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
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 discoveries mark 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 discovered The Science of
Singularics™, the study of the singularity.
We have already found a variety of important applications of Singularic Technology™,
but perhaps the most immediately useful are Neutronic Encryption™, a new theoretically
unbreakable public key encryption algorithm and Singularic Power™, a new form of clean
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
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
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
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.