Elliptic curve point multiplication python

1. How do you multiply points on an elliptic curve?
2. What is elliptic curve point addition?
3. Why is an elliptic curve secure?
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. How does an elliptic curve work?
9. Is ECC symmetric or asymmetric?
10. What is the general equation for an elliptic curve?
11. What is the key size in ECC?
12. How does ECC calculate public key?
13. How does Diffie Hellman key exchange work?
14. What is a elliptic curve group?
15. How do you find a point at infinity?
16. Why is elliptic curve a torus?

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 is an elliptic curve secure?

An elliptic curve point is singular if and only if the partial derivatives of the curve equation are null at that point. The curve is said to be singular if it possesses at least a singular point, while it is non-singular if it does not have any such points.

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.

How does an elliptic curve work?

An elliptic curve for current ECC purposes is a plane curve over a finite field which is made up of the points satisfying the equation: y²=x³ + ax + b. In this elliptic curve cryptography example, any point on the curve can be mirrored over the x-axis and the curve will stay the same.

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 is the general equation for an elliptic curve?

What is the general equation for elliptic curve systems? Explanation: 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.

What is the key size in ECC?

With a 112-bit strength, the ECC key size is 224 bits and the RSA key size is 2048 bits. The most popular signature scheme that uses elliptic curves is called the Elliptic Curve Digital Signature Algorithm (ECDSA).

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.

How does Diffie Hellman key exchange work?

In the Diffie–Hellman key exchange scheme, each party generates a public/private key pair and distributes the public key. After obtaining an authentic copy of each other's public keys, Alice and Bob can compute a shared secret offline. The shared secret can be used, for instance, as the key for a symmetric cipher.

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.

How do you find a point at infinity?

Points with new coordinates (a,b,0) are points at infinity. A line with old equation ax + by + c = 0 has new equation ax + by + cw = 0, and it has coordinates [a,b,c] or any nonzero multiple of that. The line at infinity has equation w=0, and it has coordinates [0,0,1] or any nonzero multiple of that.

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.