What is elliptic curve cryptography? ECC for dummies

Elliptic curve cryptography (ECC) is a public key cryptosystem that leverages the elliptic curve theory and the mathematical properties of elliptic curves to provide secure communication and encryption.

The elliptic curve discrete logarithm problem (ECDLP) is the foundation of security in elliptic curve cryptography. The ECDLP involves finding the exponent (or logarithm) in the elliptic curve equation when given two points on the curve.

Elliptic curve cryptography emerged in 1985 when Victor Miller and Neal Koblitz independently suggested using elliptic curves in cryptographic algorithms. The new system was more efficient and immune to attacks researchers discovered on other modern cryptosystems, such as RSA encryption. Several organizations and industry bodies have standardized various aspects of ECC for cryptographic applications in the early 2000s.

An elliptic curve is a set of points defined by a specific equation, usually in the form of y² = x³ + ax + b, where a and b are constants, determining the shape and characteristics of the curve. The equation describes the relationship between the x and y coordinates of the points that lie on the curve.

Elliptic curves exhibit a few fascinating properties:

• Horizontal symmetry. If a point (x, y) is on the curve, its reflection point (x, -y) is also on the curve. The y-coordinate of the reflection point is the negation of the original point’s y-coordinate. The operations on these points form the foundation for the cryptographic algorithms used in ECC.
• Interpolation. Any non-vertical straight line intersects the curve at a maximum of three distinct points. It helps to determine the whereabouts of a point on the curve based on other known points.
• Group structure. Elliptic curves form a mathematical group under an operation called point addition. This operation defines how you can add two points on the curve to produce a third point on the same curve.

Elliptic curve cryptography is a type of public key cryptography, so each user has a pair of ECC keys: a public key and a private key.

• The public key is shared with others. Then anyone can use it to send the owner an encrypted message.
• The private key is kept secret – only the owner knows it. They need it to decrypt the received encrypted message.

The security of the ECC encryption relies on the relationship between the key pairs and the properties and mathematical problems of the elliptic curve.

Here’s how ECC works:

1. Key generation. Bob and Alice select a specific elliptic curve with known parameters. They can then independently choose or generate random numbers as their private keys. Once Bob’s private key is ready, he computes the corresponding public key using his private key and the chosen elliptic curve.
2. Key distribution. Bob shares his ECC public key with whomever he wants to exchange messages with, let’s say his friend Alice.
3. Encryption. Once Alice knows Bob’s public key, she uses multiple calculations based on elliptic curve theory to transform a plaintext message into ciphertext.
4. Decryption. Bob receives the encrypted message and uses his valid ECC private key to obtain the original plaintext message.

It may sound complicated, but it’s a simplified overview of the steps involved in ECC.

In practice, elliptic curves used in cryptography often have additional parameters. For example, elliptic curves can be defined over finite fields rather than real numbers. A finite field is a mathematical structure with a finite set of elements and two operations, usually, addition and multiplication, used to perform computations on elliptic curve points.

In theory, elliptic curve cryptography is secure. In practice, ECC’s security deeply depends on correct implementation and the use of appropriate parameters, such as the size of the underlying elliptic curve and the ECC key length. The encryption will be vulnerable to attacks if you choose weak parameters or inadequate key size.

NSA has developed some ways to compromise certain types of ECC. However, it applies only to specific kinds of curves, and even then, it’s a demanding task that they wouldn’t sustain decrypting large amounts of data.

Quantum computing might crack ECC in the future, but it’s impossible to break elliptic curve cryptography with current computational power. So most experts consider elliptic curve encryption secure and superior to other public key encryption, such as RSA.

The elliptic curve cryptography and Rivest-Shamir-Adleman (RSA) are the most prominent and widely used public-key cryptographic systems. Here are their main similarities and differences:

• Public-key cryptography. ECC and RSA are types of asymmetric cryptography that use a pair of public and private keys for encryption and decryption.
• Mathematical foundations. ECC and RSA rely on the difficulty of mathematical problems for their security. RSA strength depends on the difficulty of factoring large numbers, while ECC relies on solving the elliptic curve discrete logarithm problem.
• Key size. ECC requires a shorter key length to achieve the same level of security. A 256-bit elliptic curve cryptography key is equivalent to a 3072-bit RSA key in terms of security strength.
• Efficiency. Since the key size is smaller, ECC also takes less computational power, bandwidth, and memory. Therefore, it’s more efficient and faster. RSA could be as efficient with a smaller key size, but that would compromise its security.
• Applications. RSA and ECC are widely used cryptosystems. RSA has been used for several decades and is well-established and standardized in many systems and applications. However, RSA is losing its spotlight as ECC has been gaining popularity as a more efficient and sustainable alternative.

Elliptic curve cryptography has many advantages, making it stand out among public key cryptosystems.

• Strong security. Elliptic curve cryptography provides the same level of security as other cryptosystems, but ECC keys are much smaller.
• Efficient performance. ECC operations require fewer computational resources, storage space, and bandwidth than most public key cryptosystems. It makes ECC suitable for devices with limited computational power, such as mobile devices and embedded systems, or for transmitting data over low bandwidth networks.
• Standardization. Since cryptographic standard organizations and industry bodies have standardized various aspects of ECC for cryptographic applications, you can find elliptic curve cryptography in many modern cryptographic libraries, protocols, and applications.
• Compatibility. Implementing ECC across different platforms and integrating it into existing cryptographic systems or protocols is possible. ECC works seamlessly alongside other cryptographic algorithms.

You can use elliptic curve cryptography for securing communication and creating digital signatures. Real-life applications of ECC include:

• Communication protocols. ECC protects the confidentiality, integrity, and authenticity of network data. Therefore, communication protocols, such as Transport Layer Security (TLS) and Secure Shell (SSH), often take advantage of elliptic curve cryptography. For example, TLS handshake uses elliptic curve cryptography algorithms for key exchange and ECC-based digital certificates for server authentication.
• Mobile devices and the Internet of Things (IoT). Because of ECC’s efficiency and compatibility, ECC can secure communication in devices with limited processing power and memory, such as smartphones, wearables, and IoT gadgets.
• Digital signatures. ECC is handy for generating and verifying digital signatures in e-commerce, financial, and other systems. It ensures the authenticity and integrity of digital documents, contracts, and transactions.
• Payment systems. ECC protects payment systems, including contactless and mobile payment solutions. From securing key exchange to encrypting transaction data and verifying the authenticity of the data’s owner, it helps to secure transactions, protect sensitive financial information, and ensure the integrity of payment processes.
• Virtual private networks (VPNs). VPNs can use ECC to establish secure and encrypted connections between clients and servers. VPNs usually use ECC for secure key exchange and server authentication while establishing a VPN connection.
• Email and messaging. Email protocols, such as Pretty Good Privacy (PGP) or Secure/Multipurpose Internet Mail Extensions (S/MIME), also use ECC. It helps to encrypt and digitally sign email messages, enabling secure communication and protecting the privacy of email content.
• Blockchain and cryptocurrencies. Many blockchain platforms and cryptocurrencies, such as Bitcoin and Ethereum, use ECC for generating and managing digital signatures, verifying transactions, and securing underlying cryptographic protocols.

Practical applications don’t end here. ECC’s security, efficiency, and compatibility have made it a preferred choice in many industries, including finance, e-commerce, telecommunications, and IoT. And ECC will likely stay a preferred choice until we are future-proofed with post-quantum cryptography.