2.4 QUIZ 1 ~ Solutions
Elliptic Curve Groups Over
Real Numbers
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 + 27b2
is 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 = (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)