A POW problem that can incentivize the energy efficient CNOT computer.

So in my previous posts, I have talked about a cryptocurrency POW which I call R5 which will incentivize the construction of the reversible computer. The POW R5 however does not incentivize the early stages of reversible computation when it is difficult to construct energy efficient implementations of reversible gates. My solution to this problem is to develop a new cryptocurrency POW problem which I shall temporarily call Slimeland (I will think of a better name in the future, I promise).

Incentivizing CNOT circuits instead of full-blown reversible circuits

So recall that the CNOT gate is the logic gate $(x,y)\mapsto(x,x\oplus y)$. Since the CNOT gate is a linear transformation, the CNOT gate is not universal for reversible computation. The goal of Slimeland is to incentivize the construction of a circuit composed mainly of CNOT gates. There are several reasons why a CNOT gate circuit should be easier to construct than a full-blown reversible circuit including the following:

  1. The CNOT gate is not universal for reversible computation.
  2. The CNOT gate acts on only two bits while universal reversible gates must act on at least 3 bits.
  3. The CNOT gate is linear.

It is therefore safe to assume that the research and development costs for developing an energy efficient reversible circuit composed almost exclusively of CNOT gates will be far less than that of developing a reversible circuit composed mainly of universal gates. Therefore, if one can construct a cryptocurrency POW problem that will incentivize the development of the energy reversible circuit composed mainly of CNOT gates, then energy efficient circuits that efficiently compute the CNOT gate will be developed very soon in order to quickly receive a return on investment.

Problem description

Suppose that $f$ is a suitable function. Suppose that $m,n$ are natural numbers and $D$ is an adjustable real number. Then he objective of the POW problem Slimeland is to find a $m$ bit hash $k$ along with an $n$ bit string $x$ so that $f(k||x)<2^{m+n}/D$. The function $f$ should be composed mostly of CNOT gates. However, since the CNOT gate is linear, the function $f$ cannot be composed exclusively of CNOT gates.

  1. The most efficient circuit for computing the function $f$ should consist solely of reversible gates without resorting to any ancilla or garbage bits.
  2. The circuit for the function $f$ should have a very high ratio of CNOT gates to non-linear gates in order to incentivize the construction of energy circuits composed mainly of CNOT gates.
  3. The circuit for the function $f$ needs to have enough non-linear gates to be cryptographically secure.
  4. The function $f$ should be as efficiently verifiable as possible.

Properties 2-4 are at odds with each other, so it is a challenge to construct a function $f$ that satisfies these conditions.

The internal testing technique

So the internal testing technique is method that I have come up with that can ensure that a cryptocurrency remains secure even if the cryptographic security of its POW problem is questionable.

Suppose that a cryptocurrency has Problem A as its POW problem. Suppose furthermore that Problem A has an adjustable level of security (the level security of symmetric cryptosystems can be adjusted simply by adding more rounds). Then a cryptocurrency applying the internal testing technique for Problem A will have two POW problems which we shall denote by Problem $A_{test}$ and Problem $A_{secure}$. Suppose that $N$ is an adjustable natural number. Then $A_{test}$ will have security level $N$ while $A_{secure}$ will have security level $3N$. Problem $A_{test}$ will also be weakened in other ways. While the security level for $A_{test}$ is decreased, the difficulty of $A_{test}$ will be increased (for example to 1000 times the difficulty of $A_{secure}$) so that it is not feasible for a miner to solve Problem $A_{test}$ using a brute force attack, and a miner must therefore solve $A_{test}$ by partially breaking its security. If $A_{test}$ is solved too often (for example, if $A_{test}$ is solved for more than one percent of all blocks), then the underlying cryptocurrency will automatically increment the number $N$ by one. As a security measure, the blocks for which $A_{test}$ is solved cannot be too close together.

Using the internal testing technique, the security of the underlying cryptocurrency POW problem will automatically be adjusted in case of any advancement in cryptanalytic techniques. By using the internal testing technique, Slimeland can remain secure even though the circuit used that solves Slimeland is mostly composed of the linear CNOT gates.

Security report for R5, the POW problem for Nebula.

