I'm looking for a quote

For the discussion of math. Duh.

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I'm looking for a quote

Postby SlyReaper » Wed Aug 13, 2008 9:21 am UTC

As per the title, I'm looking for a particular quote about number theory. I think it's by G. H. Hardy and goes goes something like "number theory is such an esoteric subject that it could never have any applications in the real world." Is anyone familiar with it?

The reason I need it is because I'm giving a presentation about cryptography.
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Re: I'm looking for a quote

Postby karst » Wed Aug 13, 2008 2:06 pm UTC

There is this one:

G. H. Hardy wrote:No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems very unlikely that anyone will do so for many years.

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Re: I'm looking for a quote

Postby SlyReaper » Wed Aug 13, 2008 2:21 pm UTC

Hmm that could be it I suppose. Thanks.
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Re: I'm looking for a quote

Postby Yakk » Wed Aug 13, 2008 5:01 pm UTC

The irony is, of course, that both have military purposes.

GPS relies on relativity to generate results that are accurate enough to be used. And GPS is, never forget, a military technology.

Number theory is heavily used in encryption security, and decryption thereof. Both of which are areas the military and intelligence branches care a lot about.
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Re: I'm looking for a quote

Postby skeptical scientist » Wed Aug 13, 2008 6:42 pm UTC

Moreover, relativity helped pave the way to nuclear weapons, which were used only a few years after Hardy wrote that essay. I'm not sure how long it took for number theory to be applied to cryptography, but that might also have taken place in WWII. I don't know crypto history very well, but the first cryptosystem I can think of based directly on number theory is RSA, which wasn't discovered until the early 1970s.
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Re: I'm looking for a quote

Postby Yakk » Wed Aug 13, 2008 6:57 pm UTC

Breaking Enigma was based around finite group theory -- I suppose modular arithmetic doesn't qualify as "number theory".
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Re: I'm looking for a quote

Postby SlyReaper » Wed Aug 13, 2008 7:37 pm UTC

Yakk wrote:Breaking Enigma was based around finite group theory -- I suppose modular arithmetic doesn't qualify as "number theory".


The enigma code was just character substitution I think (or at most, n-block substitution), but with a key that changed from time to time. It wasn't prone to frequency analysis because the content of the messages was obviously not common language (you couldn't reasonably expect E to be the most common letter for example). The interesting thing about enigma is that it was deliberately set so that no character could be encrypted to itself. So a C could never become a C in the cipher text. Apparently the Germans thought that would make it harder to decrypt. :mrgreen:

Public key encryption is most certainly based on number theory. A lot of current systems are just adaptations of RSA which make use of the properties of prime numbers. Some newer ones use elliptic curves, which I think still fall within the field of number theory. Certainly the dude who lectured elliptic curves at my university was the head honcho of number theory there. :)

Anyway yeah, the irony that another person mentioned is the exact reason I want to include it in my presentation.
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Re: I'm looking for a quote

Postby Yakk » Wed Aug 13, 2008 7:56 pm UTC

SlyReaper wrote:
Yakk wrote:Breaking Enigma was based around finite group theory -- I suppose modular arithmetic doesn't qualify as "number theory".


The enigma code was just character substitution I think (or at most, n-block substitution), but with a key that changed from time to time.

The machine had multiple rotors that changed with each character pressed, each one of which was a permutation of the alphabet.

http://en.wikipedia.org/wiki/Enigma_machine

Note that a OTP is merely a "character substitution cypher" -- the difficulty is in the randomness of the substitution. :-)
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Re: I'm looking for a quote

Postby SlyReaper » Wed Aug 13, 2008 8:15 pm UTC

Ah okay, I was almost right then. The key just changed rather more often than I thought. :lol:
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Re: I'm looking for a quote

Postby merc » Thu Aug 14, 2008 10:31 am UTC

Yakk wrote:Note that a OTP is merely a "character substitution cypher" -- the difficulty is in the randomness of the substitution. :-)

Wait, isn't a character substitution just a message encrypted via a random permutation of the alphabet? Because a one-time pad definitely isn't.

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Re: I'm looking for a quote

Postby SlyReaper » Thu Aug 14, 2008 1:49 pm UTC

Ah I found that quote. Awesome.
[...] is usually based on an incautious saying attributed to Gauss, to the effect that, if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme uselessness, the queen of mathematics
G H Hardy, A Mathematician's
Apology, 1940
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Re: I'm looking for a quote

Postby Yakk » Thu Aug 14, 2008 3:49 pm UTC

merc wrote:
Yakk wrote:Note that a OTP is merely a "character substitution cypher" -- the difficulty is in the randomness of the substitution. :-)

Wait, isn't a character substitution just a message encrypted via a random permutation of the alphabet? Because a one-time pad definitely isn't.

For each character, a OTP simply permutes the alphabet on that character. (the XOR operator using a fixed left hand side on bits is a permutation operation, as it is a bijection from a finite set to a finite set).

OTP adjacent permutations are uncorrelated completely. Enigma's permutations of adjacent characters where not uncorrelated, which is what makes it a non-OTP.
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Re: I'm looking for a quote

Postby Pathway » Fri Aug 15, 2008 2:22 pm UTC

There's "No discovery of mine has made, or is likely to make, for good or ill, the least difference to the amenity of the world." (Wikiquote) Also: "Thank God that number theory is unsullied by applications," and "Here's to pure mathematics! May it never have any use." (Elementary Number Theory, by Burton.)

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Re: I'm looking for a quote

Postby Ortho » Fri Aug 15, 2008 3:29 pm UTC

merc wrote:
Yakk wrote:Note that a OTP is merely a "character substitution cypher" -- the difficulty is in the randomness of the substitution. :-)

Wait, isn't a character substitution just a message encrypted via a random permutation of the alphabet? Because a one-time pad definitely isn't.

A mono-alphabetic substitution cypher is as you describe, if you go poly-alphabetic like the vignere cypher then if you use a random key at least as long as the message and only ever use it once. then you have a OTP

Of course OTP cyphers don't quite work like that in practice since the message and key are binary strings rather than alphabetic characters, but it is essentially the same.

The Enigma was basically a poly-alphabetic cypher with a largish key and a complicated way of using it that made decryption hard at the time, it didn't become quick to break it until they built computers to do it (not programmable computers, that came later in the war to break a diplomatic cypher). A modern computer can break the enigma fairly trivially with freely available software.

On a side note, the only cypher known that is completely unbreakable is the one time pad. I think it may have been proven that an unbreakable cypher must have a key at least as long as the message. Also if you know that 2 messages were encrypted using the same OTP then they can be broken fairly trivially.


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