• PRODUCTS

SOLUTIONS

• Blog
• Contact
What Is Elliptic-Curve Cryptography (ECC)? Definition & Example

DATE:

MARCH 22 2023

AUTHOR:

# Elliptic-Curve Cryptography (ECC) Explained

Elliptic-Curve Cryptography is a powerful and efficient method for encryption and is widely used in modern cryptographic applications. The security of ECC relies on the difficulty of solving certain mathematical problems, which makes it much more resistant to attacks than traditional cryptographic methods. ECC has become a popular choice for securing sensitive data in applications ranging from e-commerce to mobile devices.

## What is Elliptic-Curve Cryptography (ECC)?

Elliptic-Curve Cryptography (ECC) is a type of public key cryptosystem that uses elliptic-curve theory to secure data. In public key cryptography, each user has a pair of keys – a public key and a private key. The public key can be freely shared with anyone, while the private key is kept secret. Data encrypted with the public key can only be decrypted with the corresponding private key, and vice versa.

Elliptic-curves are a type of mathematical curve that have unique properties that make them ideal for use in cryptography. ECC uses these properties to perform mathematical operations that are difficult to reverse, making this trapdoor function a highly secure method of encryption. The security of ECC relies on the difficulty of solving the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is believed to be much harder than other mathematical problems used in traditional cryptographic methods.

ECC is particularly well-suited for use in mobile devices and other resource-constrained environments where traditional cryptographic methods may be too slow or memory-intensive. Additionally, ECC can provide a higher level of security with smaller key sizes, which is important in applications where storage and bandwidth are limited.

## Components of Elliptic-Curve Cryptography

Elliptic-Curve Cryptography (ECC) is made up of several components that work together to provide secure encryption. The first component is the elliptic curve itself, which is a mathematical curve that provides the foundation for ECC. The curve is defined by a set of parameters, including the coefficients of the equation that describes the curve and a base point.

The second component of ECC is the public key, which is derived from the base point on the elliptic curve. The public key can be shared freely with others, and it is used to encrypt data that can only be decrypted with the corresponding private key.

The third component of ECC is the private key, which is kept secret by the user. The private key is used to decrypt data that has been encrypted with the corresponding public key.

## The History of Elliptical Curve Cryptography

The history of Elliptic-Curve Cryptography (ECC) dates back to the mid-1980s when Neal Koblitz and Victor Miller independently proposed the idea of using elliptic-curves in cryptography. At the time, the use of elliptic curves was largely unexplored in the field of cryptography, and their work paved the way for a new generation of encryption techniques.

In the years that followed, ECC gained popularity due to its ability to provide a high level of security with smaller key sizes than traditional cryptographic methods. ECC was first standardized in 1999 by the National Institute of Standards and Technology (NIST) as a recommended encryption algorithm, and it has since become a widely used method for securing data in a range of applications.

Today, ECC is used in a wide range of cryptographic applications, including mobile devices, e-commerce, and secure messaging. ECC is commonly used for digital signatures regarding cryptocurrencies. Both Bitcoin and Ethereumblockchains apply the Elliptic Curve Digital Signature Algorithm (ECDSA) in signing transactions. Its security and efficiency have made it an important tool in the modern cryptographic toolbox, and it continues to be an active area of research and development in the field of cryptography

## How Does ECC Work?

Elliptic-Curve Cryptography (ECC) works by using the properties of elliptic-curves to perform mathematical operations that are difficult to reverse. ECC uses a public key and a private key to encrypt and decrypt data. The public key is derived from the base point on the elliptic curve, while the private key is kept secret by the user.

In the ECC cryptography the elliptic curve points, together with the generator point G form cyclic groups and subgroups, which means that a number r exists (r > 1), such that r * G = 0 * G = infinity and all points in the subgroup can be obtained by multiplying G by integer in the range [1…r]. The number r is called order of the group.

If we multiply the generator point G by certain integer k (the private key), we obtain an elliptic curve point P (the public key). Consequently, in ECC we have:

• Еlliptic curve (EC) over finite field
• G – generator point (fixed base point on the EC)
• k – private key (integer)
• P – public key (point)

To encrypt data using ECC, the sender uses the recipient’s public key to perform a series of mathematical operations on the data. The resulting ciphertext can only be decrypted using the recipient’s private key. This is because the private key is the only one that can reverse the elliptic-curve operations performed during encryption.

The security of ECC is based on the difficulty of solving the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is believed to be much harder than other mathematical problems used in traditional cryptographic methods. This means that even if an attacker were to intercept the encrypted data, they would not be able to decrypt it without the corresponding private key.

ECC also offers a number of advantages over traditional cryptographic methods. It can provide a higher level of security with smaller key sizes, which is important in applications where storage and bandwidth are limited.

## Elliptic-Curve Cryptography vs RSA

One major difference between ECC and RSA algorithms is the size of the keys required for secure encryption. ECC is known for its ability to provide a high level of security with smaller key sizes than RSA. This is because the mathematics used in ECC are more efficient and require fewer bits to achieve the same level of security as RSA. The 256-bit key in ECC provides same security the 3072-bit RSA key does. Smaller ECC key length mean that ECC is more suitable for use in resource-constrained environments, such as mobile devices.

Another difference between ECC and RSA is the mathematical problem on which their security relies. ECC is based on the Elliptic Curve Discrete Logarithm Problem (ECDLP), while RSA is based on the Integer Factorization Problem (IFP). While both problems are difficult to solve, ECDLP is believed to be much harder than IFP, which means that ECC may offer greater security than RSA.

Despite these differences, both ECC and RSA are widely used in a range of cryptographic applications. The choice between them depends on the specific requirements of the application, such as the level of security required, the resources available, and the performance constraints. Ultimately, both algorithms offer a high level of security and have their own unique advantages and disadvantages.

