What is elliptic curve theory?

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.

What is the elliptic curve equation?

If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element.

Why is the discrete log problem hard?

For the discrete algorithm problem, you normally would not write the whole group (or even its multiplication table of size n2) as the input, but only some key parameters which allow calculating the group law, as well as the element of which you want to get the logarithm.

Why are elliptic curves are important?

1) Elliptic Curves provide security equivalent to classical systems (like RSA), but uses fewer bits. 2) Implementation of elliptic curves in cryptography requires smaller chip size, less power consumption, increase in speed, etc.

Who discovered elliptic curves?

Properties and functions of elliptic curves have been studied in mathematics for 150 years. Use of elliptic curves in cryptography was not known till 1985. Elliptic curve cryptography is introduced by Victor Miller and Neal Koblitz in 1985 and now it is extensively used in security protocol.

What is discrete logarithm problem and explain it?

The discrete logarithm problem is defined as: given a group G, a generator g of the group and an element h of G, to find the discrete logarithm to the base g of h in the group G. Discrete logarithm problem is not always hard. The hardness of finding discrete logarithms depends on the groups.

What is elliptic curve discrete logarithm?

The ECDLP is a special case of the discrete logarithm problem in which the cyclic group G is represented by the group \langle P\rangle of points on an elliptic curve. It is of cryptographic interest because its apparent intractability is the basis for the security of elliptic curve cryptography.

Why is it called elliptic curve?

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 many points is an elliptic curve?

are integers modulo m, then the coordinates of P+Q will be integers modulo m, unless P+Q = ∞, provided that any division needed to add points is by a number relatively prime to m. ≡ 0 (mod m). + 3x + 4 (mod 7). There are ten points on this elliptic curve, counting ∞.

What is discrete logarithms on elliptic curves?

Discrete Logarithms on Elliptic Curves Aaron Blumenfeld Abstract. Cryptographic protocols often make use of the inherent hardness of the classical discrete logarithm problem, which is to solve g x y (mod p) for x. The hardness of this problem has been exploited in the Die-Hellman key exchange, as well as in cryptosystems such as ElGamal.

Is discrete exponentiation on elliptic curves secure?

This suggests that discrete exponentiation on elliptic curves is slightly less secure than one might hope for two reasons. First, it suggests that there’s less of a chance for a longer tail, which could prove detrimental for certain pseudorandom number generators [1]. Secondly,

What is the discrete logarithm problem?

classical discrete logarithm problem, which is to solve g x y (mod p) for x. The hardness of this problem has been exploited in the Die-Hellman key exchange, as well as in cryptosystems such as ElGamal. There is a similar discrete logarithm problem on elliptic curves: solve kB = P for k. Therefore, Die-Hellman and

What are elliptic curves?

In the past several decades, elliptic curves have entered the scene. Elliptic curves over nite \felds contain \fnite cyclic groups that we can use for cryptography. There is no factor- ization problem for elliptic curves, but what is used is the discrete logarithm problem, which is to solve kB= Pfor k.