### 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)?

s = (yP- yQ) / (xP- xQ) = (-4 - 0) / (0 - 1) = 4

xR= s2- xP- xQ= 16 - 0 - 1 = 15

and

yR= -yP+ s(xP- xR) = 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 = (3xP2+ a) / (2yP) = (3*(4)2+ (-17)) / 2*(3.464) = 31 / 6.928 = 4.475

xR= s2- 2xP= (4.475)2- 2(4) = 20.022 - 8 = 12.022

and

yR= -yP+ s(xP- xR) = -3.464 + 4.475(4 - 12.022) = - 3.464 - 35.898 = -39.362

Thus 2P = (12.022, -39.362)

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