## Advantages of Elliptic-Curve Cryptography

Elliptic-Curve Cryptography (ECC) offers a number of advantages over traditional cryptographic methods. One of the main advantages of ECC is its ability to provide a high level of security with smaller key sizes than other cryptographic algorithms. This means that ECC can offer the same level of security as traditional cryptographic methods with much smaller key sizes, making it more efficient and well-suited for use in resource-constrained environments.

Another advantage of ECC is its computational efficiency. ECC uses a smaller number of mathematical operations than other cryptographic algorithms, making it faster and less memory-intensive. This makes ECC well-suited for use in applications where speed and efficiency are important, such as mobile devices.

ECC also offers better resistance to attacks, such as side-channel attacks, which can exploit weaknesses in traditional cryptographic methods. Because ECC uses different mathematical operations than traditional methods, it is less susceptible to these types of attacks.

## Limitations of Elliptic-Curve Cryptography

While Elliptic-Curve Cryptography (ECC) offers a number of advantages over traditional cryptographic methods, it also has some limitations that should be considered.

One limitation of ECC is that it can be more difficult to implement than traditional cryptographic methods. ECC requires specialized software and hardware, which can be more expensive and time-consuming to develop and deploy.

Another limitation of ECC is its vulnerability to certain types of attacks, such as quantum attacks. While ECC is considered to be highly secure against classical attacks, it may be vulnerable to attacks from quantum computers, which could break the encryption used in ECC.

Finally, ECC also faces some interoperability challenges. Because ECC is a relatively new cryptographic algorithm, it may not be supported by all devices and software applications. This can limit the use of ECC in certain environments and may make it less accessible for some users.

## How Secure is Elliptic-Curve Cryptography?

Elliptic-Curve Cryptography (ECC) is considered to be a highly secure cryptographic algorithm. Its security is based on the difficulty of the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is believed to be much harder to solve than other mathematical problems used in traditional cryptographic methods. In addition, ECC provides a high level of security with smaller key sizes than other algorithms, making it more efficient and well-suited for use in resource-constrained environments.

However, like any cryptographic algorithm, ECC is not immune to attacks. While it is currently considered to be secure against classic attacks, it may be vulnerable to attacks from quantum computers in the future. It is important to continuously monitor the security of ECC and other cryptographic algorithms and update them as needed to ensure the continued protection of sensitive data.

## What Is an Elliptic-Curve Digital Signature?

An Elliptic-Curve Digital Signature (ECDSA) is an elliptic-curve digital signature algorithm. It is used to ensure the authenticity and integrity of digital messages and documents. ECDSA generates a digital signature by using a private key to sign a message, and the corresponding public key is used to verify the signature. ECDSA is widely used in various mobile applications due to its high level of security and efficiency.

## How Is Elliptic-Curve Cryptography Used?

Elliptic-Curve Cryptography (ECC) is used in a variety of applications to secure sensitive data and communications. One of the most common uses of ECC is in digital signatures, which are used to ensure the authenticity and integrity of digital documents and messages. ECDSA, elliptic-curve digital signature algorithm, is widely used in e-commerce, mobile devices, and secure messaging applications.

ECC is also used in key exchange protocols, such as Elliptic-Curve Diffie-Hellman (ECDH), which is used to securely exchange keys between two parties. ECDH is commonly used in secure communication protocols, such as SSL/TLS, to ensure that communications between two parties are encrypted and secure.

In addition, ECC is used in various other applications, such as secure data storage and authentication. Its efficiency, security, and resistance to attacks make it a popular choice for securing data and communications in a range of environments and applications.

## Real-World Applications of Elliptic-Curve Cryptography

One of the most common applications of ECC is in the field of finance, where it is used to secure transactions and protect financial data. ECC is also used in mobile devices, such as smartphones, to secure user data and communications.

ECC has its uses in the healthcare industry as well, to secure electronic health records and protect patient privacy. It is also deployed in the government sector to secure sensitive data and communications, such as classified information and military communications.

Other applications of ECC include secure messaging, online identity verification, and secure cloud storage.

## Conclusion

Elliptic-Curve Cryptography algorithms are widely used in various applications to secure sensitive data and communications. Its efficiency, security, and resistance to attacks make it a popular choice for securing data and communications in a range of environments and applications. As technology continues to advance, ECC and other cryptographic algorithms will continue to evolve to ensure the continued protection of sensitive information. Helenix has a unique track record of delivering cryptographic solutions for a wide range of organizational needs. To learn more about our competencies visit the Custom Development section.

## FAQ

Elliptic-Curve Cryptography (ECC) has compatibility issues with older systems that may not support ECC. Additionally, a lack of understanding and awareness of its benefits can make it difficult to convince organizations to adopt ECC.

ECC is an asymmetric encryption algorithm, meaning it uses a public and private key pair for encryption and decryption. The public key is used to encrypt messages, while the private key is used to decrypt them.

An example of ECC encryption is using the Elliptic Curve Integrated Encryption Scheme (ECIES) algorithm to encrypt and decrypt messages. ECIES combines ECC for key agreement and symmetric cypher for data encryption, making it a hybrid cryptographic system.

ECC better than RSA in terms of security and efficiency, particularly for applications with limited computing resources. ECC provides equivalent security with smaller key lengths compared to RSA, resulting in faster encryption and decryption times and lower bandwidth requirements. However, RSA has better compatibility with older systems.

ECC is used for key exchange and digital signatures, while AES is used for encrypting and decrypting data. ECC provides a way to securely exchange ECC keys between two parties, while AES ensures that the actual data is kept private and secure. Both algorithms are commonly used together in modern cryptographic systems.