An alert reader sent me a link to a really dreadful piece of drek. In some ways, it’s a

rehash of the “Nullity” nonsense from a couple of years ago, but with a new spin.

If you don’t remember nullity, it was the attempt of one idiot to define division by zero.

He claimed to have “solved” the great problem of dividing by zero, and by doing so, to be able

to do all manner of amazing things, such as to build better computers that would be less prone

to bugs.

Today’s garbage is in the

same vein: another guy, this one named Jeff Cook, who claims to have “solved” the problem of

division by zero. But this one claims that this gives him a way to prove the Reimann

hypothesis, to rapidly crack RSA public key encryption, and to devise a new “theoretically

unbreakable” encryption algorithm.

The grandiosity of this Mr. Cook is astonishing. He’s started a company (which is looking

for investors!); here’s a quote from his company’s homepage:

Great scientific discoveriesmark the milestones of human history.Such are the accomplishments achieved by the men and women of Singularics. Standing on the

shoulders of giants such as Albert Einstein and Bernhard Riemann, we have reached up through

nature’s veil and seen what lies hidden there more clearly than anyone else before us. Our

discoveries have yielded a new mathematical framework, one that provides a profound

understanding of nature’s basic mechanics. We have discoveredThe Science of, the study of the singularity.

Singularics™We have already found a variety of important applications of

Singularic Technology™,

but perhaps the most immediately useful areNeutronic Encryption™, a new theoretically

unbreakable public key encryption algorithm andSingularic Power™, a new form of clean

power generation.Neutronic Encryption, our next generation public key encryption algorithm, will play a

vital role in the digital age by ensuring that the electronic information of governments,

industry and individuals is kept secure and private in a world where cyber-terrorism is on the

rise.We have also developed a new primary power generation system capable of delivering

abundant, clean and inexpensive energy that can satisfy power requirements on any scale.

Singularic Power production technology generates zero pollution and can therefore play an

instrumental role in promoting a harmonious coexistence between human civilization and the

Earth’s fragile ecosystem.To date, our analysis of the mathematics and physics at the singularity has lead us to

eight important new inventions, most notably in the fields of information security and clean

energy. All eight inventions (patents pending), have significant and immediate application in

the global market.It is our vision to use these advances to bring about great improvements for everyone

through new technology, intelligently applied.

Mr. Cook doesn’t have too high an opinion of himself, does he?

Of course, that’s really content free hype. He’s hoping to recruit investors, and so the

grandiose claims are inevitable: no one invests in a business that says something like “We’re

an incremental improvement over our competitors!” So some amount of hyperbole is acceptable, if annoying.

The real question is, is there anything behind those grandiose claims? Does he really

have anything but hype? Is there the slightest shred of reality underlying

that hype?

Alas, no. Moving to his “cryptography” page:

Singularics has advanced the state of the art in cryptography by developing a

new theoretically unbreakable public-key algorithm called Neutronic Encryption™.Our advances in Prime Number Theory have led to a new branch of mathematics called

Neutronics. Neutronic functions make possible for the first time the ability to analyze

regions of mathematics commonly thought to be undefined, such as the point where one is

divided by zero. In short, we have developed a new way to analyze the undefined point at the

singularity which appears throughout higher mathematics.This new analytic technique has given us profound insight into the way that prime numbers

are distributed throughout the integers. According to RSA’s website, the RSA public key

encryption algorithm has an installed based of nearly one billion. Each of these instances of

the prime number based RSA algorithm can now be deciphered using Neutronic analysis . Unlike

RSA, Neutronic Encryption is not based on two large prime numbers but rather on the Neutronic

forces that govern the distribution of the primes themselves. The encryption that results from

Singularics’ Neutronic public-key algorithm is theoretically impossible to break.

There’s so much wrong with this that it’s hard to know where to start.

I suppose the best starting point is the most basic one: division by zero isn’t a problem. It’s just meaningless. When we say that it’s undefined, that’s not because we’re afraid of dividing by zero. It’s not because we’re unsure of what the answer should be. We say that it’s undefined because, simply, it’s undefined. It’s not that we haven’t given it a definition: “undefined” has a specific meaning in math.

In math, we can desribe division as a function with two parameters: D(a,b). Like every function, D has a domain (a set of inputs) and a range (a set of outputs). A function is defined for a value if and only if that value is in the domain of the function. When we say that D is undefined when b=0, what we mean is that for all values of a, (a,0) is not in the domain of D. When we say that division by zero is undefined, what we mean is that no input to division with zero as the second parameter is in the range of the division function. It’s undefined.

