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Encryption Security

Neal Stephenson on Zeta Functions 102

Posted by michael from the minor-footnote dept.
Introspective writes "Over on Cryptome they have published an Email from Neal Stephenson explaining his use of Zeta functions in Cryptonomicon. It gives a nice insight into writing about advanced cryptography ( in fiction, that is ) and the kind of reactions he gets back from his readers."
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Neal Stephenson on Zeta Functions More

Neal Stephenson on Zeta Functions

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  • I just have to laugh... (Score:5)

    by freeBill ( 3843 ) on Sunday March 11, 2001 @03:10PM (#370457) Homepage

    ...at this...

    But I can assure you that many readers of fiction underestimate just how much of a novel's content is simply made up. There is a common assumption among readers that much of what appears in a novel is thinly veiled and repackaged reality. You can imagine how provoking this is to a novelist who works so hard to invent it.
    ...especially since the brilliance of The Cryptonomicon is the degree to which it blended clearly historical facts with clearly fictional events. I found myself wondering time and again where the line was drawn between what was made up and what was not. Time and again Neal used a series of facts (say Fact A, Fact B, Fact C, Fact D, and Fact E) the first of which (Fact A) was based on history and the last of which (Fact E) clearly could not have happened. The fiction was blended so well with the fact that I couldn't tell where the transition between fact and fiction began (Fact B? Fact C? Fact D?).

    A good example of this was the account of Alan Turing and how he intuited the idea of a digital computer by contemplation of Goedel's Incompleteness Theorem: I knew Turing conceived of the idea of computers long before they were invented. I knew he was very much aware of Goedel (his Noncomputability Theorem rests entirely on Goedel's methods).

    But I had never heard it said that he had figured out that computers were possible based on the implications of Goedel. We can see that today, but we have the benefit of hindsight. Was this just something Neal made up based on that hindsight? Or did Turing really see this back then?

    There were clearly fictional parts of the book's Turing. We can safely assume the bicycle ride was invented. (Am I the only one who noticed the fact that he introduced the bike-chain explanation of why prime numbers are so key to crypto without ever doing anything with it?) The vicar's wife probably never peeked at her bowl full of balls (I think).

    Where did the Goedel inspiration go on this truth-fiction spectrum? Neal's blurring of the line makes it hard to determine from the novel. That's good. It makes for a ripping good yarn. But it also makes me less than sympathetic to the author's complaints about readers. Yeah, those readers just assume more of the novel is real than is actually the case.

    That's right. Mess with our minds and then complain that we're confused.

    For those who are wondering, yes, Enos Root did die in Sweden in 1944 only to reappear 50 years later in a prison in the Philippines. And, for those who are wondering about my question, I have found evidence that Turing's inspiration was indeed based on Goedel.

    Of course, there's always the possibility some reality hacker read the book, decided it was better than the actual story, and started spreading historical references to the Goedellian inspiration of Turing. The universe is, after all, controlled by those who have an understanding of the source code.

    Reading is FUNdamental, according to the vicar's wife. I don't think she peeked. Really.

  • Then there's the FTL drive problem (Score:5)

    by freeBill ( 3843 ) on Sunday March 11, 2001 @03:18PM (#370458) Homepage
    If I figure out a really neat idea for a faster-than-light drive but there's just one minor problem with it, I write a hard sci-fi novel based on it, glossing over the problem.

    If I figure out a really neat idea for a faster-than-light drive with no problem, I don't have time to write the novel. I'm out in the back yard building my spaceship.
  • And if you would like to achieve ever lasting fame and noteriety you could try to prove the Riemann Hypothesis which conjectures that every x satisfying Zeta(x) = 0 has real part 1/2 .

    I'd like to see a proof posted on slashdot.
  • ...for warning us about the spoilers.

    --
  • ...especially since the brilliance of The Cryptonomicon is the degree to which it blended clearly historical facts with clearly fictional events. I found myself wondering time and again where the line was drawn between what was made up and what was not.

    You should also read the Illuminatus Trilogy. Some key bizarre events in it are factual, but it's pure fiction.

    Or so They want you to think.fnord :-)

    -
  • Dr. Michael Anshel is a professor in the department of computer science at my school --
    City College of New York. He offered a little background info into the discussion over zeta functions that Mr. Stephenson is referring to.

    From: MikeAt1140@aol.com
    Date: Sun, 11 Mar 2001 19:07:28 EST

    You may find the following exchange prior to Neal
    Stephenson' letter of interest. Feel free to forward. Best.-Mike Anshel

    In a message dated 2/23/01 1:33:10 PM, schneier@counterpane.com writes:

    >

    A simple acknowledgment in future discussions of this work by the author and
    his agents that there is a cryptography based on zeta functions,introduced in
    the open literature by Michael Anshel and Dorian Goldfeld and whose patent
    rights are assigned to Arithmetica Inc would do for a starter.

