Elliptic Curve Cryptography (ECC) has become a cornerstone of modern cryptographic systems. From signing blockchain transactions to securing encrypted messaging, ECC is everywhere. But why is it so popular? What makes it different from RSA? And how can we implement it in Python? In this blog post, we’ll briefly dive into the theory of ECC and then see how to perform elliptic curve arithmetic operations using the LightECC Python library in just a few lines of code.
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Why is ECC so widely used?
ECC is heavily used in blockchain systems like Bitcoin and Ethereum—for example, in wallet address generation and transaction signing. Popular signature schemes such as ECDSA (Elliptic Curve Digital Signature Algorithm) and EdDSA (Edwards-curve Digital Signature Algorithm) are based on elliptic curves.
More recently, GnuPG has shifted away from the traditional RSA default. The latest versions now use ECDH (Elliptic Curve Diffie-Hellman) for encryption and EdDSA for signing by default, breaking RSA’s long-standing dominance.
Even though ECC is not quantum-resistant, Apple still relies on ECC in iMessage—in combination with post-quantum algorithms for added security.
ECC vs. RSA: A Matter of Efficiency
At first glance, RSA might seem easier to understand because it’s based on modular exponentiation with large integers. However, for computers, ECC is far more efficient: its operations are based on simple arithmetic—point addition and scalar multiplication—making it faster and more scalable.
When we increase RSA key sizes for stronger security, the performance cost grows exponentially. In contrast, ECC key sizes can be increased linearly in terms of performance overhead while achieving the same level of security. For instance, A 256-bit ECC key is roughly equivalent in security to a 3072-bit RSA key.
ECC Isn’t Simple—And That’s Okay
What makes ECC seem complicated is its variety of curve forms, each with their own equations and rules for arithmetic. The most common forms are:
- Weierstrass: Used in ECDSA (e.g., secp256k1 for Bitcoin). Uses the equation y2 = x3 + ax + b
- Koblitz: A special case over binary fields for hardware design. Uses y2 + xy = x3 + ax2 + b
- Edwards: Used in EdDSA, designed for faster and more secure implementations. Uses x2 + y2 = 1 + dx2y2
Graphical Interpretation
Weierstrass and Koblitz: When adding two points on the curve, we draw a line through them. This line intersects the curve at a third point, and we reflect that point over the x-axis to get the result.
Edwards curves don’t have such a graphical interpretation, but their formulas can be proven by mathematical induction. Their arithmetic is typically faster and less error-prone.
Let’s Code: ECC with Python using LightECC
You don’t need to implement elliptic curve arithmetic from scratch. With LightECC, you can define a curve and perform operations in just a few lines of code.
Install LightECC as
pip install lightecc
Define your curve as
from lightecc import LightECC
# build an elliptic curve
ec = LightECC(
form_name = "edwards", # or weierstrass, koblitz. default is weierstrass.
curve_name = "ed25519", # check out supported curves section
)
# Generator point
G = ec.G
Perform arithmetic operations as
# addition _2G = G + G _3G = _2G + G _5G = _3G + _2G _10G = _5G + _5G # subtraction _9G = _10G - G # multiplication _20G = 20 * G _50G = 50 * G # division _25G = _50G / G
Division? Not So Fast…
While LightECC supports division, elliptic curve division is not efficient. In fact, the difficulty of the Elliptic Curve Discrete Logarithm Problem (ECDLP)—that is, figuring out k from P = k * G—is what makes ECC secure. It’s a one-way function: easy to compute, hard to reverse.
Do We Really Need to Know All These Details?
Not necessarily. As developers, we often use cryptographic libraries as black boxes. But having a high-level understanding of how ECC works—and what makes it secure—can help you choose the right tools, write safer code, and even contribute to the libraries you use.
Final Thoughts
Elliptic Curve Cryptography is not only secure and efficient but also elegant once you get past its initial complexity. With libraries like LightECC, you can experiment with curve arithmetic, signature schemes, and even encryption in pure Python.
So the next time you sign a blockchain transaction or send an encrypted message, remember: behind the scenes, a beautiful little curve is working its magic.
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