Hello everyone,

Attached is the security report for R5, the POW problem for Nebula. I had to give an account on the security of Nebula since Nebula employs new kinds of cryptosystems which have not been implemented in practice so far. Keep in mind that it is ill-advised to employ a new symmetric cryptosystem in practice as soon as it is developed. It typically takes a couple of years for people to thoroughly investigate a new symmetric cryptosystem before it is employed in practice, and for public key cryptosystems it takes much longer. However, by the nature of R5, a security report that should be sufficient to ensure the security of R5 since it is much more difficult for something to go wrong with R5 that it is for something to go wrong with a symmetric cryptosystem such as a hash function.

I will at one point release an updated version of the security report for R5 since there is information about R5 which I do not want to reveal publicly at the moment (I apologize for my violation of Kerckhoffs’s principle.Don’t worry. All information about R5 will be openly available soon enough though. And my violation for Kerckoff’s principle are not for cryptographic security reasons but instead for logistical reasons).

A lesson to be learned from the recent cryptocurrency dip

Today in the news, you can find countless articles which tell us that China has just banned ICO’s (an ICO, an initial coin offering, is the way that many people fund new cryptocurrency endeavors. In an ICO, investors buy new cryptocurrency tokens since they believe that the upcoming cryptocurrency will eventually become valuable). As a result on this ban on ICOs, the total market cap for all cryptocurrencies has fallen from an all-time high of 180 billion USD (on September 2, 2:00 UTC) to 138 billion USD at the time of writing. The price of a bitcoin was almost 5000 USD but now after the dip, a bitcoin now costs 4100 USD.

One government action has the power to sway the value of cryptocurrencies so that they lose 20-25 percent of their value overnight. It is therefore necessary for those who value cryptocurrencies to make a great effort to improve the reputation of cryptocurrencies so that the world’s governments accept these cryptocurrencies. Not only are governments able to harm cryptocurrencies by passing laws against them, but these governments also have the power to launch attacks against cryptocurrencies and thus break the security of these cryptocurrencies and likely destroy these cryptocurrencies.

Cryptocurrencies currently have a mixed reputation. Since it is possible for cryptocurrencies to disrupt global economies, replace any national currency, and cause other changes that people will not want to accept, governments may want to place restrictions and regulations upon cryptocurrencies to keep them under control. Today, cryptocurrencies are not backed by gold or any other precious metals, and the process of mining new coins does not provide any product or service of value outside the cryptocurrency except for pollution. Since cryptocurrencies are not produced by providing useful products or useful services and cannot be traded for precious metals, cryptocurrencies do not have nearly as many advantages over fiat currencies as they could have, but they could improve their reputation simply by making mining useful. It is therefore necessary for cryptocurrencies to employ useful proof-of-work problems which provide benefits outside the cryptocurrency.

My solution in my upcoming cryptocurrency Nebula is to use a proof-of-work problem R5 so that in order to efficiently solve R5, one will need to construct a reversible computer and advance computational technology.
The proof-of-work R5 is nearly as efficient as a cryptographic hash function, so R5 does not suffer from negative characteristics which are present in other so-called useful proof-of-work problems. R5 is also much more useful than all other proof-of-work problems proposed so far since one cannot question the future value of reversible computation. Since Nebula will advance technology, the people and hence the governments will view cryptocurrencies more favorably (at least Nebula anyways). As a consequence, governments will be less willing to place restrictions on cryptocurrencies if some of those cryptocurrencies advance science in positive ways. The only way for cryptocurrencies to last among a skeptical population is if people only use a cryptocurrency which employs a good useful proof-of-work problem. If we keep on using the same cryptocurrencies with the same proof-of-work problems, then people will rightfully see these cryptocurrencies as useless and wasteful. I have hope that eventually, people will only use the cryptocurrencies with useful proof-of-work problems since people will rightfully perceive these new cryptocurrencies as being much more valuable than our current cryptocurrencies.

