Elliptic Curve Cryptography: An Overview

Ajay Ohri


Elliptic Curve Cryptography, commonly abbreviated as ECC, is a technique used in the encryption of data. ECC uses a mathematical approach to encryption of data using key-based techniques. ECC is often connected and discussed concerning the RSA or Rivest Shamir Adleman cryptographic algorithm. RSA completes single encryption of aspects like data, emails, and software that makes use of prime factorization. However, the ECC is profoundly a diverse mathematical method to encryption than the well-regarded RSA cryptographic algorithm.

Been in existence since the mid-1980s, the procedure of elliptic curves in cryptography was recommended individually by Mr. Neal Koblitz and Mr. Victor S. Miller in 1985, and elliptic curve cryptography algorithms were commenced to be in extensive use from 2004.

  1. What is ECC? 
  2. Elliptic Curve Algorithm
  3. Uses of ECC
  4. Benefits of ECC
  5. Example of ECC

1. What is ECC? 

What is elliptic curve cryptography often runs in a lot of people’s minds? ECC works by concentrating on specific pairs of public and private keys for encryption and decryption of web traffic.

There are other encryption methods existent such as the Diffie-Hellman and RSA cryptographic methods. These methods are grounded on the formation of keys employing enormous prime numbers that entails plenty of computational power.

The ECC cryptography is a key-based method that uses a public key encryption technique for encrypting data based on an elliptic curve theory. With the use of elliptic curve theory, faster, smaller, and more effective cryptographic keys are created.

With elliptical curve encryption, complex and mathematically robust keys are generated through the features of the elliptic curve equation in alteration to the traditional technique as the product of enormous prime numbers. The elliptic curve cryptosystem technology can be operated in concurrence with many public key encryption methods, including RSA and Diffie-Hellman. Various researches and studies suggest that ECC systems can attain a similar level of security with a 164-bit key when other techniques want a 1,024-bit key. This is mainly because the elliptic curve method supports to create equivalent security with lesser computing power and reduced battery resource usage due to which it is widely being used for various mobile applications.

2. Elliptic Curve Algorithm

An elliptic curve is an algorithm function for present ECC uses that is a plane and asymmetrical curve, which transverses a finite field comprising the points sustaining the following elliptic curve equation:

y²=x³ ax b.

Concerning the elliptic curve cryptography algorithm, this algebraic function (y²=x³ ax b) will appear like a symmetrical curve that is parallel to the x-axis when plotted.  The elliptic curve method is established on a sole one-way feature in which it simpler to complete a calculation but, at the same time, impracticable to invert or withdraw the outcomes of the calculation to find the initial numbers, unlike other forms of public-key cryptography. This property makes the elliptic curve cryptography algorithm more secure and efficient. 

Further, elliptic key cryptography takes into account and combines various mathematical operations than RSA to attain this property. To put it forward in simpler terms, for an elliptic curve, a line will pass only through three points, namely P, Q, and R, alongside the curve. By knowing two points, say, P and Q, the other point, R, can be found, but with just R, the other two points, P and Q, cannot be calculated.

A point upon the curve can be calculated by multiplying another point upon the curve with a number. However, it is not easy to discover the number that was used even though the start point and results are known. Equations based on elliptic curves are comparatively simple to execute and tremendously tough to reverse.

3. Uses of ECC

Elliptic curve cryptography encryption is one of the most generally used application techniques for digital signatures in various cryptocurrencies. Popular cryptocurrencies such as Bitcoin and Ethereum make use of the Elliptic Curve Digital Signature Algorithm (ECDSA key) particularly in signing transactions due to the security levels offered by ECC.

For digital signatures, ECC is applied in digital signatures through Elliptic Curve DSA (ECDSA key) and in key exchange through Elliptic Curve Diffie-Hellman (ECDH). These algorithms are used in different parts of the SSL standard utilizing signing SSL certificates with ECDSA instead of RSA.

Besides cryptocurrencies, ECC is also serving as the standard mode of encryption that is used widely by various web applications and is expected to grow in future times, owing to its shorter key length, security, and effectiveness.

Elliptic curve cryptography is also used in a range of functions such as:

  • Protecting the sensitive data and internal interactions by the U.S. government,
  • Maintenance and assurance of anonymity in TOR project,
  • A method or basis on which ownership is proved in respect of Bitcoins,
  • Providing signatures in Apple’s iMessage service,
  • Encoding DNS data with DNSCurve.

Elliptic Curve Cryptography ECC is also the most favored process for authentication over SSL/TLS for safe and secure web browsing.

4. Benefits of ECC

Elliptical encryption using Public-key cryptography based on algorithms is relatively easy to process in one direction and challenging to work in the reverse direction. For better understanding, ECC keys are efficient than RSA as RSA depends on the theory that multiplying prime numbers to get a greater number is simple, and factoring large numbers to return to the original primes is difficult.

The usual ECC key size of 256-bit is equal to a 3072-bit RSA key, which is 10,000 times efficient than a 2048-bit RSA key. Therefore, to remain safe and to be ahead of a hacker’s computing power, RSA keys must be long and requires keys that are 2048-bit or longer, which makes the process slower.

As ECC uses simpler and smaller keys, size is one of the prime advantages of elliptic curve cryptography. With the power to consume less energy to factor and convert more power to small mobile devices, it makes RSA’s factoring encryption weaker.

In contrast to other encryption methods, with ECC, a similar security level and high security can be attained using smaller and faster keys with less computational power.

The elliptic curve also serves security benefits and acts as the perfect alternative to RSA and DSA in situations where any threats or weaknesses are discovered in RSA, especially in cases where the remedies for the threats need a significant rise in key size.

The usage of smaller keys in ECC makes it faster as less data is transmitted from the server to the client during an SSL coordination process. Further, ECC entails lesser processing power and memory, as a consequence of which improved and faster response times are generated throughout on Web servers during usage.

The Perfect Forward Secrecy (PFS) is also an important benefit of ECC, especially for web-servers that desire Ephemeral ECDH (ECDHE) using cipher suites as they reap the advantages of both ECC and PFS.

5. Example of ECC

The elliptic curve is a graph that denotes the points created by the following equation:

y²=x³ ax b

In this elliptic curve cryptography example, any point on the curve can be paralleled over the x-axis, as a result of which the curve will stay the same, and a non-vertical line will transect the curve in less than three places.


A fast-growing and most preferred form in the field of encryption, the elliptic curve cryptography is a chief development in the cryptography used in SSL. ECC serves as a feasible alternative to the existing and traditional algorithms and provides various advantages in terms of security, speed, performance, and speed. The ability of ECC to use complex mathematical algorithms for data protection makes many researchers in the field of encryption anticipate the future of ECC to be bright and game-changing.

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