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I strongly believe that the ability to change your mind is one of the most important skills in today’s world. I have always seen the inability to engage in conversations that challenge your fundamental beliefs and strong opinions as a fatal flaw. I believe that engaging in such conversations either strengthens your beliefs or challenges them, forcing you to ask important questions about why you…
Life is ultimately a search problem. Imagine you were asked to solve the following problem. Given the points A, B, C, D, E, and F, what is the best path to go from point A to F while passing through every point exactly once. Here are the possible options: A → B → C → D → E → F A → B → D → C → E → F …. A → D → B → C → E → F There are 24 possible paths you could take. Now, let’s loosen the rules to…
The need to make systems and algorithms faster so as to make them more practical has been ever constant in the world of software engineering and cryptography at large, and lattice-based cryptography is not left out. Number-theoretic Transform (NTT) was used in ML-KEM (Module-Lattice-Based Key-Encapsulation Mechanism), a lattice-based cryptographic algorithm used in establishing a shared secret key…
During the weekend I spent time with a friend. We talked at length about work and life in general. One idea that really stood out was Death Ground —the notion that while war is terrible for people, it often accelerates progress for nations because it forces rapid innovation and adaptation in ways peace rarely does. The story of Hernán Cortés illustrates this idea well. As narrated by ChatGPT: In…
This is the first post in the series: A Deep Dive on Lattice-Based Cryptography . We focus on lattices as an independent mathematical concept and its properties. However, we focus on aspects that would be useful in the study of Lattice-based Cryptography. Throughout this series, we use 2-dimensions and 3-dimensions because they are easy to work with. In practice we use sufficiently higher…
“Mathematical intuition is the ability to see the truth without first having gone through a formal process of reasoning.” — Henri Poincaré The goal of this post is to force you to think about the chain rule a bit deeply. The only prerequisite for reading this is understanding the power rule(i.e $\dfrac{d}{dx}[x^n] = nx^{n-1}$). Understanding differentiation intuitively is recommended.…
For $y = f(x)$, differentiation is the ratio of how $y$ changes as $x$ changes . This is the rate of change of the function. For a straight line, this would be constant. For example, the derivative of $f(x)=2x + 1$ is 2. This is the good old slope of a graph(the ratio of change in $y$ to the change in $x$: $\dfrac{y_2 - y_1}{x_2 - x_1}$) But then, how do we find the rate of a change of a function…