Consider a variation (simplified and broken version) of the dh key exchange algorithm using a public-key protocol: two parties a and b wanting to communicate first agree on a public key (n, g) where n is a (large) prime number, and g a fixed number, 2 \leq g \leqn - 1. to establish a secret key between a and b, party a chooses a private key (random number) x, calculates x = g* x (mod n), then sends it to party b. similarly, party b chooses its own private key y, calculates y = g*y (mod n), then sends it to party a. when a receives the value y = g* y, it calculates the secret key k = g*x + g*y (mod n). similarly, party b after receiving x = g*x (mod n) from a, calculates g* y + g* x (mod n), then use it as the secret key. note that an eavesdropper e monitoring the communications can capture both x and y (i. e., both g* x and g* y, modulo n) although e does not know the two private keys (random numbers) x and y. (e does know the public key (n, now answer the below questions regarding this new secret key exchange algorithm.

(a) show/explain why the above algorithm works in the sense that it allows the communicating parties a and b establish a (common) secret key.

(b) suppose the eavesdropper e, after capturing the value g x (mod n), uses the below algorithm to calculate/crack the private key x:

step 1. set k = 1
step 2. compute g* k (mod n)

step 3. test if g* k = (the captured value) g* x (mod n): if equal, then found the private key k, declare "success" and stop