For the following questions, assume the use of the field GF(23). The field is described here using polynomial representation with the irreducible polynomial x3 + x + 1. A generator for the field is g = (010), and the powers of g are:

g1 = (010) g2 = (100) g3 = (011) g4 = (110) g5 = (111) g6 = (101) g7 = (001) = 1

a) Does the elliptic curve equation y2 + xy = x3 + g5x2 + g6 define a group over GF(23)?

b) Do the points P(g3, g6) and Q(g5, g2) lie on the elliptic curve y2 + xy = x3 + g2x2 +g6 over GF(23)?

c) What are the negatives of the following elliptic curve points over GF(23)? P(g3,g6), Q(g,0), R(0,g3)

d) In the elliptic curve group defined by y2 + xy = x3 + g2x2 + g6 over GF(23), what is P + Q if P = (g2,g6) and Q = (g5,g5)?

e) In the elliptic curve group defined by y2 + xy = x3 + g2x2 + g6 over GF(23), what is 2P if P = (g3, g4)?