    Michael Anshel

  • ... (Score:3)

    by ruin ( 141833 ) on Sunday March 11, 2001 @07:48PM (#370463) Homepage
    Now I know why Cryptonomicon was such a thick book...
    --
  • Sorry about that... lameness.

    Here is a link [cuny.edu] to the full email I got from Dr. Anshel giving a little background to the story.

  • Re:I just have to laugh... (Score:5)

    by schulzdogg ( 165637 ) on Sunday March 11, 2001 @03:49PM (#370465) Homepage Journal
    There were clearly fictional parts of the book's Turing. We can safely assume the bicycle ride was invented. (Am I the only one who noticed the fact that he introduced the bike-chain explanation of why prime numbers are so key to crypto without ever doing anything with it?) The vicar's wife probably never peeked at her bowl full of balls (I think).

    Incorrect! Turing's bicycle did have a broken chain! He did bury silver as a shore against accupation. Read his biography. It was amazing how realistic some of those things were...

  • s needs to be greater than 1 for the series to converge.

    The series Sum 1/n from n=1 to n=inf does not converge although it diverges very slowly. This can more easily be seen by looking at the integral S 1/n dn which is like log n, which diverges. And more so, for s less than one, the integral really diverges.
  • Duke Mathematical Journal is a pretty good journal, so it must be a good paper.

  • The real history of WWII cryptanalysis (Score:4)

    by Animats ( 122034 ) on Sunday March 11, 2001 @07:52PM (#370468) Homepage
    The book to read is "Battle of Wits, The Complete Story of Codebreaking in WWII" [amazon.com]. This has a broader overview of the entire operation, including pictures of codebreaking machines not previously published. The bombes are of course shown, but there are also film comparators that used 70MM film, as well as a building-sized machine built out of telephone switching equipment by Bell Labs. There's also considerable discussion of the use of standard and modified IBM punched card equipment.

    Most writers on the subject haven't really captured the scale of the operation. This wasn't done by a few smart people. WWI cryptanalysis was like that, but WWII made it into an industrial operation. Friedman was the one who first used IBM gear for cryptanalysis (in 1934, and it was a really tough sell getting the money during that period). Once the operation really got going, tens of thousands of people, and thousands of machines of various types, were involved.

    Cryptanalysis on that scale had ever been done before. The Germans and Japanese had cryptanalytic operations, but at the "small group of smart people" level. Small groups would never have decrypted enough stuff to seriously affect the war. But the industrial-strength effort mounted by the Allies made a real difference.

    It's instructive to look at the pictures. The stuff built by National Cash Register looks like the innards of a cash register. The stuff built by IBM looks like IBM tabulators. The stuff built by Bell Labs looks like a telephone central office. The Colossus machine, though, does have a vague resemblance to an early tube computer, although the big endless loops of paper tape clearly indicate its special purpose nature.

    Colossus was actually based on some prewar British Telephone experiments with electronic switching. And nothing that came out of the crypto work worked anything like a general-purpose computer. All the crypto stuff was very special-purpose. This really isn't where computers came from. Babbage actually had a much more computer-like architectural concept.

    The problem wasn't theoretical. It was that nobody had yet developed a useful high-speed data storage device that didn't involve moving parts. Using two tubes to store one bit was too expensive and bulky to be used for a general purpose computer. Delay line memory came after the war, and was an outgrowth of some radar gear that used delay lines. The stuff during the war stored its state in relays, tubes, paper tape, or punched cards. The hardware for a useful, programmable, general purpose computer just wasn't available yet.

  • For those who are wondering, yes, Enos Root did die in Sweden in 1944 only to reappear 50 years later in a prison in the Philippines.

    You know, I didn't get ths part at all. Did he fake his own death? Was I reading this part too late at night and I missed something? Mod me off-topic, just answer me :)
  • There really is a National Puzzle Center [nsa.gov] run by the NSA. Typical question: Which of the following palettes represents a possible PNG palette?
  • ...especially since the brilliance of The Cryptonomicon is the degree to which it blended clearly historical facts with clearly fictional events. I found myself wondering time and again where the line was drawn between what was made up and what was not. Time and again Neal used a series of facts (say Fact A, Fact B, Fact C, Fact D, and Fact E) the first of which (Fact A) was based on history and the last of which (Fact E) clearly could not have happened. The fiction was blended so well with the fact that I couldn't tell where the transition between fact and fiction began (Fact B? Fact C? Fact D?).

    This is supposed to be brilliant? They do this in just about every Star Trek episode: "Genghis Khan, Hitler, Stalin, and Sporkon of Cygnus 9..." (etc.)

