Suppose to build RSA crypto system you picked primes "p" and "q" as 3 and 7 and "e" as 5 what are the public and private keys? What is the encryption of a message whose integer equivalent is 12. Show that the decryption gives back the message 12.

1) Public key of the receiver is (e, n) is (5, 21)  and Private key of the receiver (d, n)   is (5 , 21) ,

2) the encryption of a message whose integer equivalent is 12 is 3

3) Decryption of the message ⇒ P = C^d mod n  

⇒ P = 3⁵ mod 21

⇒ P = 243 mod 21

⇒ 12

Explanation:

Given that,

p = 3

q = 7

e = 5

1)

Now, n = pq = 3 × 7 = 21

Ø(n) = (p-1) × (q-1)  = 2 × 6 = 12

Public key of the receiver is (e, n) is (5, 21)

and private key of the receiver is (d, n)

we have to find 'd' by using the expression

ed = 1 + kmodØ(n)

d = 1 + kmodØ(n) / e

now to get 'd' , we need to choose the least positive integer 'k', by substituting different values of ‘k’ from 0,

so for k =0 , d = (1+0) / 5 = 0.2 not an integer.

for k =1 , d = (1+12) / 5 = 13/5 = 2.6 not an integer.

for k =2 , d = (1+24) / 5 = 5 , now 5 is an integer

So k = 2 and d = 5

Private key of the receiver (d, n)   is (5 , 21)

2)

Now the encryption of a message whose integer equivalent is 12?

Encryption of the message ⇒ C = P^e mod n

⇒ C = 12⁵ mod 21

⇒ 248832 mod 21

⇒ 3

3)

Also the decryption gives back the message 12.

Decryption of the message ⇒ P = C^d mod n

⇒ P = 3⁵ mod 21

⇒ P = 243 mod 21

⇒ 12

answer:

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