Question

5. Consider the RSA encryption scheme, Alice wants to send a message to Bob. Both Alice and Bob have p= 17,9 = 19. Alice has e=31 and Bob has e=29. a. What is the public key pair used in the transmission? 2 marks b. What is the secret key pair used in the transmission? 4 marks c. Encrypt the message m=111. 4 marks d. Decrypt the resulting ciphertext. 4 marks e. What's the security problem between Alice and Bob? How to solve this problem. Explain your answer. 3 marks

Answer 1

Given:

p=17

q=19

a) below given are public keys pairs

e=31 ( alice public key)

<31,323>

e=29 ( bob public key)

<29,323>

b) we have given public keys but we need to find private keys.

Alice private key:

n=pq =17*19 = 323

Phi(n) = (p-1)(q-1)

=> 16*18 = 288

Let d is private key

=> (e*d)mod phi(n) = 1

=> (31*d) mod 288 = 1

Substituting d=511, above equation is satisfied

So, private key pair of alice is <511,323>

Bob private key :

Similarly,

=> (e*d)modphi(n) = 1

=> (29*d) mod 288 =1

Substituting d= 437, above equation is satisfied.

So, private key pair of bob is <437,323>

c)

Bob encrypt using public key of alice :

M^e mod n

111^31 mod 323 = 144144144

Alice encrypt using bob public key :

111^29 mod 323 = 121121121

d) decryption

Alice side:

144144144^511 mod 323 = 111

Bob side:

121121121^437 mod 323 =111

e)

Alice and bob has chosen p and q same which can be a security problem. Alice and bob should choose different p and q.

Values of p and q are very small, so its vulnerable to attackers. Value of p and q should be large enough.