Suppose the two primes in the RSA schema are p 13 and q 23

Suppose the two primes in the RSA schema are p = 13 and q = 23. Find the numbers r, e, and d. And show that a plain text, say 225, can be recovered using public key P = (e, n) and secret key S = (d, n). Do the same as (1) for the two primes p = 23 and q = 29.

Solution

1)       Given p = 13 and q = 23, so n = pq = 299

Now, r= Euler totient( n ) = (p-1)(q-1) = 264

Let e = 5, it should be valid as gcd( e, 264) = 1

Then d = e^-1( mod 264 )   ( = should have been written as actually three -- instead of two )

we can find d to be, 53   and we can verify that 53*5 mod 264 is indeed 1

So, now we got d and e

Testing on 225:      m = 225    ( our message )

Encrypted = (m^e)mod n = (225)⁵mod 299 = 49

Decrypted = (49)^dmod n = (49)⁵³mod 299 = 225     ( got the message back )

2)     Given p = 23 and q = 29, so n = pq = 667

Now, r= Euler totient( n ) = (p-1)(q-1) = 616

Let e = 5, it should be valid as gcd( e, 616 ) = 1

Then d = e^-1( mod 616 )

we can find d to be, 493 and we can verify that 493*5 mod 616 is indeed 1

So, now we got d and e

Testing on 225:      m = 225    ( our message )

Encrypted = (m^e)mod n = (225)⁵mod 667 = 187

Decrypted = (187)^dmod n = (187)⁴⁹³mod 667 = 225     ( got the message back )