In one of my papers, I needed an upper bound on the fractional moments of the geometric distribution. The integer moments have a nice-looking upper bound involving factorials. In addition, Mathematica showed that the same upper bound holds for fractional moments once we use Gamma function instead of factorials. The problem was, I could not prove it. Worse, I could not find a proof after much looking on the Internet.
This post contains a proof that for any real , the th moment of a geometric random variable with success probability is at most .
Continue reading “Fractional Moments of the Geometric Distribution”
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 a blockchain protocol such as Bitcoin, the users see the world as a sequence of states. A simple yet functional view of this world, for the purpose of analysis, is a Boolean string of zeros and ones, where each bit is independently biased towards favoring the “bad guys.”
A bad guy is activated when for some . He may try to present the good guys with a conflicting view of the world, such as presenting multiple candidate blockchains of equal length. This view is called a “fork”. A string that allows the bad guy to fork (with nonnegligible probability) is called a “forkable string”. Naturally, we would like to show that forkable strings are rare: that the manipulative power of the bad guys over the good guys is negligible.
Claim (, Bound 2). Suppose is a Boolean string, with every bit independently set to with probability for some . The probability that is forkable is at most .
In this post, we present a commentary on the proof that forkable strings are rare. I like the proof because it uses simple facts about random walks, generating functions, and stochastic domination to bound an apparently difficult random process.
Continue reading “Forkable Strings are Rare”
In this post, I am going to review two erasure codes: the Blaum-Bruck-Vardy code and the BASIC code (also here). These are erasure codes, which means, their purpose is to encode a number of data disks into a number of coding disks so that when one or more data/coding disks fail, the failed disk can be reconstructed using the existing data and coding disks.
A strength of these codes is that although the algebra is described on extension fields/rings over , the encoding/decoding process uses only Boolean addition/rotation operation and no finite field operation. These codes are also MDS (Maximum Distance Separable), which means they have the largest possible (minimum) distance for a fixed message-length and codeword-length.
(Recall that if a code has data components and parity components in its generator matrix in standard form, its distance is at most by the Singleton bound. Hence the code is MDS if and only if it can tolerate arbitrary disk failures.)
The BASIC code does the following things in relations to the BBV code:
- Adds a virtual parity bit after each disk, giving each disk an even parity
- Does polynomial arithmetic modulo instead of as in the case of BBV code
- Shows equivalence to the BBV code by making a nice observation via Chinese Remainder Theorem
- Proves MDS property for any number of coding disks when is “large enough” and has a certain structure
Open Question: What is the least disk size for which these codes are MDS with arbitrary distance?
Continue reading “Two MDS Array Codes for Disk Erasures: the Blaum-Bruck-Vardy Code and the BASIC Code”