3.4 QUIZ 2 ~ Solutions
Elliptic Curve Groups over
Fp
1. Does the elliptic curve equation y2 = x3 + 10x + 5 define a
group over F17?
No, since:
= 4(10)3 + 27(5)2 mod 17
= 4675
mod 17
= 0
Thus this elliptic curve does not define a group
because 4a3 + 27b2 mod p is 0
2. Do the points P(2,0) and Q(6,3)
lie on the elliptic curve y2 = x3 + x + 7 over F17?
The point P(2,0) is
on the elliptic curve since both sides of the equation agree:
(0)2 mod 17
= (2)3 + 2 + 7 mod 17
0 mod 17 = 17 mod 17
0 = 0.
However,
the point Q(6,3) is not on the elliptic curve since the equation is
false:
(3)2 mod 17 = (6)3 + 6 + 7 mod 17
9 mod 17 = 229 mod
17
9 = 8, does not agree.
3. What are the negatives of the
following elliptic curve points over F17?
P(5,8) Q(3,0) R(0,6)
The
negative of a point P = (xP, yP) is the point -P = (xP, -yP mod p).
Thus
-P(5,9) -Q(3,0) -R(0,11)
4. In the elliptic curve
group defined by y2 = x3 + x + 7 over F17, what is P + Q if P = (2,0) and Q =
(1,3)?
s = (yP - yQ) / (xP - xQ) mod p = (-3) / 1 mod 17 = -3 mod 17 =
14
xR = s2 - xP - xQ mod p = 196 - 2 - 1 mod 17 = 193 mod 17 =
6
yR = -yP + s(xP - xR) mod p = 0 + 14*(2 - 6) mod 17 = -56 mod 17 =
12
Thus P + Q = (6,12)
5. In the elliptic curve group
defined by y2 = x3 + x + 7 over F17, what is 2P if P = (1, 3)?
s = (3xP2
+ a) / (2yP ) mod p = (3 + 1) * 6-1 mod 17 = 4 * 3 mod 17 = 12
xR = s2 -
2xP mod p = 144 - 2 mod 17 = 142 mod 17 = 6
yR = -yP + s(xP - xR) mod p =
-3 + 12 * (1 - 6) mod 17 = -63 mod 17 = 5
Thus 2P = (6,5)