GATE | GATE CS Mock 2018 | Question 51

In a RSA cryptosystem, a participant A uses two prime numbers p=13 and q=11 to generate his public and private keys. If the public key of A is 37, then the private key of A is _____.

(A) 18
(B) 23
(C) 11
(D) 13


Answer: (D)

Explanation: Public Key = (n,e)
Private Key = (n,d)
n = pq = 143
z = (p-1)(q-1) = 120
Given e = 37
Using ed mod z = 1
Candidates for 1 mod z = 121 241 361 481 601 721 841 961 1081 1201 1321 1441 1561 1681 1801 1921 2041 2161 2281 2401 2521 2641 2761 2881 3001 3121 3241 3361 3481 3601
Smallest number which is a multiple of 37 is 481
and 481/37 = 13
So the private key is 13.

So, option (D) is correct.

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