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For the following questions, assume the use of the field GF(2 3 ). The field is described here using polynomial representation with the irreducible...
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)?
