4.4 Quiz 3

For the following questions, assume the use of the field F₂3. 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 

1. Does the elliptic curve equation y² + xy = x³ + g⁵x² + g⁶ define a group over F₂3? 

2. Do the points P(g3, g6) and Q(g5, g2) lie on the elliptic curve y² + xy = x³ + g²x² + g⁶ over F₂3? 

3. What are the negatives of the following elliptic curve points over F₂3? 

P(g³,g⁶) Q(g,0) R(0,g³) 

4. In the elliptic curve group defined by y² + xy = x³ + g²x² + g⁶ over F₂3, what is P + Q if P = (g²,g⁶) and Q = (g⁵,g⁵)? 

5. In the elliptic curve group defined by y² + xy = x³ + g²x² + g⁶ over F₂3, what is 2P if P = (g³, g⁴)? 

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