    This letter from Stephenson proves yet again what a clever guy he is, and what a good communicator. But I remain totally underwhelmed by Cryptonomicon, which I don't see as being much more than the average espionage potboiler. Tom Clancy probably does as much research as Stephenson does. Just because Stephenson researches "stuff that matters" (to you), doesn't make him any more of a genius.
    --

  • Slightly OT, but the Zeta function, in addition to being of significance to primes [v-wave.com] is important to astronomy. That's because to derive the thermal emission from eg. a star as a function of Temperature you need to integrate the Planck Blackbody Function [treasure-troves.com], which gives Zeta(4) = pi^4/90 The result is known as the Stephan-Boltzmann Law [treasure-troves.com]
  • that's why you read it in the new 'scientist' as opposed scietific american or nature...
  • prove the RP, make a quick million... http://www.claymath.org/prizeproblems/index.htm
  • I don't know about you but I lose a little respect for people that insult their audience. I guess we're all stupid enough to give him our money but not smart enough to understand that it's fiction?
    You know, there is a big difference between saying that we are all not smart enough to understand that its fiction, and saying that there is somebody who will misunderstand where the line between reality and fiction lies in a novel that was deliberately constructed to obscure that line.
  • The Xanth series after number 9 and the Adept series after the first three all sucked.
  • I liked it. I heartily recommend it to anyone who likes all his other books.
  • Ummm... Mute, Ox/Orn/Omnivore, Viscous Circle (series), and a bunch of others. Early Xanth, Incarnations, and Adept books.

    I liked those ones much more than his later work which I stopped reading. I partly outgrew him, but he also got a lot more childish in later books.

    IMHO the 80s were his strong period.
  • Soylent Green is people. Whoops, sorry.
  • His books all seem to suffer from "Send In The Cavalry!" syndrome. One guy struggles against all odds for the whole book, and then suddenly his friends and the all-powerful mob/police/army/EPA/Allies come save the day.
  • I think there was a production problem, and multiple "leaves"(? Can't remember what the individual page bundles are called in publishing)

    They're called 'signatures'.

    I just wonder how unique it is...?

    I'll give you US$100 for it.

  • Re:Is this an appropriate posting? (Score:2)

    by Anonymous Coward
    Some e-mail has come into my box recently that appears to be a fragment of an exchange between you, or some friend or associate of yours, and Bruce Schneier. The subject is zeta function cryptography in my novel CRYPTONOMICON. The e-mail has been bounced back and forth a few times and so it is not entirely clear to me who was holding down your side of the exchange. I am going to send this message to you in the hopes that you find it of interest and that you will
    forward it to anyone you think is interested.
    Emphasis mine. Sigh. Please read more carefully.
  • Think of every mass-market book or movie that touches on a topic you know well (e.g. computers, hang-gliders, math, Egyptian hieroglyphs, rockets, ancient stone-work.) They ALL misexplain the topic. Any rational author will pick a few good concepts and gloss over the details... audiences do not want a hundred pages of footnotes about the current state of the art.

    Live with it, and pray your product isn't mentioned by name. Do you really expect "Sneakers" to provide cryto info, or "Dr Strangelove" to explain nuclear strategy? Any item more complex than a felt-tip pen should be made non-company specific by a rational author/screenwriter.

  • Hope for the Clueless? (Score:3)

    by Alien54 ( 180860 ) on Sunday March 11, 2001 @02:00PM (#370484) Journal
    I liked this bit:

    ... I can assure you that many readers of fiction underestimate just how much of a novel's content is simply made up. There is a common assumption among readers that much of what appears in a novel is thinly veiled and repackaged reality. You can imagine how provoking this is to a novelist who works so hard to invent it. Furthermore, since my novel actually does contain an original cryptosystem, readers are even more inclined than usual to assume that all of the crypto mentioned in the book is real.

    I have to laff at all this. Obviously some folks really need to get out more often. Sometimes the reality check bounces. Sometimes paranoia pays, and sometimes it becomes a self-fulfilling prophecy.

    Never seen anything like that around here, of course.

  • My wife has read it, and tells me that it is Really Bad. In fact, she says that it is Late 80's Piers Anthony bad. Maybe even worse.

    Don't buy it to read ... buy it because you collect neil stephenson.

    --

  • Blockquoth the poster, quoting someone else:
    A simple acknowledgment in future discussions of this work by the author and his agents that there is a cryptography based on zeta functions,introduced in the open literature by Michael Anshel and Dorian Goldfeld and whose patent rights are assigned to Arithmetica Inc would do for a starter.
    Um, is every sci fi author using nuclear power supposed to reference Fermi, Oppenheimer, Einstein, Curie....?

    The fact of the matter is, zeta functions are fair game, mathematically, and their possible application to cryptography is not all that inobvious. Sure, the company has the patents on one particular system, but it's pretty clear that the system in the book is not that one.

    If Stephenson wants to be nice, he can mention it. But there's no obligation, legal or moral, that he give Arithmetica some free advertising.

    Since the system described in the book is pretty primitve (and eminently breakable) by modern standards, I'm not entirely sure why Arithmetica wants to be associated with it anyway.

