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# Elliptic curve cryptography blockchain

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Symmetric cryptography has been around for thousands of years. Originally the secret information was simply the steps involved in scrambling or unscrambling the message; however, this turned out to be susceptible to attacks called cryptanalysis because there are only so many ways to scramble a message so that it can be unscrambled, and someone can usually guess how the scrambling works using statistics and by guessing at what the unscrambled message is and trying all the possible scrambling techniques to see which one fits.

In modern symmetric cryptography, the steps involved in scrambling are public and the secret information is a number which is scrambled into the message. Only someone who knows the secret number can unscramble the message to its original form. Of course, anyone could get lucky and guess the secret number, so the secret number is always very large hundreds of digits long. Someone could theoretically be lucky enough to guess it, but modern cryptography is based on the prediction that no one is that lucky.

The problem with symmetric cryptography is that the secret number called the key is simultaneously confidential and shared. It tends to be difficult for two people to agree on this secret number without leaking it to anyone else. Enter asymmetric cryptography.

Asymmetric Cryptography Unlike symmetric cryptography, asymmetric works without the sender and receiver knowing the same key. Instead, the receiver creates a private key and calculates a public key from it. He then publishes the public key and anyone who wishes to send him a message can encrypt it with the public key, but only the receiver can decrypt it with his private key. In this way, the two complement each other, providing both security and speed.

This approach is known as hybrid cryptography. Asymmetric cryptography has a wider use than just sharing symmetric keys though: signatures. ECC is a type of asymmetric cryptography, so it uses key-pairs a private key and public key. There are two main ways ECC is used. For later elliptic-curve-based protocols, the base assumption is that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" ECDLP.

The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points. The size of the elliptic curve, measured by the total number of discrete integer pairs satisfying the curve equation, determines the difficulty of the problem. RSA is still used for providing authentication.

A disadvantage is an additional effort for creating and maintaining the EC key. Elliptic curve ciphers were first proposed independently by Victor Miller and Neal Koblitz in the mids. At a high level, they are analogs of existing public-key cryptosystems in which modular arithmetic is replaced by operations defined on elliptic curves.

As with all public-key cryptosystems, the security of elliptic curve cryptosystems relies on difficult mathematical problems at the core. This problem is often called the elliptic curve discrete logarithm problem. Currently, general methods of calculating discrete logarithms of elliptic curves are much less efficient than traditional methods of factoring or calculating discrete logarithms. Elliptic curves are not ellipses. They are named that way because they are represented by expressions similar to the cubic equations used to calculate the circle of an ellipse.

The characteristic of the finite field GF 2 is 2, and the characteristic of real and complex numbers is infinity Integer Factorization Using Elliptic Curves In , Hendrik Lenstra published the landmark paper that introduces and analyzes the Elliptic Curve Method ECM , which is a powerful algorithm for factoring integers using elliptic curves.

For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieves, and the fastest is the general number field sieve.

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