Hi, I’m Saad, I love maths and I love to backpack in the woods. Welcome to my blog! Here’s my story and if you are technically inclined, here’s my notebook.

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# Saad Quader

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# Our Paper on Confirmation Time for Proof-of-Stake Blockchains

# Our Paper on Realizing a Graph on Random Points

# My Takeaways from “Endure,” by Alex Hutchinson

# My Presentation on Proof-of-Work vs. Proof-of-Stake Blockchain Protocols

# Notes on the PCP Theorem and the Hardness of Approximation: Part 1

## PCP Theorem: the Interactive Proof View

# Things I Want to Remember

“The magic of mathematics and theoretical computer science is all the unexpected connections.'' ~~ Robert Endre Tarjan

**What is the Confirmation Time for Blockchains?**

When you are selling a pizza and accepting Bitcoin, usually you wait “for a while” before you “confirm the transaction” and deliver the pizza. You are cautious because a fraudulent user may “double spend”—use the same bill to pay two different vendors.

If you wait for a while before confirming, the block having your transaction will get deeper and deeper in the blockchain, making it harder for a fraudulent buyer to double-spend the money you received.

But how long should you wait?

Waiting too long means that only a fraction of the transactions get confirmed in a given timeframe—not good for the hungry buyers or the network when we compare it against the “almost instant” confirmation time for regular credit card transactions.

Continue reading “Our Paper on Confirmation Time for Proof-of-Stake Blockchains”

Our paper, titled How to Realize a Graph on Random Points, has gone up in the Arxiv. I’ll write more about it in future posts.

**Abstract:**

Continue reading “Our Paper on Realizing a Graph on Random Points”

I finished listening to the book “Endure” by Alex Hutchinson. It is one of the most important books that I have read. I have had multiple important realizations from it.

Brain sets an expectation. Giving up is almost always a choice.

**Brain sets an expectation.** You know it or not, feel it or not, your brain sets an expectation at a physiological as well as a psychological level. Try as we may, the brain controls your actions (again, at both the physiological and psychological levels) so that you meet the expectation, but don’t exceed it.

Therefore, **these expectations matter**. Sometimes these expectations are conscious, in that we can spell them out. Other times, they are at a subconscious or even unconscious level. Are you aware of what you expect of yourself? Expect does not mean hope: what do you really, really expect of yourself?

Continue reading “My Takeaways from “Endure,” by Alex Hutchinson”

I gave a talk in our seminar about the proof-of-work vs. the proof-of-stake blockchain paradigm. Although I don’t have an audio/video recording, here is a Google Slides rendering of my original Powerpoint slides. Some of the animations are out of place/order, but in general, it feels okay.

I intended this talk to be accessible in nature, so I intentionally skipped many details and strived not to flaunt any equation in it.

**Advertised Summary: **Bitcoin is a blockchain protocol where finalized transactions need a “proof of work”. Such protocols have been criticized for a high demand for computing power i.e., electricity. There is another family of protocols which deals with a “proof of stake”. In these protocols, the ability to make a transaction depends on your “stake” in the system instead of your computing power. In both cases, it is notoriously difficult to mathematically prove that these protocols are secure. Only a handful of provably secure protocols exist today. In this talk, I will tell a lighthearted story about the basics of the proof-of-work vs. proof-of-stake protocols. **No equations but a lot of movie references**.

Please enjoy, and please let me know your questions and comments.

In this note, we are going to state the PCP theorem and its relation to the hardness of approximating some NP-hard problem.

Intuitively, a PCP (Probabilistically Checkable Proof) system is an interactive proof system where the verifier is given random bits and he is allowed to look into the proof in many locations. If the string is indeed in the language, then there exists a proof so that the verifier always accepts. However, if is not in the language, no prover can convince this verier with probability more than . The proof has to be short i.e., of size at most . This class of language is designated as **PCP[r(n), q(n)].**

Theorem A (PCP theorem).Every NP language has a highly efficient PCP verifier. In particular,.

Continue reading “Notes on the PCP Theorem and the Hardness of Approximation: Part 1”