  • I wonder what kind of ass chewing Stephenson would have given Comstock if he whined about using his name as a bogus key.
  • Has anyone heard any rumours on Quicksilver, it's
    been about 2 years since Cryptonomicon was released. Seems about time for the next one.
  • My wife has read it, and tells me that it is Really Bad.

    It's not *that* bad! The plot's a bit loose, the writing might not be up to his usual standards, but it's really, really, funny. I'm almost tempted to read it again....

    --Bruce Fields


  • I'll be honest, I found it a very tough book to read. The narrative was poor, and it wasn't easy to read without putting thought into it.

    I'm sorry, if I want thought, I'll read philosophy or texts; when I read a novel, I'm after relaxation and entertainment.

    Snow Crash and The Diamond Age both drew me in with fantastic narrative and were truly great books. Even The Big U was better than Cryptonomicom.

    ~Cederic
  • >>Vidi, Vici, Veni

    More like...
    Veni,vidi, visa

    I came, I saw, I shopped

  • Sadly, it has a typically poor Stephenson ending. He really needs to learn how to write a graceful ending that ties up some of the loose ends he's spent the whole book generating. I find it very frustrating to read about characters for hundreds of pages and develop some empathy for them and then have the book rudely chopped off just before finding out how their personal situations were resolved.

  • >works in Gotham City, instead of New York--by putting him in Gotham City, the creators afforded themselves the creative license to put buildings in different places, etc. I always thought Metropolis was New York and Gotham is somewhere around where Chicago should be... that said, with DC comics continuity who knows?
  • I always thought that Neal was using a clever pun in choosing FINux as the name of the OS. Fin, obviously due to Linus Torlvalds heritage.

  • To Quote:

    For those who are wondering, yes, Enos Root did die in Sweden in 1944 only to reappear 50 years later in a prison in the Philippines.

    They faked his death. N.S just takes us along for the ride. Just before Shaftoe departs for manila there's a description of Bishoff and R.von.H walking someone covered entirely in blankets in to the back of a car. I think. It was one of the things I didn't catch the first time I read the book.

    I want to know more about Societas Eruditorium as well. But hey.

  • Re:So what is the zeta function ? (Score:1)

    by Anonymous Coward
    There is a million dollar prize for proving the Riemann Hypothesis. See the Clay Institute Page [claymath.org].

  • If you want facts about Turing, I would recommend the excellent biography "Alan Turing: The Enigma" by Andrew Hodges

  • Not quite. The conjecture is that every non-trivial zero of the Riemann Zeta function lies on Re(s)=1/2.

    One can show that \zeta(s) = 0 whenever s = -2k, for any k > 0. (\zeta(0)=-1/2.) This follows from the functional equation, which in pidgin-TeX is:

    \zeta(s) = 2^{s} \pi^{s-1} \sin(\pi s/2) \Gamma(1-s) \zeta(1-s).

    This says that once we know \zeta(s) for Re(s)>1 (and this is the region for which the Dirichlet series \sum_n 1/n^s converges), then we know \seta(s) for Re(s) 1, it follows that \zeta(-2k) must be zero to cancel the pole of the Gamma function.

  • That's weird. I thought a book was an instance of art and as such, one of its functions was to generate some sort of activity in the reader's brain.

    This is not just aimed at the parent comment. I find the ubiquity of such anti-intellectual stance astonishing. Try any alt.books.* newsgroup and all people discuss is whether the story had a "good ending". Sigh.
  • Keyword. Late eighties. All those were early eighties.

    --

  • Thanks for the info - I don't plan on selling, my actual goal is to get NS to sign both of them, and keep them.

    Worldcom [worldcom.com] - Generation Duh!
  • You know, I didn't get ths part at all. Did he fake his own death? Was I reading this part too late at night and I missed something? Mod me off-topic, just answer me :)

    I dunno. That secret society Root was in seemed to know a thing or two about healing (remember when Amy was about to lose her leg from an infection, and Root did something which she refused to discuss but healed her). And we know Rudy is a member of the same society.

    And remember that Shaftoe witnessed the death. He was a fscking marine, he should probably be able to tell when someone is dead (or close to it).

    My theory is that Root was not-quite-but-almost dead, the society has some technique or technology (nano-tech in the 40's?) that can heal people, and that's what brought him back. Guess we'll find out.

  • This appears to have been a private communicate so I'm not totally sure it was appropriate for them to publish it on their web page, or slashdot to refer to it here. Interesting though...
  • Yeah, that confused me. And the fact that the Societas Eruditorium was poised to actually play a role in the plot, but didn't.

    It's my understanding that Cryptonomicon is the first part of a series of books. Makes sense, when you consider the vast amount of material which was vaugely referenced but never used (who were the black and Indian guys with Rudy and Otto on the boat (he never even gives their names!)? What's up with that secret society? What was Root doing working for the NSA during the 50s? If he faked his death, why? If not, how is he still alive? Not to mention the data haven; plenty of interesting things could happen there).
  • If you haven't read Cryptonomicon yet, I heartily recommend you do so. It's three quarters of a good novel, and the last quarter has enough twists in it that you can overlook the cheese and the fanboy religious overtones.