Unfortunately, it is too late for Bitcoin to start using a useful proof-of-work problem since most bitcoins have already been mined and switching a proof-of-work will require a destructive hard-fork, so one will need to start using new altcoins instead. Fortunately, in recent months, altcoins have come to dominate over half of the total market cap of all cryptocurrencies, and new altcoins are constantly being created. In a few years, the dominant cryptocurrencies will likely be ones which have not been launched yet but which will employ new innovations. Nebula is one of these cryptocurrencies with a new innovation that challenges the status quo in the cryptocurrency community.

A message to the skeptics

There have been many people in the pure mathematics community who have denied the possibility that reversible computing may be more energy efficient than conventional (by conventional I mean irreversible) computing. I hope you know that if conventional computing can potentially be infinitely efficient, then one could construct a computer that could decrease entropy within a closed system. You are denying the second law of thermodynamics. You are denying everything that people know about statistical mechanics. Stop denying science.

There are also other skeptics who have denied the value of cryptocurrencies altogether. To those of you I will say that you need to read and understand the original paper Bitcoin by Satoshi Nakamoto.

Generalizations of Laver tables is posted (140 pages)

The full version of the paper Generalizations of Laver tables is now posted.

In the paper, I have focused on building the general theory of Laver tables rather than solving a major problem with regards to the Laver tables. In fact, one should consider this paper as an account of “what everyone needs to know about Laver tables” rather than “solutions to problems about Laver tables.” This paper lays the foundations for future work on Laver tables. Since there is only one paper on the generalizations of Laver tables as of August 2017, an aspiring researcher currently does not have to go through many journal articles in order to further investigate these structures. I hope and expect that this paper on Laver tables will incite a broad interest on these structures among set theorists and non-set theorists, and that further investigation on these structures will be made possible by this paper.

Researching Laver tables

If you would like to investigate Laver tables, then please investigate the permutative LD-systems, multigenic Laver tables, and endomorphic Laver tables instead of simply the classical Laver tables. Very little work has been done on the classical Laver tables since the mid 1990’s. The classical Laver tables by themselves are a dead-end research direction unless one investigates more general classes of structures.

The most important avenue of further investigation will be to evaluate the security and improve the efficiency of the functional endomorphic Laver table based cryptosystems. Here are some ways in which one can directly improve functional endomorphic Laver table based cryptography.

  1. Try to break these cryptosystems.
  2. Compute $A_{96}$.
  3. Find compatible linear orderings on Laver-like LD-systems.
  4. Find new multigenic Laver tables and new Laver-like LD-systems.

It usually takes about 15 years from when a new public key cryptosystem is proposed for the public to gain confidence in such a cryptosystem. Furthermore, people will only gain confidence in a new public key cryptosystem if the mathematics behind such a cryptosystem is well-developed. Therefore, any meaningful investigation into large cardinals above hugeness and the Laver tables will indirectly improve the security of these new cryptosystems.

While people have hoped for a strong connection between knots and braids and Laver tables, the Laver tables so far have not produced any meaningful results about knots or braids that cannot be proven without Laver tables. The action of the positive braid monoid is essential for even the definition of the permutative LD-systems, so one may be able to apply the permutative LD-systems to investigating knots and braids or even apply knots and braids to investigating permutative LD-systems. However, I would regard any investigation into the application of Laver tables to knots and braids to be a risky endeavor since so far people have not been able to establish a deep connection between these two types of structures.

If you are a set theorist investigating the Laver tables and you are not sure if you will stay in academia for your entire career, then I recommend for you to work on something that requires extensive computer programming. This will greatly improve your job prospects if you ever leave academia for any reason. Besides, today nearly all respectable mathematicians need to also be reasonably proficient computer programmers. You do not want to be in academia trying to help students get real-world jobs when you do not yourself have the invaluable real-world skill of computer programming.

My future work

I will not be able to work on Laver-like algebras too much in the near future since I am currently preoccupied with my work on Nebula, the upcoming cryptocurrency which will incentivize the construction of the reversible computer. I am already behind on my work on Nebula since this paper has taken most of my time already, so I really need to work more on Nebula now. Since developing and maintaining a cryptocurrency is a full-time job, I will probably not be able to continue my investigations on Laver tables.

