3. Consider the RSA cryptosystem, where

$p \stackrel{\text{def}}{=} 25525635435900842730349748303929424117$,

$q \stackrel{\text{def}}{=} 259965242284515788826732110240207250949$.

(a) Compute the modulus *n*.

(b) Compute (*p* - 1) * (*q* - 1).

(c) Which of the two possible public-key exponents *e* is legitimate, 3 or 31?

(d) Take the one legitimate *e* from the previous item and compute the secret-key exponent *d*. If need be, use the extended Euclid algorithm of Exercise 2.19-1.

(e) Encrypt the message 19857367.

(f) Decrypt the message 27.