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)
Let E : y 2 = x 3 +ax+b be an elliptic curve over Z/(n).
Let P be a point on the E all of whose coordinates are in (Z/(n)) and let X = (x0, y0) be the natural lift of P to Z. Consider the family of curves over Z F(E, X) = {E(α, β)||α|, |β| ≤ 3n 3 , E passes X and E reduces to E modulo n} where E(α, β) is defined by the equation y 2 = x 3 + αx + β. Then for sufficiently large n, a random curve in the family has rank 1 with probability greater than some constant. In light of the general heuristic assumption [1, 2, 21], it would be actually reasonable to expect that a random curve in F(E, X) has rank 1 with probability around 1/2, and rank 2 with probability around 1/2. How do we sample a random curve in the family? To do this, we choose a random integer i < n and set α = a+in, then set β = y 2 0 −x 3 0 −αx0. It is easy to see that E passes through X. x3+ax+b is never showing zero for any X element Fp. so E always have a odd number.
Let E be an elliptic curve involving the equation x3 + ax + b = y2 over the finite field Fp. Supp...
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
3. Let E be the elliptic curve y2-x3+x 6 over ZI1 1) Find all points on E by calculating the quadratic residues like the one demonstrated in the lecture 2) What is the order of the group? [Do not forget the identity element 0] 3) Given a point P - (2, 7), what is 2P? [point doubling] 4) Given another point Q (3, 6), what is PQ? [point addition] 3. Let E be the elliptic curve y2-x3+x 6 over ZI1...
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1 < x2 < X3. Suppose that y = ax + by + c is a parabola passing through the three points (x1, yı), (x2, y), and (x3, Y3). We have that a, b, and c must satisfy i = ax + bx + C V2 = ax + bx2 + c y3 = ax} + bx3 + c Let D = x X2 1....
Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal...
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
Suppose is a closed curve in the plane and that -Y dr + 2? + y2 2 dy = 671 z? + y2 How many self-intersection points must have, at least? By "self-intersection point", I mean a point where the curve intersects itself other than its endpoints. For example, a simple closed curve has zero self-intersection points, and a figure 8 has one self-intersection point. Hint: If a curve has self-intersection points, then it can be divided up into a...
G. Shorter Questions Relating to Automorphisms and Galois Groups Let F be a field, and K a finite extension of F. Suppose a, b E K. Prove parts 1-3: 1 If an automorphism h of K fixes Fand a, then h fixes F(a). 3 Aside from the identity function, there are no a-fixing automorphisms of a(). [HINT: Note that aV2 contains only real numbers.] 4 Explain why the conclusion of part 3 does not contradict Theorem 1. G. Shorter Questions...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...