Cryptography Multiple Choice Questions on “Elliptic Curve Arithmetic/Cryptography”.
1. What is the general equation for elliptic curve systems?
a) y3+b_1 xy+b_2 y=x33+a_1 x2+a_2 x+a_3
b) y3+b_1 x+b_2 y=x2+a_1 x2+a_2 x+a_3
c) y2+b_1 xy+b_2 y=x3+a_1 x2+a_2
d) y2+b_1 xy+b_2 y=x3+a_1 x2+a_2 x+a_3
Answer: d
Clarification: The general equations for an elliptic curve system is y2+b_1 xy+b_2 y=x3+a_1 x2+a_2 x+a_3.
2. In Singular elliptic curve, the equation x^3+ax+b=0 does ____ roots.
a) does not have three distinct
b) has three distinct
c) has three unique
d) has three distinct unique
Answer: a
Clarification: In Singular elliptic curve, the equation x^3+ax+b=0 does not have three distinct roots.
3. How many real and imaginary roots does the equation y2=x3-1 have
a) 2 real, 1 imaginary
b) all real
c) all imaginary
d) 2 imaginary, 1 real
Answer: d
Clarification: On solving the equation we get 2 imaginary and 1 real root.
4. How many real and imaginary roots does the equation y2=x3-4x have
a) 2 real, 1 imaginary
b) all real
c) all imaginary
d) 2 imaginary, 1 real
Answer: b
Clarification: On solving the equation we get all real roots.
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)?
a) (15, -56)
b) (-23, -43)
c) (69, 26)
d) (12, -86)
Answer: a
Clarification: P=(x1, y1)= (0,-4)
Q=(x2, y2)= (1,0)
From the Addition formulae:
λ= (0-(-4)) / (1-0) = 4
x3= = 16 – 0 – 1 = 15 and
y3= 4(0 – 15) –(-4) = -56
Thus R=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)?
a) (12.022, -39.362)
b) (32.022, 42.249)
c) (11.694, -43.723)
d) (43.022, 39.362)
Answer: a
Clarification: From the Doubling formulae:
λ = (3*(4)2+ (-17)) / 2*(3.464) = 31 / 6.928 = 4.475
x3= (4.475)2- 2(4) = 20.022 – 8 = 12.022 and
y3= -3.464 + 4.475(4 – 12.022) = – 3.464 – 35.898 = -39.362
Thus 2P = (12.022, -39.362).
7. “Elliptic curve cryptography follows the associative property.”
a) True
b) False
Answer: a
Clarification: ECC does follow associative property.
8. “In ECC, the inverse of point P =(x1, y1) is Q = (-x1, y1). “
a) True
b) False
Answer: b
Clarification: The inverse of point P =(x1, y1) is Q =(x1, -y1).