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🍊 Sat Slice #1: "Mozart of Math" on Anonymity and Satoshi

🍊 Sat Slice #1: "Mozart of Math" on Anonymity and Satoshi

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For this first installment of Sat Slices, we ventured deep into the archives of the Bitcoin Talk forum, frequented by Satoshi Nakamoto back in 2010. What we found was unexpected: a fascinating little snippet––no more than a few paragraphs long––written in 2012 by a renowned mathematician named Terence Tao.

Who is Terence Tao?

Before we dive into the piece, it's important to understand how well-respected Terence Tao is.

Here's an excerpt from his Wiki page:

Tao has been the author or co-author of over three hundred research papers. He is widely regarded as one of the greatest living mathematicians and has been referred to as the "Mozart of mathematics."

Wow! That's high praise.

Tao has also won countless awards for his contributions to the field, including the Fields Medal, considered the Nobel Prize of mathematics.

So, it's very interesting that someone as highly respected as Tao within the mathematics community was not only aware of bitcoin as early as 2012, but also briefly commented about bitcoin's creator, Satoshi Nakamoto.

Online Anonymity is "Very Fragile"

According to Tao, "anonymity on the Internet is a very fragile thing".

Specifically, he notes that every online identity is only about 31 bits of information away from being completely exposed. "Bits" are elements of one's identity, such as gender, nationality, profession, and so forth. Any time one of these bits is exposed, a person gets closer to losing their anonymity.

Once 31 bits are revealed, it should theoretically be possible to deduce a person's identity.

The Basics of Shannon's Entropy

To understand how Terence Tao calculated the number 31, let's jump over to the world of Information Theory and learn the basics of Shannon's Entropy.

In computing, "bits" are binary states of either 1 or 0, so one bit has two possible outcomes. Expressed as a power of 2, two outcomes is 2^1, so one bit has a Shannon entropy of 1. Likewise, two bits has entropy 2 and four possible outcomes (2^2), three bits has entropy 3 and eight possible outcomes (2^3), and so on.

Applied to the real world scenarios, a coin flip has two possible outcomes (heads or tails) which is 2^1, so it has an entropy of 1. A race between 8 equally matched people has eight possible outcomes which is 2^3, so it has an entropy of 3.

31 Bits Explained

Now that we understand bits and entropy, let's see how Tao's math checks out:

Tao wrote in 2012, when he estimated there were roughly 2 billion Internet users globally. Simply put, 2 billion possible identities can be expressed as log2(2 billion) = roughly 2^31. Hence, 31 bits!

In Tao's own words:

Initially, all one knows about an anonymous internet user is that he or she is a member of this large population, which has a Shannon entropy of about 31 bits.

Terence Tao on Satoshi Nakamoto

In response to his post on Google+, one bitcoiner asked Tao point blank: "Satoshi? O_o"

Tao replied:

If you are referring to the Bitcoin founder, this is an excellent example of someone who has very carefully controlled the information about his (or her) online identity (and in cutting off all activity from that identity after a limited amount of time), and in particular keeping it well below the 31 bit threshold.

(This is not to say that there was no information emitted whatsoever; for instance, it is clear that Satoshi is quite skilled in computer science and cryptography, which is already worth a half-dozen or so bits of information. His supposed Japanese origin may well be disinformation, though.)

Per Tao, Satoshi Nakamoto has stayed anonymous largely because of achieving the following two objectives:

  1. Satoshi likely revealed less than 31 bits of information about their identity, despite having an online presence for several years.
  2. Satoshi likely published disinformation about themselves, which helped to reduce the likelihood of having their identity compromised.

The Importance of Disinformation for Anonymity

In the following excerpt, Tao provides a mathematical justification for the importance of disinformation in protecting one's anonymity online:

...suppose that one reveals 100 independent bits of information about oneself. Ordinarily, this would cost 100 bits of anonymity (assuming that each bit was a priori equally likely to be true or false), by cutting the number of possibilities down by a factor of 2^100; but if 5 of these 100 bits (chosen randomly and not revealed in advance) are deliberately falsified, then the number of possibilities increases again by a factor of (100 choose 5) ~ 2^26, recovering about 26 bits of anonymity.

In practice one gains even more anonymity than this, because to dispel the disinformation one needs to solve a satisfiability problem, which can be notoriously intractible computationally, although this additional protection may dissipate with time as algorithms improve (e.g. by incorporating ideas from compressed sensing).

Several things Satoshi Nakamoto revealed about him/herself may be false, including things like gender (male), number (individual), nationality (Japanese), and so on.

Updating Tao's Calculation for 2023

Given he wrote this over a decade ago, we should probably update Tao's original calculation to account for present day Internet user population statistics.

In 2023, there are roughly 5.19 billion Internet users globally. Expressed as a power of 2, that's log2(5.19 billion) = 2^32.

So in 2023, it would theoretically take about 32 bits of information to compromise someone's anonymity.

Original Post from Terence Tao

You can check out the original post from Terence Tao here.


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