Elliptic

# Elliptic Curve Point Addition Question

## How do you add two points on an elliptic curve?

In order to add distinct points, construct the line between them and determine the third point of intersection with the curve. The sum of the two points is then the reflection of the third point about the axis of symmetry, which is the axis for the case illustrated here.

## What is elliptic curve point addition?

Elliptic curve groups are additive groups; that is, their basic function is addition. The addition of two points in an elliptic curve is defined geometrically. The negative of a point P = (xP,yP) is its reflection in the x-axis: the point -P is (xP,-yP).

## What is point addition?

Point addition

With 2 distinct points, P and Q, addition is defined as the negation of the point resulting from the intersection of the curve, E, and the straight line defined by the points P and Q, giving the point, R.

## What is point at infinity elliptic curve?

When in (projective) Weierstrass form, an elliptic curve always contains exactly one point of infinity, ( 0 , 1 , 0 ) ("the point at the ends of all lines parallel to the -axis"), and the tangent at this point is the line at infinity and intersects the curve at ( 0 , 1 , 0 ) with multiplicity three.

## Why are elliptic curves important?

Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization.

## Which elliptic curve is used in Bitcoin?

Secp256k1 is the name of the elliptic curve used by Bitcoin to implement its public key cryptography. All points on this curve are valid Bitcoin public keys.

## How do you find the 2P of an elliptic curve cryptography?

If x2 = x1 and y2 = −y1, that is P = (x1,y1) and Q = (x2,y2) = (x1,−y1) = −P, then P + Q = O. Therefore 2P = (x3,y3) = (7,12).

## How do you solve the elliptic curve cryptography?

Elliptic curves are currently behind practically most preferred methods of cryptographic security. Elliptic curves are also a basis of very important factorization method. If the line through two different points P1 and P2 of an elliptic curve E intersects E in a point Q = (x, y), then we define P1 + P2 = P3 = (x, −y).

## Is an efficient algorithm to count point on the elliptic curve over finite field?

-We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known.

## Who invented elliptic curves?

Elliptic curve cryptography was introduced in 1985 by Victor Miller and Neal Koblitz who both independently developed the idea of using elliptic curves as the basis of a group for the discrete logarithm problem. [16, 20].

## Why is elliptic curve a torus?

After adding a point at infinity to the curve on the right, we get two circles topologically. ... Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram (in fact a square in this case) with the sides glued together i.e. a torus.

## What is the sum of three points on an elliptic curve that lie on a straight line?

The answer is zero. People who have answered your question previously mistook elliptic curve to ellipse,I'll be adding a screenshot check that out. Originally Answered: what is the sum of three points on an elliptic curve that lie on a straight line? A straight line can not intersect an elliptic curve at 3 points.

## Why are elliptic curves called elliptic?

So elliptic curves are the set of points that are obtained as a result of solving elliptic functions over a predefined space. I guess they didn't want to come up with a whole new name for this, so they named them elliptic curves.

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