Slides from talk at BLAST 2017 and a rant on giving talks about the classical Laver tables

Here are my slides for my talk at the BLAST 2017 conference at Vanderbilt University in Nashville, Tennessee.

As a side note, I just noticed this other conference. All of the talks at that other conference on Laver tables are woefully outdated (i.e. 1995 or so). They only talk about the classical Laver tables. As an analogy, only talking about the classical Laver tables is like only talking about the cyclic groups of order $3^{n}$ and then claiming that they some how represent group theory as a whole. If you are going to give a talk about Laver tables or write a paper on the Laver tables, then please read the abridged version of my paper before you do so.

The classical Laver tables by themselves are a rather dead-end research area that have not been active within the last 20 years (one can probably try to analyze the fractal structure obtained from the classical Laver tables but such an analysis will probably be difficult and incremental). In order to advance further research in this area, one needs to consider the generalizations including Laver-like algebras, multigenic Laver tables, and functional endomorphic Laver tables. The classical Laver tables do not explain what the subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generated by multiple elements look like (one cannot even show that $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite without using the multigenic Laver tables). The classical Laver tables do not have any cryptographic applications. The classical Laver tables are just one sequence of structures, and it is hard to advance mathematics simply by looking at only one kind of structure with limited complexity. There is no reason at all to look at the classical Laver tables without looking at more general structures.

It is better to call the structures $A_{n}=(\{1,…,2^{n}\},*_{n})$ “classical Laver tables ” instead of simply “Laver tables.” There are other structures to consider.

How to give a classical Laver table talk.

The first step to giving a presentation on the classical Laver tables is to make sure you give your talk to the proper audience. The best audience to give a classical Laver table talk to is an audience of middle schoolers or maybe high schoolers (it is not that hard to fill out the multiplication table of a classical Laver table). Once you have your audience of middle schoolers present, you should get them to fill out an $8\times 8$ classical Laver table and then a $16\times 16$ classical Laver table. After they fill out the $16\times 16$ classical Laver table. And yes, middle schoolers are completely capable of filling out classical Laver tables. It is not that hard. After they are done filling out the tables, you can show them pictures that arise from the classical Laver tables on the projector and hint about how these objects come from infinity.

A remedial public service announcement about the second law of thermodynamics.

So it appears that some people in the mathematical community have some trouble with the second law of thermodynamics since they do not believe in science. They instead believe in perpetual motion machines, the world is flat (or a torus, or a projective sphere), and that computers, refrigerators, TV’s, cars, and airplanes are powered by demons rather than gasoline and electricity. Let me therefore give you a few experiments that you can do at home on the second law of thermodynamics.

WARNING: All of these experiments are AT YOUR OWN RISK. I cannot be held responsible for any electric shocks, burns, other injuries, dumb patents, or any other damages as a result of these experiments.

A scientific experiment on perpetual motion machines

So let me give you a do-it-yourself experiment that you can do that will hopefully convince you that perpetual motion machines do not exist.

Step 1: Find an extension cord. Extension cords can typically be found attached to electrical outlets or around equipment that requires electricity such as televisions or computers. If you cannot find an extension cord, then you may be able to buy one at Wal-Mart or Staples.
Extension cord

Step 2: Plug the extension cord into itself.

Extension cord plugged into itself

Step 3: Turn the extension cord on and then plug other things into the extension cord.

Step 4: Does the light on the extension cord turn on and do the things plugged into the extension cord turn on and/or power up? If the light on the extension cord turns on and your appliances receive power, then, congratulations, you have created perpetual energy. Go and patent your invention and make trillions of dollars. If the light does not turn on, then perpetual energy does not exist, and the second law of thermodynamics is true.

An experiment on the second law of thermodynamics

Step 1: Make a cup of some hot coffee. Do not drink the coffee.


Step 2: Measure the temperature of the coffee with a thermometer and the temperature of the room which the coffee is in. If you do not have a thermometer, then you may find one at Wal-Mart or some other store. The temperature of the coffee should be much hotter than the temperature of the room. Please make sure that you have a thermometer and not a moisture meter.


Step 3: Wait 24 hours. Do not move the coffee. Store the coffee in a regular cup. Do not store the coffee in a thermos. Make sure that your mother does not heat up your coffee again since that will ruin this experiment.