The fact that division by zero is undefined is *not* a matter of accident, or of

ignarance. It is not just a trivial little thing. It’s actually *important*. If

division by zero were defined, then a lot of what we consider standard math would completely

*stop working*. The fact that division by zero is undefined is a fundamental part

of the structure of our system of numbers; it’s one of the basic field axioms that define

the basis of how we understand the real numbers. Take that axiom away, and suddenly things

stop working.

It’s not impossible to create a system in which division by zero is defined. But if you do

that, *you’re starting from scratch*. Almost every theorem about real numbers relies on

the field axioms, and will therefore be invalid in your new system. So you’ll need to

re-derive almost everything. And it’s not just a matter of finding a different derivation;

many of the things that we take for granted *will not work* in your new system.

There are serious mathematicians who’ve played with the idea of defining division by zero.

(For example, someone – I think Conway himself – played with the idea of defining division by

zero in a variant form of the surreal numbers.) One good way of recognizing a crank is by

looking at what they do with their new division-by-zero defining system. A serious mathematician starts working out what affect their definition has on the basic axioms,

and what still works. A crank defines division by zero, and then proceeds to continue working as if they haven’t broken anything.

Mr. Cook has a paper on his page about his wonderful system and how it allows him

to prove the Riemann hypothesis. In the paper, he just blithely proceeds on

as if he hasn’t broken anything. He doesn’t show the slightest awareness that he’s

relying on axioms that he’s invalidated.

Moving on, let’s look at the next silly claim. Not only does this guy claim to have “solved” division by zero, but he claims to have developed a “theoretically unbreakable” public key encryption system.

Sounds great, doesn’t it? Using his brilliant division-by-zero techniques,

he can crack RSA; and he’s even got a replacement, which is an *unbreakable* public key crypto system.

Except, of course, “theoretically unbreakable public-key encryption” is basically a

very complicated non-sequitur.

It’s possible to create an unbreakable cryptosystem. In fact, it’s *easy* to

create an unbreakable cryptosystem. One of the easiest crytographic

systems is based on something called a *one time pad*. In a one-time pad, the

two parties to the communication share a very long secret – typically a book of

random numbers. To encrypt a message, you convert each character in the message

to a number, and add it to the number from the pad. To decrypt, you just subtract. Each

number in the pad is used exactly once, and there’s absolutely no pattern to the numbers. Once

a page from the pad has been used to encrypt or decrypt a message, it’s torn out and destroyed. If you don’t have a copy of the pad, you can’t decrypt the message. It doesn’t matter how much computer power you have available; it doesn’t matter how clever you are; it doesn’t matter what brilliant algorithms you can think up. Without the numbers on the pad, there is absolutely *no way* to decrypt the message.

Public key – aka asymmetric encryption – is an entirely different story. There

is *no way* to build an unbreakable public key system. You can make a system in

which it’s incredibly difficult to crack it – which is exactly the RSA model. But *any* public key system can be cracked by a brute force attack. It’s the nature of the system: you can’t possibly avoid that. For *any* possible public key cryptosystem, there’s

a straightforward brute-force attack:

ct = Encrypt(pt, public_key) for i in PossibleKeys do attempt = Decrypt(ct, i) if attempt = pt then print "Private key = " + i

If you’ve got a public key cryptosystem, that attack *will* work. Period. There’s

no way around it. It might take a *very* long time. But it *will* work

eventually. Further, every public key cryptosystem is based on some fundamental

relationship between the public and private keys. A cryptanalyst can study that

relationship, and use it to refine the attack above. There is simply no such thing

as an unbreakable public key cryptosystem.

So without even knowing anything about how his “Neutronic” encryption purportedly

works, I can say that his claim is absolutely nonsense – worse than nonsense, it’s

an idiotic claim that demonstrates that Mr. Cook really doesn’t understand how

public key encryption works.

Further – he claims that he’s developed a way of cracking RSA. But nowhere on the site

does he do anything to support that claim. Mr. Cook and his fledgling company haven’t

demonstrated that. They’ve got no explanation of *how* their alleged break of

RSA works, beyond the division by zero rubbish – and that’s all built up on

incredibly, stupidly bad math.

Mr. Cook and his coworkers is a con-artist, trying to convince people to give him

their money. His claims range from unsupportable rubbish to nonsensical word salad.