Blockchains and Mathematical Freedom


Bitcoin and other crypto-currencies are a type of money secured by the fact that finding them requires an incredible amount of computer power. Letting people use and trust this security requires a very special piece of mathematics that lets a structure called a blockchain make it so that anyone can check the origin and transfer of ownership of each and every piece of cyber-money in the system. The special math is a trapdoor function In today’s post, Occupy Math wants to talk about some things besides setting up a crypto-currency you can do with blockchains.


We start with the math. A trapdoor function is a series of computations that change an input into an output, but also make it incredibly hard to turn the output back into a copy of the input. These functions not only make bitcoin work, they are also the core of public key codes, which are codes that have different keys for encoding and decoding. You give everyone your encoding key. Trapdoor technology means that the messages encoded by your correspondents can only be read by someone with the secret decoding key. In the picture above the tan circle is the messages that have not yet been encoded and the gray one is the messages that only you can read with the secret key.

You may have heard that bitcoin is having a bad day. This is true, but it’s for reasons unrelated to the math it uses. The trapdoor function in bitcoin are used to solve the problem of spending a piece of digital money more than once — the encryption puts a record of your spending where anyone can see it so that, if you copy your digital money and try to spend it again, you will be caught.

If you want a technical explanation of blockchains, the wonderful video series 3Blue1Brown has one of the best explanations of blockchains Occupy Math has ever read. If you don’t have time for the video, what blockchains do is to let you set up a distributed, open, secure ledger system. This is a way of recording transactions where everyone can both see them and have confidence that what is recorded is really what happened.

Open, distributed, secure ledger? What’s that good for?


The notorious mob boss Al Capone went to jail for eleven years because his accountant let law enforcement have access to his financial ledger. Following his money is the “red line” that President Trump said that the investigation into election-tampering must not cross. Cooking the books with bad math is the proximate cause of the Great Recession in 2008 — banks got greedy and it all eventually fell in ruins. Ledgers are how you keep track of a financial institution or network. An open distributed ledger that anyone can verify would have made thousands of horrible, costly crimes far more difficult to commit.

Let’s take a specific example, home owner’s insurance. You create a blockchain and software to help people use it and declare that as soon as $10,000,000 in policies are sold the insurance will go active. People buy in and get the insurance activated. To make a claim, just like in standard insurance, you need fire and police reports and an inspector’s report that the claim is valid. The inspector works for (is paid by) the money pool in the blockchain — he has an access type that tells the blockchain that he is an insurance investigator and filing a report results in a payment.

  • Since everything reported by every inspector is visible, suspicious conduct is very risky. Someone approving fake claims will be much more visible in this system.
  • The people in the chain that validated the reports, claims, and payouts are visible to everyone. If you’re going to cheat you have to do it in public.
  • It may be, with member agreement at the level of bylaws, that even the state insurance inspector, or whatever the equivalent office is, will have access and can perform inspections whenever they wish.

What the blockchain enables is insurance without an expensive, and sometimes corrupt, company taking a large cut. An inspector who is too freely approving claims or who files fake inspection reports, and people that over-insure or try other forms of cheating, are all visible to everyone. The visibility alone will reduce fraud.

This system is a bit cumbersome. There has to be a very careful list of rules for what types of transactions are allowed, maximum levels of insurance, how to get registered as an inspector and so on. There will be a small administrative structure, mostly run by member volunteers and hired experts. It is also completely against the law (now) in many countries to run insurance this way and so there would need to be changes in regulations to make blockchain secured insurance possible.

The potential of blockchains is to reduce cost and corruption.

The distributed blockchain insurance company described above has a lot of missing details and needs new laws to even be permitted. That said, it would make insurance fraud much harder. Shady inspectors or people that are over-insuring their home would be doing it much more openly. Members have the right to examine all transactions and a financial motive to do so. You might even have “designated monitors” who get a rake-off on their premiums. The key point is this: getting rid of the huge company at the center of things saves lots of money, and the system itself makes it impossible for executives to get up to no good, because there are no executives.


Another potential use of blockchain technology is charity. Some charities are run to make money for themselves, making only nominal payments to those they are supposed to help. Others are outright scams. A charity does need some money to run, but this rake-off for overhead can grow to scam levels, even with the best of intentions. Large salaries and perks for executives or even simple inefficiency can keep help from those that need it. Government welfare programs often fall into this area. When most of the dollars to help the disadvantaged are spent on the people administering the program and checking to make sure the disadvantaged are not getting away with anything, then the government is running a huge charity scam.

A blockchain system for making and tracking donations and payouts could reduce (not eliminate) these sorts of problems. Clients of the system might have debit cards that the system refills at appropriate intervals. The system will let donors see exactly where their contributions end up and document the exact rake-off that is used to keep the servers running and administer member votes on rules and policy changes. Making your votes proportional to your recent contributions might even encourage giving — if the members approve that rule. The key thought is that everything happens in the open according to rules that are open and under the control of members.

Blockchains have the potential to increase freedom.


Transparency and open dealing are a pillar of democracy. The on-line masthead of the Washington Post declares that Democracy Dies in Darkness, a view with which Occupy Math heartily agrees. The ability of blockchains to create secure open ledgers has the potential to increase openness and transparency in areas where democracy is currently taking the most damage.

One of the things blockchains will do is make certain types of companies — insurance firms, stockbrokers, and banks, for example — less important and powerful. This means that there will be incredible push back against regulations making blockchain technology a way to do what these companies do now. There will be fake news, advertisements about how blockchains are coming to steal your life savings, worried looking mothers holding small babies and speaking about the future in vague and scary terms. This is another example of why an informed citizen is a better citizen.

Occupy Math is excited about the potential of blockchains. They are a place where subtle and beautiful math can impact the world in a big way. As always, the technology can be mis-applied and used to support immoral and criminal behavior, but it can also do enormous good. I urge you to watch the video on blockchains before the nonsense merchants set up their shop. Occupy Math agrees that there is peril in using blockchain technology, but that is just another way of saying that we are trying to run the world with human beings. Any thoughts on all this? Please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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