority. Also, the system will make use of a universally available secure hash function. Then, if user Alice wishes to register in order to be able to receive encrypted data she presents her identity (e.g. email address) to the authority. In return she will be given a private key with properties. A user Bob who wishes to send encrypted data to Alice will be able to do this knowing only Alice’s public identity and the universal system parameters. There is no need for a public key directory. There were some methods developed in which administer verifies the information, such as RSA. In recent years, because of the development of public key cryptography functions, a new method of verification is accordingly being created. When comparing RSA with Pell’s Equation, Proposed scheme is more secure and faster than RSA Public Key Cryptography to achieve the same level of security as Pell’s Equation. II. PROPOSED SCHEME 1.1 Pell’s equation based encryption system Our scheme is based on Pell’s equation, and to begin with, we briefly introduced this scheme. We can divide Pell’s schemes into four parts, Pell’s equation, Key Management, Encryption and Decryption. This scheme also needs a trusted third party can decide the contents of the published information, whereas others can only download without permission to modify it. 1.2 Pell’s Equation In number theory, for any constant integer D the equation x2 −D y2 ≡ 1 is called the Pell’s equation[1]. This has many applications in various branches of science. The set of all pairs(x, y) describes cyclic group Gp over the Pell’s equation x2 −D y2 ≡ 1 (mod P), where P is an odd prime. The properties are also found in the group GN over the Pell’s equation x2 −D y2 ≡ 1 (mod N), where N is a product of two primes. This group GN then developed to be a public key crypto scheme based on Pell’s equations over the ring ZN * . From the group GN, we find a group isomorphism mapping f : GN → ZN * such that a solution (x, y) of the Pell’s equation x2 − D y2 ≡ 1 (mod N), can easily be transformed to unique element u in ZN * . This implies that the plain texts/cipher texts in the in the group GN can easily transformed to the corresponding plain texts/cipher texts in the RSA scheme[2].
  • 2. An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol www.ijres.org 18 | Page Let p be an odd prime and D be a non zero quadratic residue element modulo p, which we denote if with Fp. Gp the set of solutions (x, y)  Fp  Fp to the Pell equation x 2 − D y 2  1 (mod P). We then define an addition operation “  “ on Gp as follows. If two pairs ( 1x , 1y ), ( 2x , 2y )  Gp, then the third pair ( 3x , 3y ) can be computed as ( 3x , 3y )  ( 1x , 1y )  ( 2x , 2y )  ( 1x 2x + D 1y 2y , 1x 2y + 2x 1y ) (mod P). It is easy to verify that the Gp together with the operation “  “ is an abelian group with the identity element by (1, 0) and the inverse of the element (x, y) by (x,-y). Further it can be proved that <Gp,  > is a cyclic group of order p-1. Now we want to prove that the group Gp is isomorphic to Fp*, Where Fp*, is all non- zero elements of Fp denotes a multiplicative group of Fp. Theorem1: Gp together with the operation “  “ is a cyclic group of order p-1. Now we want to prove that the group Gp is isomorphic to Fp * , where Fp * denotes a multiplicative group of Fp. Theorem2: Two groups Gp and Fp * are isomorphic Now we define another operation “  ” as follows : ( , ) ( , ) ( , ) ......... ( , )i x y x y x y x y     i times over Gp . If ( ix , y i ) = i  ( ix , y i ) , we expand the above expression and have ix = 0 k i kiseven    ( i k )D /2k   x i-k yk ; iy = 0 k i kisodd    ( i k )D /2k   x i-k yk According to the definition of the mapping f, we have f (( ix , y i ))  ix - a iy  0 k i kiseven    ( i k )D /2k   x i-k yk - a 0 k i kisodd    ( i k )D /2k   x i-k yk  0 k i kiseven    ( i k )D /2k   x i-k yk + 0 k i kisodd    ( i k )D /2k   x i-k (–a y)k  (x-ay)i Because Gp is a cyclic group of order p-1, we have that if k  1 (mod P-1), then (x,y) = k  (x, y), for all (x, y)  Gp Let N be a product of two large primes p and q. Zn * denotes a multiplicative group of ZN. From the above theorem, it is easy to develop the following theorem. Theorem3: The mapping f : GN  ZN * satisfying f ((1,0))  ( 1 mod N ), f ((x, y))  x - ay (mod N), where (x, y)  GN and a2  D (mod N), is a group isomorphism . Its inverse mapping f -1 (1)  (1,0) (mod N), f -1 (u)  ((u + u-1 )/2, (u-1 - u)/2a (mod N), where u  ZN * . Theorem 4: If ( iX , y i ) = i  (x, y), over GN , we have ix - a iy ≡ (x - ay)i ( mod N). Theorem 5: If k ≡ 1 mod (l .c .m (p-1, q-1)), we have (x, y) = k  (x, y), for all (x, y)  GN.
