Question

Given two prime numbers 17 and 19. Compute the encryption and the decryption keys using RSA algorithm.

Answer 1

In RSA we have Two prime Number i.e. p and q

Steps to get Keys

1. Compute n=pq and  ϕ=(p−1)(q−1)

2.Choose an integer e, 1<e<ϕ such that gcd(e,ϕ)=1

3. Compute the secret exponent d, 1<d<ϕ, such that e*d ≡ 1 mod ϕ

Now We have Two keys in RSA

1. Encryption Key i.e. e that we find in Step 2

2. Decryption Key i,e. d that we find in Step 3

Process followed by RSA

Sender send set of ( n ,e ) which the  the Public key along with the encrypted data and when receiver gets the data then it used n and e to get the private key d and use that key to decrypt the message .

Now lets calculate keys using given prime number

p = 17 , q=19

n=pq = 17 * 19 = 323

ϕ=(p−1)(q−1) = 16 * 18 = 288

Now We get e as

gcd(e, 288 )=1

They are many possible so but we have to take any less than 288 .

So lets we have 215

Encryption Key = 215

Now Lets find the decryption key

e*d ≡ 1 mod ϕ

Now e = 151

215 * d mod 288 = 1

We have to find modular inverse to get d

Hence we get d = 71

Hence we get Two keys as 215 and 71

Note : Answer may be differ and we have many options to choose the encryption key i.e. e but for every e , d is fixed and unique,

So if u choose any other encryption key then follow the same process to get d by using modular inverse

Thank You

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