Step 4: Measure the temperature of the coffee and the temperature of the room which you have left the coffee in.

Step 5: If the temperature of the coffee is equal to the temperature of the room, then the second law of thermodynamics is correct.
If you find that the temperature of the coffee is still near a boiling temperature, then you may have been using a moisture meter instead of a thermometer. In this case, you should (at your own risk of course) stick your finger into the coffee. If you do not get burned, then the second law of thermodynamics is correct after all. If you do get burned, then you need to ask your mother if she heat up the coffee again. If she denies heating up your coffee again, then the second law of thermodynamics is false. You should therefore publish your results in the top physics journal. If they reject you, then they are persecuting you.

A computer experiment on the second law of thermodynamics

Here is another remedial experiment which you can perform to convince you that not only does the second law of thermodynamics apply to our physical universe, but the second law also applies to any time reversible universe.

Step 1: Click here

Step 2: Draw an image in the field. You may draw any image that you like.

image in cellular automata

Step 3: Select a rule or input a rule yourself. You may use any rule that you like. After you select a rule, then you should run the rule for at least 100,000 generations.

Step 4:

If the resulting image looks something like this,
then try running your cellular automaton under a different rule and repeat steps 1-4 a few times. If the image still looks orderly every time, the perhaps the second law of thermodynamics does not hold for all universes.

On the other hand, if the resulting image looks like this,


then the second law of thermodynamics works not just in our physical universe but also in any hypothetical or simulated universe such as a cellular automaton as long as that universe is time reversible.

An experiment on the second law of thermodynamics

With this experiment, you will be able to tell whether the universe will spontaneously violate the second law of thermodynamics and make all of the oxygen molecules migrate to the other side of the room while the nitrogen molecules stay on your side of the room.

Step 1: Gather the materials. For this experiment, you will need an oxygen mask and a full tank of oxygen. Please make sure you know how to use the oxygen mask before starting this experiment. You should also bring a fidget spinner to entertain yourself since this experiment may take a long time to complete and you may get bored. Oh. And make sure to pack some lunch and dinner since this experiment may take a while.

Fidget spinner


Step 2: Once you have the required materials, go to a room and lock the door. Please prepare to stay in that room for up to 10 hours since it may take a while for the second law of thermodynamics to be violated.

Step 3: If you feel like you cannot breathe, then all of the oxygen molecules have migrated to the other side of the room in violation of the second law of thermodynamics. In this scenario, immediately put on your oxygen mask since it may take a while for the violation of the second law of thermodynamics to wear off. If the second law of thermodynamics is still being violated as your oxygen tank is about to be depleted, then please leave the room and get some fresh air. At this point, you should probably determine whether this phenomenon is a true violation of the second law of thermodynamics or whether your house is haunted. In order to tell if your house is haunted, please get it inspected. Any good home inspector will be able to tell whether evil spirits live in your home or not. If you paid 40,000 dollars for a house that normally costs 500,000 dollars, then your house is probably haunted; I mean, why would anyone sell a non-haunted house for 1/10th of its worth?

Step 4: If nothing happens and you do not have to put your oxygen mask on after 10 hours, then the second law of thermodynamics is probably correct.

Satire notice: This post is satire. I wrote this post in order to address the anti-scientific beliefs which I have found among the mathematical community, namely the denial of Landauer’s principle.

Nebula-The cryptocurrency that will produce the reversible computer

So I have just posted the paper outlining the proof-of-work problem for my upcoming cryptocurrency Nebula. Here is the link for the paper. I hope to launch Nebula as soon as possible.

The idea behind Nebula is to use a reversible computing optimized proof-of-work (RCO-POW) problem instead of an ordinary proof-of-work problem (if you do not know what I am talking about, I suggest for you to read the original paper on Bitcoin). An RCO-POW problem is like an ordinary proof-of-work problem except for the fact that the RCO-POW problem can be solved by a reversible computing device just as easily as it can be solved using a conventional computing device.

