Elliptic curve point multiplication example

Here is a simple example of point multiplication. Let P be a point on an elliptic curve. Let k be a scalar that is multiplied with the point P to obtain another point Q on the curve i.e. to find Q = kP. If k = 23 then kP = 23.

1. How do you multiply points on an elliptic curve?
2. What is elliptic curve point addition?
3. Why are elliptic curves important?
4. How do you solve the elliptic curve cryptography?
5. What is point at infinity elliptic curve?
6. Which elliptic curve is used in Bitcoin?
7. What is elliptic curve discrete logarithm?
8. What is a elliptic curve group?
9. Why is it called an elliptic curve?
10. How does ECC calculate public key?
11. Is ECC symmetric or asymmetric?
12. What are the recommended key lengths for elliptic curve cryptography?

How do you multiply points on an elliptic curve?

Given a curve, E, defined along some equation in a finite field (such as E: y² = x³ + ax + b), point multiplication is defined as the repeated addition of a point along that curve. Denote as nP = P + P + P + … + P for some scalar (integer) n and a point P = (x, y) that lies on the curve, E.

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).

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.

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).

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.

Which elliptic curve is used in Bitcoin?

The elliptic curve used by Bitcoin, Ethereum, and many other cryptocurrencies is called secp256k1. The equation for the secp256k1 curve is y² = x³+7. This curve looks like: Satoshi chose secp256k1 for no particular reason.

What is elliptic curve discrete logarithm?

The elliptic curve discrete logarithm problem (ECDLP) is the following computational problem: Given points P, Q ∈ E(Fq) to find an integer a, if it exists, such that Q = aP. ... We focus on the case of elliptic curves, but occasionally this involves mention of higher genus curves and their divisor class groups.

What is a elliptic curve group?

Elliptic curves are curves defined by a certain type of cubic equation in two variables. The set of rational solutions to this equation has an extremely interesting structure, including a group law. The theory of elliptic curves was essential in Andrew Wiles' proof of Fermat's last theorem.

Why is it called an elliptic curve?

Anyway, since Jacobi's functions started off with ellipse arc length problem, they are called elliptic functions. ... These curves are called elliptic curves. So elliptic curves are the set of points that are obtained as a result of solving elliptic functions over a predefined space.

How does ECC calculate public key?

The number of integers to use to express points is a prime number p. The public key is a point in the curve and the private key is a random number (the k from before). The public key is obtained by multiplying the private key with the generator point G in the curve.

Is ECC symmetric or asymmetric?

ECC is an approach — a set of algorithms for key generation, encryption and decryption — to doing asymmetric cryptography. Asymmetric cryptographic algorithms have the property that you do not use a single key — as in symmetric cryptographic algorithms such as AES — but a key pair.

What are the recommended key lengths for elliptic curve cryptography?

It has been noted by the NSA that the encryption of a top-secret document by elliptic curve cryptography requires a key length of 384 bit. A key length of the same size by RSA would deliver no where near the same level of security.