Bob and Alice have agreed that Bob would answer Alice's invitation using ElGamal with the following parameters: ( prime p = 29, e1= 3, d = 5 and the random r = 7 ) find first the set of public and private keys. Bob replies in pairs of C1,C2 as follows: (12, 27), (12, 19), (12, 13), (12, 22), (12, 0), (12, 2), (12, 25), (12, 19), (12, 1), (12, 22), (12, 3), (12, 23), (12, 1), (12, 4). Please decipher the response that Bob sent to Alice.

540, 380,260,440, 0, 40, 500, 380, 20, 440, 60, 460, 20 and 80.

Explanation:

So, we are given the following parameters or data or information which is going to assist us in solving the question above, they are;

(1). "prime p = 29, e1= 3, d = 5 and the random r = 7"

(2). C1C2 reply; "(12, 27), (12, 19), (12, 13), (12, 22), (12, 0), (12, 2), (12, 25), (12, 19), (12, 1), (12, 22), (12, 3), (12, 23), (12, 1), (12, 4)".

So, let us delve into the solution to the question;

Step one: determine the primitive modulo 29.

These are; 2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27.

Step two: Compute V = k ^c mod p.

Say k = 2.

Then;

V = 2^7 mod 29 = 128 mod 29.

V = 12.

Step three: determine the Public key.

Thus, (p,g,y) = (29,2,12)

Private key = c = 7.

Step four: decipher.

Thus for each code pair we will decided it by using the formula below;

(1). (12,27).

W = j × b^(p - 1 - c) mod p.

W= 27 × 12^(29 -1 -7) mod 29. = 540

(2). (12, 19).

19 × 12^(29 - 1 - 7) mod 29.

( 12^(29 - 1 - 7) mod 29 = 20).

= 19 × 20 = 380.

(3).(12, 13) = 13× 20 = 260.

(4). (12, 22) = 22 × 20 = 440

(5). (12, 0) = 0 × 20 = 0.

(6). (12, 2) = 2× 20= 40.

(7). (12, 25) = 25 × 20 = 500.

(8). (12, 19) = 19 × 20 = 380.

(9).(12, 1) = 1 × 20 = 20.

(10). (12, 22) = 22 × 20 = 440.

(11). (12, 3) = 3× 20 = 60.

(13). (12, 23) = 23 × 20 = 460.

(14). (12, 1) =1 × 20 = 20.

(15). (12, 4) = 4 × 20 = 80.

540, 380,260,440, 0, 40, 500, 380, 20, 440, 60, 460, 20 and 80.

Explanation:

So, we are given the following parameters or data or information which is going to assist us in solving the question above, they are;

(1). "prime p = 29, e1= 3, d = 5 and the random r = 7"

(2). C1C2 reply; "(12, 27), (12, 19), (12, 13), (12, 22), (12, 0), (12, 2), (12, 25), (12, 19), (12, 1), (12, 22), (12, 3), (12, 23), (12, 1), (12, 4)".

So, let us delve into the solution to the question;

Step one: determine the primitive modulo 29.

These are; 2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27.

Step two: Compute V = k ^c mod p.

Say k = 2.

Then;

V = 2^7 mod 29 = 128 mod 29.

V = 12.

Step three: determine the Public key.

Thus, (p,g,y) = (29,2,12)

Private key = c = 7.

Step four: decipher.

Thus for each code pair we will decided it by using the formula below;

(1). (12,27).

W = j × b^(p - 1 - c) mod p.

W= 27 × 12^(29 -1 -7) mod 29. = 540

(2). (12, 19).

19 × 12^(29 - 1 - 7) mod 29.

( 12^(29 - 1 - 7) mod 29 = 20).

= 19 × 20 = 380.

(3).(12, 13) = 13× 20 = 260.

(4). (12, 22) = 22 × 20 = 440

(5). (12, 0) = 0 × 20 = 0.

(6). (12, 2) = 2× 20= 40.

(7). (12, 25) = 25 × 20 = 500.

(8). (12, 19) = 19 × 20 = 380.

(9).(12, 1) = 1 × 20 = 20.

(10). (12, 22) = 22 × 20 = 440.

(11). (12, 3) = 3× 20 = 60.

(13). (12, 23) = 23 × 20 = 460.

(14). (12, 1) =1 × 20 = 20.

(15). (12, 4) = 4 × 20 = 80.

Explanation:

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