It is very rare for a problem to be solvable by a reversible computing device using just as many steps as it is solvable using a conventional computing device. In general, it takes more steps to solve a problem using a reversible computation than it takes to solve the same problem using conventional computation. Therefore, since reversible computation has this computational overhead and since reversible computers currently do not exist, chip manufacturers do not have much of an incentive to manufacture reversible computing devices. However, since RCO-POW problems are just as easily solved using nearly reversible computational devices, chip manufacturers will be motivated to produce energy efficient reversible devices to solve these RCO-POW problems. After chip manufacturers know how to produce reversible devices that can solve these RCO-POW problems better than conventional devices, these manufacturers can use their knowledge and technology to start producing reversible devices for other purposes. Since reversible computation is theoretically much more efficient than conventional computation, these reversible computing devices will eventually perform much better than conventional computing devices. Hopefully these reversible computational devices will also eventually spur the development of quantum computers (one can think of reversible computation as simply quantum computation where the bits are not in a superposition of each other).

Nebula shall use the RCO-POW which I shall call R5. R5 is a POW that consists of five different algorithms which range from computing reversible cellular automata to computing random reversible circuits. I use the multi-algorithm approach in order to ensure decentralization and to incentivize the production of several different kinds of reversible devices instead of just one kind of device.

The only thing that will be different between Nebula and one of the existing cryptocurrencies is the POW problem since I did not want to add features which have not been tested out on existing cryptocurrencies already.

Cryptographic applications of very large cardinals-BLAST 2017

In August, I will be giving a contributed talk at the 2017 BLAST conference.

I am going to give a talk about the applications of functional endomorphic Laver tables to public key cryptography. In essence, the non-abelian group based cryptosystems extend to self-distributive algebra based cryptosystems, and the functional endomorphic Laver tables are, as far as I can tell, a good platform for these cryptosystems.

ABSTRACT: We shall use the rank-into-rank cardinals to construct algebras which may be used as platforms for public key cryptosystems.

The well-known cryptosystems in group based cryptography generalize to self-distributive algebra based cryptosystems. In 2013, Kalka and Teicher have generalized the group based Anshel-Anshel Goldfeld key exchange to a self-distributive algebra based key exchange. Furthermore, the semigroup based Ko-Lee key exchange extends in a trivial manner to a self-distributive algebra based key exchange. In 2006, Patrick Dehornoy has established that self-distributive algebras may be used to establish authentication systems.

The classical Laver tables are the unique algebras $A_{n}=(\{1,…,2^{n}-1,2^{n}\},*_{n})$ such that $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$ and $x*_{n}1=x+1\mod 2^{n}$ for all $x,y,z\in A_{n}$. The classical Laver tables are up-to-isomorphism the monogenerated subalgebras of the algebras of rank-into-rank embeddings modulo some ordinal. The classical Laver tables (and similar structures) may be used to recursively construct functional endomorphic Laver tables which are self-distributive algebras of an arbitrary arity. These functional endomorphic Laver tables appear to be secure platforms for self-distributive algebra based cryptosystems.

The functional endomorphic Laver table based cryptosystems should be resistant to attacks from adversaries who have access to quantum computers. The functional endomorphic Laver table based cryptosystems will be the first real-world application of large cardinals!

Where are your manners?

So if you remember, I recently made this post calling out the trolls, haters, and science denying crackpots on another network. I have removed all the content from this post since things have been getting out of hand and because that post has been getting too many views. I have a couple questions though.

If you have noticed, this post had very little content to it. Why does a content-free post get so much more attention than a post which actually has content to it? It seems like the people here are attracted to contentless click-bait posts much more than they are to actual content. This is really pissing me off. Your attraction to click-bait articles and false and hateful rumors is disgusting and deplorable. You should be interested in new ideas instead of immediately rejecting them out of hatred. People have been ruining my reputation since they are more interested in stupid rumors than in truth.

I realize that some of my posts are quite technical, but some of them are not. Some of them just announce some new program that I have written which you can launch in your browser where the only thing you have to do is read a couple directions and click a few buttons and produce a few pictures that you can hang on your wall. You do not even have to do anything.

P.S. So today, the logicians have been extremely rude. Please stop it.