ISRO2014-24

Question

In a system an RSA algorithm with $p=5$ and $q=11$, is implemented for data security. What is the value of the decryption key if the value of the encryption key is $27?$

• A. $3$
• B. $7$
• C. $27$
• D. $40$

Comments

but (3+27)%40=30 so how could be the ans as option (a).

 

Please explain that.

Answer 1

RSA Algorithm   http://courses.cs.vt.edu/~cs5204/fall00/protection/rsa.html   and example https://www.cs.utexas.edu/~mitra/honors/soln.ht

 Answer of above ,

     p=5 , q=11 , encryption key(e) = 27

     n =  p*q = 5*11= 55

     φ(n) = (p - 1) * (q - 1) = 4 * 10 = 40

     Given[e = 27], d such that (d * e) % φ(n) = 1

     decryption key (d) --> (d * 27) % 40 = 1

     so , when  d =3  than L.H.S = R.H.S

     Answer option (a).

Answer 2

n = 5 * 11 = 55

∅(n) = (p-1)(q-1) = 40

e = 27,  d = ?

ed mod( ∅(n) ) = 1

27d mod 40 = 1

try all option for d,

let, d=3  ------- (27*3)mod40 = 81 mod 40 = 1

option A the answer

Answer 3

Answer: (a)

Explanation: RSA algorithm is is an asymmetric cryptographic algorithm. RSA involves a public key (encryption key) and private key (decryption key).

• Choose two different large random prime numbers p and q. Here already given p = 5, q =11.
• Calculate n = p*q where n is the modulus for the public key and the private keys. Here n = 55.
• Calculate the totient: ϕ = (p − 1) * (q − 1). Here ϕ = 40.
• Choose an encryption key integer e such that 1 < e < ϕ and e is co-prime to ϕ
  i.e. e and ϕ share no factors other than 1 or we can say gcd(e, ϕ) = 1.
  Here e is already given that is e = 27.
• Compute an decryption key d to satisfy the congruence relation d * e ≡ 1 mod ϕ.
  Here we require to find out the value of decryption key.
  d * 27 = 1 mod 40 => d = 81

Reference: https://simple.wikipedia.org/wiki/RSA_(algorithm)