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Data Security 1 El_Gamal Cryptography

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Data Security2 Introduction El_Gamal is a public-key cryptosystem technique El_Gamal is a public-key cryptosystem technique was designed by Dr. Taher Elgamal. was designed by Dr. Taher Elgamal. El_Gamal depends on the one way function, means that the encryption and decryption are done in separate functions. El_Gamal depends on the one way function, means that the encryption and decryption are done in separate functions.

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Data Security3 Introduction The encryption process requires two modular exponentiations (extra time). The encryption process requires two modular exponentiations (extra time). A disadvantage of El_Gamal encryption is that there is message expansion by a factor of 2. That is, the ciphertext is twice as long as the corresponding plaintext. A disadvantage of El_Gamal encryption is that there is message expansion by a factor of 2. That is, the ciphertext is twice as long as the corresponding plaintext.

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Data Security4 Key generation Receiver A must do the following: 1- Generate a large random prime number (p) 2- Choose a generator number (a) {show in slide 7 } 3- Choose an integer (x) less than (p-2),as secret number. number.

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Data Security5 Key generation 4- Compute (d) where d= a x mod p d= a x mod p 5- Determine the public key (p, a, d) and the private key (x) and the private key (x)

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Data Security6 Key generation Example : let p = 11 and a = 2 and x = 5 let p = 11 and a = 2 and x = 5 calculate d = 2 5 mod 11 = 10 calculate d = 2 5 mod 11 = 10 public key = (11,2,10) public key = (11,2,10) private key = (5) private key = (5)

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Data Security7 Generator number How to test (a) generator or not : 1- (a) must be between 1 and p-1 2- Find Ø = p-1 3- Find the all factors of Ø {f1,f2,….,fn} – { 1 }

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Data Security8 Generator number 4- Find {q1,q2,…..,qn} where qi = fi qi = fi for the redundant factors for the redundant factors qi = fi freq qi = fi freq 5- (a) generator number if and only if wi= a Ø/qi mode p <> 1, for all qi wi= a Ø/qi mode p <> 1, for all qi

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Data Security9 Generator number Example 1 : let p= 11, a=2,test a is generator number or not ? let p= 11, a=2,test a is generator number or not ? sol: sol: Ø= p-1 = 10, factors of 10 = {2, 5} Ø= p-1 = 10, factors of 10 = {2, 5} q1 = 2,q2 = 5 q1 = 2,q2 = 5 w1 = 2 10/2 mod 11 = 10 <> 1 w1 = 2 10/2 mod 11 = 10 <> 1 w2 = 2 10/5 mod 11 = 4 <> 1 w2 = 2 10/5 mod 11 = 4 <> 1 i.e a generator number. i.e a generator number.

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Data Security10 Generator number Example 2 : let p= 11, a=3,test a is generator number or not ? let p= 11, a=3,test a is generator number or not ? sol: sol: Ø= p-1 = 10, factors of 10 = {2, 5} Ø= p-1 = 10, factors of 10 = {2, 5} q1 = 2,q2 = 5 q1 = 2,q2 = 5 w1 = 3 10/2 mod 11 = 1== 1 w1 = 3 10/2 mod 11 = 1== 1 w2 = 3 10/5 mod 11 = 9 <> 1 w2 = 3 10/5 mod 11 = 9 <> 1 i.e a not generator number. i.e a not generator number.

