3. Let E be the elliptic curve y2-x3+x 6 over ZI1 1) Find all points on E by calculating the quad...
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod 19). (Hint: Corresponding to any given x , points (x,y) and (x,-y) can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is a quadratic residue mod 19. Recall that a value v ∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p). If v is indeed a quadratic residue, we can calculate the two...
Let E be an elliptic curve involving the equation x3 + ax + b = y2 over the finite field Fp. Suppose you have the additional information that x3 + ax + b is never zero for any X ∈ Fp. Show that E must have an odd number of points. (Hint: don’t forget O)
Considering the ECDSA based on the elliptic curve E: y2 =x3+ax+b over GF(p) where 0<a,b<p, assume that the size of the elliptic curve group is 160 bits, then the size of an ECDSA is a 640 bits b 80 bits c 320 bits d 160 bits
Consider the elliptic curve y^2 = x^3 - 10x + 6 over the real numbers. (a) Verify that the points P = (3, -1.732) and Q = (0.562, 0.7467) are actually on the curve. (b) Show that an elliptic curve group can be formed by verifying that 4a^3 + 27b^2 notequalto 0. (c) Calculate P + Q in the elliptic curve group using a geometric method (i.e show the curve in the Cartesian plane). (d) Calculate P + Q in...
3. [10] (quadrifolium) Let (a2 + y2) = (2 -)2 be a curve. Find the points on the curve where the normal line is parallel to y 0. re2y, find the normal line at 4. [4] Let (1,0). [0, 10] with f(0) f(10) 0 and 5. 5 Let f(a) be continuous and differentiable on f(5) 4. Mark TRUE or FALSE for the following statements and JUSTIFY. (No points will be given without the correct justification) (A) There is some c...
Let S denote the sphere x2 y2 2 = 1. Given two points P(1,0,0), (a) Find the distance between P and Q. Lets call this Euclidean distance. (b) Find the plane that goes through O, P, Q. What is the intersection of this plane with the sphere? (Hint: use OP × OQ as the the normal vector) (c) Observe that the length of the arc PQ is 0 the angle between OP,0Q in radians. (Hint: You know how to find...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
Find a normal vector and an equation for the tangent plane to the surface: x3 - y2 - z2 - 2xyz + 6 =0 at the point P : (−2, 1, 3). Determine the equation of the line formed by the intersection of this plane with the plane x = 0. [10 marks] (b) Find the directional derivative of the function F(x, y, z) = 2x /zy2 , at the point P : (1, −1, −2) in the direction of...
Consider the following. z = x2 + y2, z = 36 − y, (6, -1, 37) (a) Find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point. x − 6 12 = y + 1 −2 = z − 37 −1 x − 6 1 = y + 1 12 = z − 37 −12 x − 6 = y + 1 = z − 37 x − 6 12 =...
2. The following table gives points on the Lorenz curve for the U.S. in 1970. 0 .6 8 .2 .4 L(x) 0 .323 .568 1 041 .149 a) Based on the table, the poorest 20% of the population in 1970 earned what percent of the total income? b) The richest 20% of the population earned what percent of total income in the US, in 1970? Use a calculator or computer to find a quadratic curve best fit with form L(x)...