SupposeEllipticcurveEisdefinedbyy ${?}^2=x^3+x+6$ overZ $1039.$ Since $Z_{1039}$ is small, you can program to slove all

the following problems.

How many points are over E$.$

What is the lexically largest point over E$.$ Here lexically larger point

means to order the points by the first coordination first and then the

$1$

second coordination, E$.$g$.,(320,100)>(319,500)$ and

Does point $(1014,291)$ belong to $E?$

Suppose $=(799,790)$ is a generator and $=(385,749).(E,,)$ is the ElGamal public key. Given the

plaintext value $(575,419)$ and random $K=100,$ what is the ciphertext value? Given the ciphertext

value $((873,233),(234,14),$ what is the plaintext value $($Please note: if you re$-$encrypt the resulting

plaintext value with $K=100,$ you may not get back $((873,233),(234,14)),$ that is correct. As you recall,

for whatever $K$ is selected for encryption, the decrypted value is the same.$)$

Suppose E and a generator $=(818,121)$ are public. Alice and Bob achieve a shared secret by doing

Diffie$-$hellman key exchange. Alice sends Bob a value $(199,72),$ and Bob sends Alice a value

$(815,519),$ what is the secret they achieve?