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Data Security11 Generator number Example 3 : let p= 41, a=2,test a is generator number or not ? let p= 41, a=2,test a is generator number or not ? sol: sol: Ø= p-1 = 40, factors of 40 = {2, 2, 2, 5} Ø= p-1 = 40, factors of 40 = {2, 2, 2, 5} q1 = 2 1 = 2,q2 = 2 2 = 4,q3 = 2 3 = 8 q1 = 2 1 = 2,q2 = 2 2 = 4,q3 = 2 3 = 8 q4 = 5 q4 = 5 w1 = 2 40/2 mod 41 = 0.98 <> 1 w1 = 2 40/2 mod 41 = 0.98 <> 1 w2 = 2 40/4 mod 41 = 40 <> 1 w2 = 2 40/4 mod 41 = 40 <> 1

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Data Security12 Generator number w2 = 2 40/8 mod 41 = 32 <> 1 w2 = 2 40/5 mod 41 = 10 <> 1 i.e a generator number

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Data Security13 Encryption Sender B must do the following : 1- Obtain the public key (p, a, d ) from 1- Obtain the public key (p, a, d ) from the receiver A. the receiver A. 2- Choose an integer k such that : 2- Choose an integer k such that : 1 < k < p-2 1 < k < p-2

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Data Security14 Encryption 3- Represent the plaintext as an integer m where 0 < m < p-1 4- compute (y) as follows : y = a k mod p y = a k mod p 5- compute (z) as follows : z = (d k * m ) mod p z = (d k * m ) mod p

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Data Security15 Encryption 6- Find the ciphertext (C) as follows : C= ( y, z ) C= ( y, z ) 7- The sender B send C to The receiver A.

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Data Security16 Decryption Receiver A must do the following : 1- Obtain the ciphertext (C) from B. 2- compute (r) as follows : r = y p-1-x mod p r = y p-1-x mod p 3- Recover the plaintext as follows: m = ( r * z ) mod p m = ( r * z ) mod p

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Data Security17 Example Let p = 11 and a generator number = 2 and select integer number x = 5 calculate d = 2 5 mod 11 = 10 calculate d = 2 5 mod 11 = 10Then public key = ( 11, 2, 10) public key = ( 11, 2, 10) private key = (5) private key = (5)

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Data Security18 Example Plaintext = Age Represent the plaintext as integer value as follows: The new plaintext = ( 1 7 5 )

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Data Security19 Example Encryption (sender): y = a k mod p, z = (d k * m ) mod p y = a k mod p, z = (d k * m ) mod p Choose an random integer value k = 6 Choose an random integer value k = 6 y A = 2 6 mod 11 = 9 y A = 2 6 mod 11 = 9 z A = (10 6 *1) mod 11 = 1 z A = (10 6 *1) mod 11 = 1

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Data Security20 Example Choose an random integer value k = 4 y g = 2 4 mod 11 = 5 y g = 2 4 mod 11 = 5 z g = (10 4 *7) mod 11 = 7 z g = (10 4 *7) mod 11 = 7 Choose an random integer value k = 7 y e = 2 7 mod 11 = 7 y e = 2 7 mod 11 = 7 z e = (10 7 *5) mod 11 = 6 z e = (10 7 *5) mod 11 = 6 Ciphertext = (9,1) (5,7) (7,6) The sender B send the ciphertext to the receiver A.

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Data Security21 Example The receiver decrypt the ciphertext as follows : Compute (r) and (m) where Compute (r) and (m) where r = y p-1-x mod p, m = ( r * z ) mod p r = y p-1-x mod p, m = ( r * z ) mod p r1= 9 11-1-5 mod 11 = 1 r1= 9 11-1-5 mod 11 = 1 m1= (1*1) mod 11= 1 m1= (1*1) mod 11= 1

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Data Security22 Example r1= 5 11-1-5 mod 11 = 1 m2 = ( 1 * 7 ) mod 11 = 7 r1= 7 11-1-5 mod 11 = 10 m3 = ( 10 * 6 ) mod 11 = 5

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Data Security23 Example The receiver find the plaintext ( 1 7 5 ) Convert the plaintext to letters = Age

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Data Security24 End For more information about EL_Gamal cryptosystem please visit : For more information about EL_Gamal cryptosystem please visit : http://en.wikipedia.org/wiki/Taher_ElGamal http://en.wikipedia.org/wiki/Taher_ElGamal

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