Elliptic Curve Cryptography
Elliptic Curve Cryptography (ECC) is a powerful and efficient way to secure digital communications.
It is widely used in everything from encrypted messaging apps to securing websites with HTTPS. ECC provides the same level of security as traditional methods (like RSA) but with much smaller keys, making it faster and more efficient—a huge advantage for modern computing.
Math on a curve
Instead of using massive prime numbers (like in RSA), ECC is based on elliptic curves, which are special mathematical shapes defined by an equation of the form:
If you plot this equation, you get a smooth, looping curve. Mathematical operations on this curve provide a secure way to encrypt and decrypt information.
One key operation is point addition, where adding two points on the curve always results in another point on the curve. Another is scalar multiplication, where a point is repeatedly added to itself—this forms the foundation of ECC encryption.
Why is ECC Secure
ECC relies on a problem called the Elliptic Curve Discrete Logarithm Problem (ECDLP). This problem is incredibly hard to solve—much harder than breaking RSA encryption of the same key size.
For example, a 256-bit ECC key provides about the same security as a 3,072-bit RSA key. This means that ECC can offer stronger security with smaller keys, which require less computational power (a substantial benefit for mobile devices).
Where is ECC Used?
ECC is widely used in modern security systems, including:
- HTTPS & TLS - Securing websites
- Bitcoin & Cryptocurrencies - Digital signatures for transactions
- End-to-End Encrypted Messaging - Used in Signal, WhatsApp, and iMessage
- Smart Cards & IoT Devices - Efficient security for small devices
There are several different standards covering selection of curves for use in elliptic-curve cryptography (ECC):