# Elliptic curve in cryptography

## What is Elliptical Curve Cryptography (ECC)?

Elliptic Curve Cryptography (ECC) is public key cryptography based on elliptic curve theory. It can be used to create smaller, faster and more efficient public key cryptography. Elliptic Curve Cryptography is an alternative of Rivest Shamir Adleman (RSA) cryptography. This algorithm is used for creating digital proposals. It makes the digital signature for cryptocurrencies like Bitcoin and Ethereum etc. It also establishes the cryptocurrency for data, software and encrypted emails.

An Elliptical curve is a representation of a looping line that is intersected by two axes, and these lines indicate the position of points on the graph. This curve is entirely symmetric in nature, and it can be mirrored along with the x-axis of the graph.

Elliptic Curve Cryptography (ECC) is the public key cryptography that follows a mathematical concept that combines the two keys and gives the output of encrypting and decrypting data. The result contains the public key that is known to everyone and a private key that is known only to the sender and receiver.

Elliptic Curve Cryptography (ECC) generates the encrypted and decrypted data elliptic curve equation and does not follow the traditional method to generate the data.  The points of the graph can be calculated by the following formula.

`y²=x³ + ax + b`

Elliptic Curve Cryptography (ECC) is similar to other encryption or decryption methods, such as the Diffie-Hellman and RSA methods. Each public critical cryptographic process follows the concept of a one-way method. This means, with the help of one mathematical formula, it is straightforward to compute the point value of the graph. We can calculate the value of point A to point B without knowing any public or private keys.  In the Elliptic Curve Cryptography (ECC) method, we can derive the value of any third-party value by computing the first-party and second-party value simultaneously. The diagram below shows that there are three points, A, B, and C, in the elliptical graph. Taking point C, we can create the mirror image on the opposite side of the chart. From that point, we can draw a line that intersects point E, as shown in the below graph. This process can be completed the maximum number of times to define a max value. The n is known as the private key. The value of n indicates how many times the process is going to be run.

## RSA vs Elliptical Curve Cryptography

Elliptic Curve Cryptography (ECC) provides a very secure computing feature of encrypting and decrypting the data compared to Diffie Hellman and RSA public key cryptography. Elliptic Curve Cryptography (ECC) can provide security with very low computing power and uses a low battery compared to Diffie-Hellman and RSA public key cryptography. Elliptic Curve Cryptography (ECC) can be widely used in the field of the Internet of Things (IoT) and mobile applications.

ECC offers several benefits compared to RSA:

• It operates on devices with low CPU and memory resources.
• It encrypts and decrypts faster.
• Larger key sizes can be used without significantly increasing the key size or CPU and memory requirements.

## How Secure is Elliptical Curve Cryptography?

ECC is thought to be highly secure if the key size used is large enough. The U.S. government requires the use of ECC with a key size of either 256 or 384 bits for internal communications, depending on the sensitivity level of the information being transmitted.

But ECC is not necessarily any more or less secure compared to alternatives such as RSA. The primary benefit of ECC is the inherent efficiencies gained when encrypting and decrypting data.

## History of Elliptical Curve Cryptography

The properties and functions of elliptic curves in mathematics have been studied for more than 150 years. Their use within cryptography was first proposed in 1985, separately by Neal Koblitz from the University of Washington and Victor Miller at IBM.

ECC was first developed by Certicom, a mobile e-business security provider, and was then licensed by Hifn, a manufacturer of integrated circuitry and network security products. Vendors, including 3Com, Cylink Corp., Motorola, Pitney Bowes, Siemens, TRW Inc. (acquired by Northrop Grumman) and Verifone, supported ECC in their products.

The use of ECC in public and private sectors has increased over the past few years. While RSA continues to be more widely used and is easier to understand than ECC, ECC's efficiency benefits make it appealing for many enterprise use cases. These include speeding up secure access to Secure Sockets Layer-encrypted websites and streaming encrypted data from IoT devices with limited computing power.