The Enigmatic Beauty of Elliptical Curves
Ramakrishna Prasad Nori (RK)
Founder - Head AI Research & Solutions | November 1, 2024
First and foremost, let's clarify that an elliptical curve isn't precisely an ellipse. The most prominent and widely discussed application of elliptical curves resides in the realm of cryptocurrency, powered by blockchain technology. Elliptical Curve Cryptography, as it's known, plays a pivotal role in creating unbreakable public keys for blockchain transactions, at least for the time being. The potential impact of quantum computing on this encryption method is a topic for another conversation entirely.
From my earliest school days, I had a passion for ellipse, parabola, but none captivated me quite like the elliptical curve. The mere sight of this curve is aesthetically pleasing. It possesses symmetry over the horizontal axis, the x-axis, and boasts an inherently beautiful shape. However, like all things beautiful, it conceals numerous mysteries, making it a superstar in the world of cryptography.
One intriguing property of the elliptical curve is that any random point on it can be reflected over the x-axis, and remarkably, the reflected point also resides on the same curve. Another captivating feature is that any non-vertical line will invariably intersect the curve in at least three distinct points.
However, the most crucial feature, particularly in cryptography, pertains to point addition on the curve. Consider two arbitrary points on the curve; you can reach a final point by repeatedly adding the point to itself 'n' times. Determining 'n' with knowledge of only the first and final points is an insurmountable challenge. In simpler terms, you can move from the initial point to the last, but you can't reverse the journey.
For a more visual analogy, let's delve into a key moment from the Mahabharata, the Battle of Kurukshetra. On the 13th day of the battle, Guru Dronacharya's well-devised diversion plan leads Arjuna and Lord Krishna away from the battle's epicentre. The Kauravas gain the upper hand with the formation of the Padmavyuham. To regain their advantage, Abhimanyu takes the lead and enters the Padmavyuham. He knows how to get in but not how to get out.
In mathematical terms, this scenario embodies the concept of a trapdoor function. A trapdoor function, in the context of computer science and cryptography, is a mathematical function that is easy to compute in one direction but computationally difficult to reverse, unless you possess specific "trapdoor" information that allows you to perform the reverse computation efficiently.
The elliptical curve's trapdoor function proves remarkably resistant to classical computers' attempts at decryption. It's worth noting that while quantum computers may eventually break these encryption keys, for now, we can continue to revel in the captivating beauty of this curve.