    You can read the prologue [cryptonomicon.com] online and decide for yourself. Try before you buy, and see some of the zeta functions Stephenson is talking about.
  • If I figure out a really neat idea for a faster-than-light drive with no problem, I don't have time to write the novel. I'm out in the back yard building my spaceship.

    If I'm not wrong, a FTL vehicle would go back in time, hence you would have lots of time to write. You could write it yesterday for example.
    __
  • Any mathematicans care to explain it ?
  • That's a great story in the Dayton Daily News. It mentions that the NCR people had been working on high-speed electronic counters, and seriously looked at building an all-electronic bombe using gas-tube technology. But they decided it was just too complex and too big a step to make happen fast, so they chose to build an electromechanical machine, something NCR could produce in quantity with their existing plant.

    That was the right decision. The mass-produced NCR bombes had a real effect on the war effort. ENIAC, on the other hand, didn't work until after the war.

    The key point here is that using electronics wasn't a conceptual problem. It's that good parts weren't available yet. Much of the early history of vacuum-tube computers revolved around getting better tubes made. That was basically solved by 1950; by the time the UNIVAC I was built, operational tube failures weren't a problem.

    (Why? At every UNIVAC I power-up, the machine was run on "high margins" for a while, with the voltages slightly high. This caused any tubes near failure to fail. Failures were easily detected, and they were then replaced, allowing a day with no tube failures. Tube failures in operation were very rare, probably rarer than Windows crashes. But that was 1950 tube technology, not 1942 tube technology.)

  • Stephenson's points about not linking his fiction too closely to real-world companies are interesting - I wonder if this has any bearing on an oddity in Cryptonomicon : Linux is disguised (barely) under the name Finux - however, if I recall correctly, Windows and Be are identified by name. I'd be interested to know the rationale behind this.
  • My reality alarm went off when Stephenson claimed that the language the Tagalog tribe wanted to foist off on the rest of the Philippines was called "Pilipina" This actually refers to a female of the species (perhaps a side reference to "Glory"), rather than "Pilipino" which refers to the language.

  • The BIG U is in print (Score:4)

    by YIAAL ( 129110 ) on Sunday March 11, 2001 @01:12PM (#370512) Homepage
    This is kinda offtopic, but Stephenson's first novel, "The Big U" is now back in print. I just bought a copy. For the many who've been looking for it in used bookstores, it's now available.
  • A year or so back when _The Big U_ was pulling down absurd bucks on Ebay Neal made very clear he didn't consider it worth the fuss.

    That warning made, I'll probably buy it myself because I'm such a fanboy.
  • Although explicit permission was given in this case, John Young of Cryptome [cryptome.org] has a habit of publishing things that other people don't want published. He's gotten in semi-serious trouble for publishing classified documents before (and they're still on-line). Ironically enough, however, he took down the DeCSS code because 'enough other people were mirroring it' (paraphrase, can't find the link right now).
  • Also from Neal's web page:
    Persons who wish to interfere with my concentration are politely requested not to do so, and warned that I don't answer e-mail.
  • Wish I had though of this before:
    "All your zero are belong to 1/2 + b*i"

    -Chris

  • Zeta functions...mmmm... (Score:3)

    by Telal ( 314917 ) on Sunday March 11, 2001 @02:34PM (#370517)
    I'd function Catherine Zeta Jones anytime. :-)
  • Actually, this wasn't acknowledged by the international scientific comunity, but Mai (no last name), a papua mathematical savant who developed the equivalent of 2500 years of Western number theory from first principles using only the bones of defeated and eaten enemies, has developed an entire proof of the Riemann hypothesis, which has been photographed into microfilm by anthropologist Lucius Zingelberger, and is currently being stored in the local library at the village of Ikai, 200 miles into the deep woods of the island. As far as I know, it has no telephone number or Internet connection, so interested parties should visit Ikai Library in person; I believe the daily fee for borrowing microfilm from it is 8000 human bones (how one will acquire this amount is none of my business). Good luck on the trip, and don't forget to take your shots! (And your shotgun. White man's meat is very much appreciated in Papua New Guine. Just ask Prof. Zingelberger.)

    -- Kaufmann
  • I keep seeing references to this or that algorithm being used to generate a one-time pad. The whole point of a one-time pad is that it's generated randomly so that it cannot be uniquely decrypted to any particuar plaintext without knowing the pad used.

    What Stephenson describes is a stream cipher using the zeta function to generate the bits, and using the date as the key. It's no more a one-time pad than would be, say, RC4.

  • You missed it - they already made a movie [imdb.com] out of it.... :)

  • If you're into gaming (of the war- or RP- variety), or hang out with those who are, you will kill yourself laughing reading this book.

