2.4 Quiz 1 - Solutions

1.

Does the elliptic curve equation y²= x³- 7x - 6 over real numbers define a group? 

Yes, since 

4a³+ 27b²= 4(-7)³+ 27(-6)²= -400 

The equation y²= x³- 7x - 6 does define an elliptic curve group because 4a³+ 27b²is not 0. 

2.

What is the additive identity of regular integers? 

The additive identity of regular integers is 0, since x + 0 = x for all integers. 

3.

Is (4,7) a point on the elliptic curve y²= x³- 5x + 5 over real numbers? 

Yes, since the equation holds true for x = 4 and y = 7: 

(7)²= (4)³- 5(4) + 5 

49 = 64 - 20 + 5 

49 = 49 

4.

What are the negatives of the following elliptic curve points over real numbers? 

P(-4,-6), Q(17,0), R(3,9), S(0,-4) 

The negative is the point reflected through the x-axis. Thus 

-P(-4,6), -Q(17,0), -R(3,-9), -S(0,4) 

5.

In the elliptic curve group defined by y²= x³- 17x + 16 over real numbers, what is P + Q if P = (0,-4) and Q = (1,0)? 

From the Addition formulae: 

s = (y_P- y_Q) / (x_P- x_Q) = (-4 - 0) / (0 - 1) = 4 

x_R= s²- x_P- x_Q= 16 - 0 - 1 = 15 

and 

y_R= -y_P+ s(x_P- x_R) = 4 + 4(0 - 15) = -56 

Thus P + Q = (15, -56) 

6.

In the elliptic curve group defined by y²= x³- 17x + 16 over real numbers, what is 2P if P = (4, 3.464)? 

From the Doubling formulae: 

s = (3x_P²+ a) / (2y_P) = (3*(4)²+ (-17)) / 2*(3.464) = 31 / 6.928 = 4.475 

x_R= s²- 2x_P= (4.475)²- 2(4) = 20.022 - 8 = 12.022 

and 

y_R= -y_P+ s(x_P- x_R) = -3.464 + 4.475(4 - 12.022) = - 3.464 - 35.898 = -39.362 

Thus 2P = (12.022, -39.362) 

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