  • 3. An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol www.ijres.org 19 | Page Legendre symbol Let a be an integer and p > 2 a prime, Define the Legendre Symbol (a/p)=0,1,-1 as follows a p = 0 if p divides a 1 if a is QR mod p −1 if a NQR mod p Jacobi Symbol If p is a positive odd integer with prime factorization P = 1 r i  pi ai . The Jacobi symbol ( n / p) is defined for all integers n by the equation ( n / p) = 1 r i  (n / pi)ai where (n / pi) is the legendre symbol. 1.3 Key Management In this step, the Bob chooses the Secrete Key Generation[5], and computes and publishes some auxiliary information on the bulletin. The Bob will perform the following steps: 1.3.1 Key Generation: Assume that the Bob to send messages E (cipher text) and Alice case receive the messages E and M. 1. The Bob chooses two primes p and q (p  q) 2. The Bob compute p  = p mod 4 and q  =q mod 4 where p  , q   {1,-1} 3. The Bob find non square integer D > 0 such that Legender symbols (D/p) = - p  and (D/p) = - q  4. The Bob compute n = p * q and m= (p+ p  ) (q+ q  ) / 4 5. The Bob select a integer value for S such that the jacobi symbol ( ( S2 - D)/n) = -1. As there are closely to ( ) / 2n  such values of S. 6. The Bob select a integer value for e such that ( e, m ) =1. and makes {n, e, S, D} as public. 7. The Bob compute d * e  (m+1)/2 mod m for d and keeps as private key 8. D is Key for the Cipher Text. 1.4 Encryption: The Bob changes the messages M to E. 1. The Bob get the M be a message to communicate / encrypt 2. The Bob compute j1 = (M2 -D)/n 3. The Bob compare If j1 = 1 go to step (4) else go to step (6) 4. The Bob compute x  (M2 +D) / ( M2 -D) (mod n ) And y  2M / (M2 -D) (mod n) 5. Go to step (8) 6. The Bob compare If j1 = -1 go to step (7) else go to stop(). 7. The Bob compute x  ((M2 +D)(S2 + D) + 4DMS) / (( M2 -D) (S2 - D) ) (mod n ) And y  (2S (M2 +D) + 2M (S2 + D)) / (( M2 -D) (S2 - D) ) (mod n ) 8. The Bob compute j2 = x (mod 2) where j2 {0, 1} (nothing that x2 - Dy2 =1(mod n) for these values of x, y and assume that (y, n) =1) 9. The Bob Put Xl = x and Yl = y 10. The Bob compute (Xl+1, Xl) (mod n ) such that if i  j
  • 4. An Identity Based Encryption Scheme based on Pell’s Equation With Jacobi Symbol www.ijres.org 20 | Page Xi+j = Xj + D Yi Yj and Yi+j = Xi Yj + Xj Yi if i = j X2i = Xi 2 +D Yi 2 or 2 Xi 2 – 1 and Y2i = 2 Xi Yi 11. The Bob compute E = DYXl (Xl+1 – x Xl)-1 mod n (here E is the cipher text) with 0<E<n 12. Send the {E, j1, j2 ) = cipher text. 1.5 Decryption After receiving the ciphertext (E, j1,j2), the Alice checks that x2 −D y2 ≡ 1 If yes, Alice can continue 1. The Alice compute X2l = (E2 + D) / (E2 - D) (mod n) and i. Y2l = (2E / (E2 - D)) (mod n ) 2. The Alice compute Xd (X2e)  X2de (x)(mod n) and i. Xd+1 (X2e)  X2de+2e (x)(mod n) ii. We have X2ed =  x (mod n) and j2  x (mod 2) 3. The Alice compute  and therefore determines x (mod n) And find y   Y 2de   (X2de+2e - X2e Xd ) / DX2e mod n we have t  x + y D mod n Compute t1 such that t1 = t if j1 = 1 else if j1 = -1 t1 = t (S- D ) / (S + D ) And t1 = (M + D ) / (M - D ) mod n 4. The Alice compute M  (t1 +1) ( D ) / ( t1 -1) (mod n) The Bob sends message (M) to key generation function that produces a secure private key (D). This private key is then encrypted with public key cryptosystem using the Bob’s private key to form the result. Both the message and the result are prepended and then transmitted. The Alice takes the message (M) and produces a secure private key (D). The Alice also decrypts the result using the Bob’s public key. If the calculated secure private key (D) matches the decrypted results, the result is accepted as valid. Because only the Bob knows the private key and only the Bob could have produced a valid result. III. CONCLUSION In this paper, we have proposed an Identity based encryption scheme using Pell’s equation with Legendre and Jacobi symbols. This scheme is a multi-use scheme, which means every Alice only needs to keep one private secrete. The Alice just use the public information and their own private information to get the information back. When then identity is changed, the Bob only needs to adjust the private key. And one of the benefits of this scheme is that the Bob will not ask for some extra information from all the Alice while in the information updating stage. We also show how to verify the information which comes from other Alice. Furthermore, because this scheme is constructed on the pells equation, we attain a much higher level of security comparing with the original public key problem in the same bits length. REFERENCES [1]. Chen C.Y., Chang C.C. Yang W.P.,"Fast RSA type cryptosystem based on Pell equation”, Proceeding of International Conf. On Cryptology and Information Security Taiwan, Dec.1-5, 1996 [2]. C Cocks Split “Generation of RSA Parameters with multiple participants” proceeding of 6th IMA conference on Cryptography and Coding, Spring LNCS 1355. [3]. A. Shamir, “Identity- Based Cryptosystems and Signature schemes advances in Cryptology” – Proceeding of Crypto’84. [4]. D. Boneh and M. Franklin, “Identity-based encryption from the Weil paring,” Advances in Cryptology- Crypto’2001, Lecture Notes on Computer Science 2139, Springer-Verlag (2001), pp.213-229. [5]. Michael J.JacobSon and Jr.Hugh C.Williams 2009 “Solving the Pell Equation”, pages 353-359.