Suppose Elliptic curve E is defined by y$2=$x$3+$x$+6$ over Z$1039.$ Since Z$1039$ is small, you can program to solve 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 second coordination, E$.$g$.,(320,100)>(319,500)$ and $(320,100)<(320,101).$

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

Suppose $\backslash$alpha $=(799,790)$ is a generator and $\backslash$beta $=(385,749).($E$,\backslash$alpha $,\backslash$beta $)$ 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 $\backslash$alpha $=(818,121)$ 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?