    If you ever stayed in a college dorm, you are also in mortal danger of fatal mirth.

    I enjoyed this book. It's not a masterpiece, but it is good fun.

  • n^0 is 1 for all n AFAIK.

    So the first term will be 1.

  • put me in context here... name a few 80s piers anthony books... everything i've ever read by him i absolutely loved
  • Speaking of bombe's - there was a recent series of articles in the Dayton Daily News about the people behing the NCR / Navy's Enigma cracking machine here [activedayton.com]. Nice read.
  • That's right about Turing's bicycle. Please moderate the previous posting up.
  • One reason why everyone cares about the Riemann Hypothesis (which, on the face of it, is not all that exciting) is that the zeta function is very closely related to the way the prime numbers are distributed. There's some deep magic behind this, to do with contour integration and generating functions and stuff.

    The zeta function is supposed to show the distribution of primes approximately; however, I seem to recall reading that the prime numbers are shown to be distributed in a random manner and that the zeta function also predicts a somewhat random distribution. If the primes truly are distributed randomly then one would think that you could not predict them since they lie randomly about the natural numbers.

    Well one might ask of what benefit is this? Well aren't some crypto systems based on large prime numbers? If you could predict which numbers are prime then some crypto systems would be easily breakable.

    Also I seem to recall a New Scientist article that claimed that the universe is created from randomness and this randomness was also somehow inextricably linked to the randomness of prime numbers. So in some sense mathematics and physics may be linked together in their foundations.

  • "The zeta function is defined as an infinite sum:
    zeta(s) = Sum from n=0 to infinity of 1/n^s"

    Are you sure, because if s is positive then the first term in the sum would be 1/0

    Is it the sum from 1 to inf?
  • Nevertheless, it, or things much like it, were used many times during WWII, by both sides. If it takes a month to figure out what function they're using, and what it's keyed off (Using the date was to make it easier to break, it could have just as easily been the tenth word in the London Time from exactly two weeks ago.), then you can safely use it to coordinate a battle. As long as the function is unknown to the other side, they can't decode it, and once it is known, they still have to figure out what your key is.

    It's a very clever solution before the invention of public keys and whatnot, and it's called a psuedo one time pad. In fact, it is still used today, by spys who aren't able to access computers. They just go to a library, and grab last weeks New York Times and do an easy cipher based off line X on page Y in section Z, and vary those (and the paper that was last week's paper changes too, of course) based on some easy formula, like add one to X and subtract three from Y each week. It's not that secure, because it is one piece of information, and once the people you're hiding stuff from knows it, they can decode all your previous and future messages, but it works for short amounts of time. There simply are too many possibly places to get the seed from.

    -David T. C.

  • I have two copies of the book, one in fully readable condition, and one "strange". I assume the "strange" one to be due to a publishing error, but I wonder how many got out of the publisher, and how many were kept (ie, not returned)?

    Anyhow, my GF got the book for me a couple of xmases ago. She bought it off Amazon, and when I received it, I immediately began reading it. About a third of the way through, the book "repeated" - I thought I was losing my mind, but the text did repeat. I scanned farther forward, and it "repeated" again, never getting more than 50-75 pages "forward". I think there was a production problem, and multiple "leaves"(? Can't remember what the individual page bundles are called in publishing) got inserted. Funny thing was, the bundles weren't from near the end of the area I was at, but instead were from the mid-beginning, from a point I was well past.

    Anyhow, it made the book unreadable, so I had my GF ask for another from Amazon - they complied, but never asked for the original back in return. I just wonder how unique it is...?

    Worldcom [worldcom.com] - Generation Duh!
  • I find it very frustrating to read about characters for hundreds of pages and develop some empathy for them and then have the book rudely chopped off just before finding out how their personal situations were resolved.
    Well, that's the way real life is; very rarely is everything all wrapped up and resolved at once. Any given day leaves lots of unresolved issues; in fact, most us us die with a lot of the "plot points" in our lives still hanging.

    Tom Swiss | the infamous tms | http://www.infamous.net/

  • Yes. The series sum_{n=1..infty} n^{-s} only converges for s in (1, infty).

    However, the zeta function can be analytically continued to the entire complex plane as follows:

    pi^(-1/2 s) Gamma(1/2 s) zeta(s) = 1/(s[s-1]) + int_1^infty x^{1/2 s-1} + x^{-1/2s - 1/2} omega(x) dx (1)
    Gamma is a function which satisfies Gamma(n) = (n-1)!; it is defined as
    Gamma(s) = int_0^infty e^{-t} t^{s-1} dt
    as least for positive real x, and analytically continued over the whole complex plane.

    Finally, omega is defined by

    omega(x) = sum_{n=0..infty} e^{-n^2 pi x}
    So what you do is that you prove that for real s in (1, infty), equation (1) gives zeta agreeing with the sum_{n=1..infty} n^{-s} definition; and you prove that equation (1) is an analytic function, well-defined for the whole complex plane.

