This is the first post in the series: Introduction to 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 dimensions. The only prequisite to start this series is to understand the following: $\mathbb{R}$ represents the set of all real numbers. $\mathbb{Z}$ represents the set of all integers. $\mathbb{Q}$ represents the set of all rational numbers. $\mathbb{R}^n$ represents the set of all n-dimensional real vectors

Understanding Why Chain Rule Works "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. The chain rule is a differentiation method for composite functions. It is defined as follows for a function $f(u)$ where $u = g(x)$ $$\dfrac{d}{dx}[f(u)] = \dfrac{d}{du}f(u)*\dfrac{d}{dx}(u)$$ which implies differentiate $f(u)$ with respect to $u$ treating $u$ as an independent variable, and differentiate $g(x)$ with respect to $x$ and then multiply both.

Intuition behind Differentiation 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 that's a curve? We could decide to find the slope using $\dfrac{y_2 - y_1}{x_2 - x_1}$ but this only tells us the rate of change between certain intervals in $x$. It doesn't represent change in the whole function. This might be useful in some cases but how do we get the rate of change in the function as a whole. We know that the slope can't be constant because it's a curve so we resort to finding the instantaneous rate of change. We may not be able to find the rate of change of the whole curve but can find the rate of change of the curve in an instant.

Purpose: Summarizing your activity during the program, reflect on your project and participation. Self-Introduction You deserve to know more than just my name. I am Godspower Eze, I am from Anambra, Nigeria and currently based in Lagos, Nigeria. And, I have been programming professionally for 3 years. Contributions Elixir Consensus Client Link - Project Proposal

This is a summary of my week 16 and last dev update. We have come a long way! What did I work on for week 16? I worked on process_sync_aggregate and enabling incoming requests for beacon blocks by range in the Elixir Consensus Client. Helpful Resources Sync aggregate processing https://eth2book.info/capella/part3/transition/block/#sync-aggregate-processing BeaconBlocksByRange - https://github.com/lambdaclass/lambda_ethereum_consensus/blob/main/docs/specs/p2p-interface.md#beaconblocksbyrange What's next?

Hey, there! This is a summary of my week 15. What did I work on for week 15? I worked on the function that allowed validators to withdraw stakes or rewards from the Elixir Consensus Client. How do withdrawals work? Every Proof-of-stake system needs a way for validators to withdraw their stakes and rewards. This was added in the Capella updrade; the first upgrade after the merge.

Hey, there! This is a summary of my week 12, week 13 and week 14. What did you do in the past three weeks? I have been focused on contributing to the Elixir Consensus Client by LambdaClass. I got 4 pull requests merged and more are in the works. What is the most fun issue you worked on so far?

Hey, there! This is a summary of my week 10 and week 11. How was the last two weeks? A bit rough tbh. But, guess who's here now? Me! What did you do in the past two weeks?

Hey! I am Godspower Eze and I am working towards implementing a verkle library in the Nim Programming Language with a focus on Pedersen Commitment and Lagrange Interpolation. I am also contributing to Consensus Clients This is a summary of week 8 and week 9. What did you do in the past two weeks? Read the book: Why and How ZKSNARKS. That helped me understand why we use Polynomials in cryptography Implemented two flavors of Polynomial Commitments; Using Modular Exponentiation and Using Elliptic Curve Operations. Made multiple PRs to the Elixir Consensus Client by Lambdaclass and they were merged.

Hey! I am Godspower Eze and I am working towards implementing a verkle library in the Nim Programming Language with a focus on Pedersen Commitment and Lagrange Interpolation. I am happy to share that I am officially a fellow. It's a recognition for the efforts I have put into the program in the past months as a permissionless participant and I don't take it for granted. I am grateful. This is a summary of week 6 and week 7. How do you feel? I feel great to say the least.

Hey there 👋🏾 I am Godspower Eze and I am working towards implementing a verkle library in the Nim Programming Language. This is a summary of how week 2 and week 3 went. What did I learn in the past two weeks? The past two weeks has been challenging and fun. I went from confused to a bit less confused. For complex topics, you will find that this a very big win. I finally understand the gist of Elliptic Curve Cryptography(ECC).

I am Godspower Eze and I am working towards implementing a verkle library in the Nim Programming Language with a focus on Pedersen Commitment and Lagrange Interpolation. This is a summary of week 4 and week 5. How has the last two weeks been? It's been great! I successfully implemented Elliptic Curve Diffie-Hellman (ECDH) and Elliptic Curve Digital Signature Algorithm (ECDSA). This is really huge for me and I am so proud of myself! I would have loved to explain them to you but I have already given detailed explanations on both implementations. So, just check them out.

Hey there 👋🏾 I am Godspower Eze and I am working towards implementing a verkle library in the Nim Programming Language. This is how week 1 went. What did I learn this week? I learnt a lot! I started out by completing my study on the Verkle Tree paper. And then, watched a video by Dankrad feist which cemented my fundamental knowledge of how a verkle works.

Introduction Hey 👋🏾 I am Godspower Eze and I am committed to participating in the Ethereum Protocol Fellowship as a Permissionless Participant. I have always been fascinated by the inner workings of Ethereum and this is an oppurtunity to learn and contribute to it. For the next four months, I want to learn as much as possible as partaining to my project of choice and gain enough experience to work in core engineering. Project of Choice