    Now, the so-called trivial zeros of the zeta function occur where s is of the form -2k, and k is a natural number. There are no other zeros of the zeta function anywhere but in the "critical strip", which is the area where 0 < Re s < 1. There are infinitely many zeros in the critical strip, and the Riemann hypothesis says that they all occur with Re s = 1/2.

  • He still did get some niggling details wrong though, which in all honesty doesn't really matter. If you try to explain any subject, even relatively simple ones, that you're not yourself well versed in an expert in that field will be able to pick it up.

    I try not to be a pedant, if the stories good I don't bother with small details. I work with a lot of people who like to tear apart every little problem in a book or movie. It's just a mental circle jerk to me.

  • Orginally, Siegel and Schuster based Metropolis on Toronto. Btw, check out 'Watchmen' for accuracy in streets. There's an intersection that re-occurs throughout the series, and they actually built a cardboard model of it to get the perspectives right from any angle.
    -----------------
  • When I was in Grad School there was a certain amount of folklore about books or movies that had accurate Mathematics in them. To my knowledge there is only one, a movie called "It's My Turn", about a Math professor, played by Jill Somebody. She gives an accurate proof of the Snake theorem during a lecture scene.

    Maybe Springer-Verlag should start a film studio - Arnold Schwarzennegger as "The Last Topology Hero", etc.

  • Hm. I'd guess by the fact that the answer is in Neal's FAQ his plan didn't work, and he just recieved vast numbers of email from Linux fans demanding to know the reasons for the change instead, but it's interesting to know the reason. Thanks.
  • I thought the point was that Enoch *did* die. I mean, he gets shot; coughing up blood he's babbling about getting to the church to get his gear; R.von.H. shows up with his old cigar box - a box related to his status in this ancient secret 'Societas Eruditoium'. Lods of foreshadowing on this box -- Shaftoe even has smack dreams about the contents of the box. Anyway, R.von.H shows, uses the mystical contents of the box, revives Enoch, drives him away.

    Secondary effects: the woman inherits Enoch's military death benefits & takes his son to England. Enoch is legally dead -- no one is going to come looking for him while he does his secret shit in the Phillipines. So, even if I'm wrong, and it was just a fake death, there are your reasons.

  • yeah - Finux: for industrial strength typesetting!

  • Yes (Score:3)

    by alewando ( 854 ) on Sunday March 11, 2001 @01:16PM (#370538)
    I'd explain it myself, but this page [v-wave.com] does a pretty good job, and I'd hate to duplicate efforts. It's an important function in number theory, particularly concerning prime numbers.
  • yeah, that's the link I found, right AFTER I had preordered the book..that's ok, though, as I am in the middle of reading it again (just got to the second semester, in which things stop making sense)
  • The zeta function is defined as:

    Zeta(x)=Sum(n=1 to inf){1/(n^x)}

    For instance, Zeta(2) is

    1/1 + 1/4 + 1/9 + 1/16 + ...

    MathWorld probably explained it much better than I could. Sigh.

  • I read books for many reasons:
    - to relax
    - to learn something new
    - because I am bored
    - etc

    When I am reading a novel, it is usually so that I can relax. I find Gerald Seymour books extremely easy to read, they have fantastic stories written superbly, are highly entertaining, and also cause me to ponder their content long after I read them. They do cause brain activity, and they often don't have a happy ending.

    Snow Crash blew me away with its vision. Extremely well written, fast paced, exciting concepts and also something that took little effort to read. I enjoyed it greatly, and it broadened my mind.

    Cryptonomicon however was a bore to read. I found it drudgery. Sure, none of the concepts mooted are beyond me (I hope) but to properly understand the story I had to stop and figure out little puzzles, or think through an issue before I could continue. That's not what I'm reading a novel for.

    If I want something to read that is going to make me think, I have work related texts to read, or I'm currently halfway through "Secrets and Lies" - that book requires brain activity, but is thus not a purely entertainment based relaxation aid.

    To link this all back to your comment: If books are an instance of art, then Cryptonomicon is bad art, unless its function was to generate brain activity involving disappointment, boredom and tiredness.

    ~Cederic
  • "Ironically enough, however, he took down the DeCSS code because 'enough other people were mirroring it' (paraphrase, can't find the link right now)."

    Well, only because in that same document he published the DeCSS source!!!

    http://cryptome.org/dvd-hoy-reply.htm
  • >Sadly, it has a typically poor Stephenson ending.

    I thought the same of Snow Crash, but i assumed he was keeping it open for a sequel.
  • Having been in this business for quite a few years now I can assure you that the annoyance of people who are left out of novels is nothing compared to the fury of those who fancy that they have been inserted into novels without having given their permission.

    Bit of a passing reference to the reason he feels that the Big U is a bad book, perhaps? :)

  • "The coding starts with a continuously generated string of random numbers, say from a satellite put up to broadcast them or from some other source."

    Maybe that's what the 'number stations' are for...

  • Re:Cryptonomicon (Score:5)

    by Ryan J. Evans ( 29421 ) on Sunday March 11, 2001 @01:20PM (#370546)
    From Neal's webpage: [well.com]

    Since Finux was the principal operating system used by the characters in the book, I needed some creative leeway to have the fictitious operating system as used by the characters be different in minor ways from the real operating system called Linux. Otherwise I would receive many complaints from Linux users pointing out errors in my depiction of Linux. This is why Batman works in Gotham City, instead of New York--by putting him in Gotham City, the creators afforded themselves the creative license to put buildings in different places, etc.
  • Michael Chrichton did quite a bit of detailed explaining in Airframe, and used quite a few business names; but I guess that's the exception.

    --
  • For real fan-boy collectors, I have the gold-cover edition of "Cryptonomicon" signed by Neal. Bidding on Ebay starts soon... :)
    -----------------
  • Yeah, that confused me. And the fact that the Societas Eruditorium was poised to actually play a role in the plot, but didn't.

    A number of Stephenson's books seem to have a problem where the reader is never quite sure if a character died or not, as his prose in these parts is perhaps deliberately vague.

    Plus his endings always leave the reader danglingt, which is too bad, because the rest of the text is awesome.
  • Rosebud was a sled. Whoops! Forgot to warn you. Sorry about that.
  • This crowd is going to have a pretty good bogosity filter for the crypto stuff. Stephenson is an excellent BS artist. As an environmental engineer, I've lost count of how many technically astute folks have asked me why the EPA hasn't instituted a PCB remediation program like the one described in Zodiac [amazon.com].

    Stephenson's assertion that many readers of fiction underestimate just how much of a novel's content is simply made up can't be overemphasized...

  • I don't know about you but I lose a little respect for people that insult their audience. I guess we're all stupid enough to give him our money but not smart enough to understand that it's fiction?

  • And don't forget that Root then travels to Manila with Otto and Rudy. It's never adequately explained why they faked his death, actually.

    Remember that Cryptonomicon is but one volume (and I do mean volume!) in a multivolume work, and that the next volume takes place in the 1500's or so. I suspect that we're going to hear a lot more about Societas Eruditorum in that book. The thing that I didn't catch until the second time around was those perforated gold sheets that they found in the wreck of the V-Million. These were the Leibniz-Archiv that Rudy had mentioned in Sweden, and that he had apparently sweet-talked out of Hermann Goering. Hopefully we'll hear more about those in the next volume too.
  • I enjoyed the historical stuff in Stephenson's book so much, that it really made me wonder where he drew the line between fiction and reality. When I finished The Cryptonomicon (after rolling my eyeballs at its typical Stephensonian over-the-top ending), it left me quite curious to know more about Turing's life.

    The Cryptonomicon provoked me to read the new American edition of "Alan Turing: The Enigma" by Andrew Hodges [turing.org.uk]. It was out of print for the longest time, but the American edition was just recently published. It's an excellent book, entertaining while being both historically and scientifically accurate, and it's gotten straight 5 star reviews on Amazon (although neither the author nor the subject were straight). Tom Jennings [wps.com] [inventor of FidoNet and founder of the Little Garden ISP] wrote the first review of the original edition, and he rates it as one of the most important books he's ever read. So I bought a bunch and gave them out as xmas presents!

    -Don

  • Re:So what is the zeta function ? (Score:5)

    by ElJefe ( 41718 ) on Sunday March 11, 2001 @01:20PM (#370557)
    The zeta function is defined as an infinite sum:
    zeta(s) = Sum from n=0 to infinity of 1/n^s

    Here, s can be any real or complex number. For example, zeta(2) = Pi^2 / 6, and zeta(.5 + 14.134i) = 0.

    The Reimann hypothesis is that all the zeroes of the function lie are of the form .5+b*i. where b is some real number. To date, this hasn't been proven and remains one of the great unsolved problems of math.

    From what I've been told, the zeta function also shows up a lot in number theory and quantum mechanics, but I don't really know much about it...

    -Chris
    (I'm an applied mathematician, dammit.)

  • Note that the sum for n=[0, infinity] of 1/n^s does not actually converge (is not a complex number) if s is not positive real, if I remember correctly. Fortunately, the sum does behave nicely (is an analytic function) when s is positive real, and you can thus use techniques from complex analysis to extend it uniquely over the entire complex plane. Chris's statement about the Riemann Hypothesis is correct; what we do have are a set of bounds which say that the (nontrivial) zeros of the Riemann zeta function can only occur in certain regions of the complex plane. (re: nontrivial: there are a few zeros of the Riemann zeta function on the real line as well; they are trivial zeros).

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