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Chaff in the Channel (Stealth PGP work)
by tcmay＠got.net 17 Dec '03

17 Dec '03

At 9:32 PM 2/29/96, Bruce Zambini wrote: >Well, that's what I want to avoid; I think the issue is that as long as >stego is predictable, there's a problem, ie a message to a certain party >can be shown to exist, even if it's not readable. This might prove more >than ample evidence in certain circumstances. > >You shouldn't be able to recover the stego'd message without special >knowledge. This isn't addressed by current software, to my knowledge. Sorry if I haven't been following the latest "stego" messages too closely. If it is desired that an image, say, carry a steganographic message that is "undetectable" to adversaries, then much more than just stripping off the PGP markers (headers, identifying bits, whatever) must be done: the LSB bit plane, if this is the stego channel, must have statistics which are indistinguishable from "normal" LSB bit planes of images. (Not an easy thing to define or to implement, but there you go.) So, when the Khmer Rouge People's Enforcement Division looks at the image they have confiscated from your computer and examines the LSB bit plane for evidence of human rights files encrypted steganographically, that bit plane had better not have unusual statistics...it had better not look "too" random, as real life LSB randomness may not have nearly the entropy of PGP randomness, say. What can be done? One emergent standard could be to the following: - when images are sent, or stored, replace the true LSB bit plane (I say "true" to distinguish the actual "grey levels" of one or more of the color bit planes from RGB encodings in which the nominal LSB is not at all the minimum brightness changes) with a "PGP chaff image." - this PGP chaff image could be randomly generated, or chosen from a library, or (surprise, surprise) actually be an encoded message. - the point is that some percentage of all images would have this chaff present, so that mere possession of an image with the offending statistics would not ipso facto be proof of possession of an encrypted/stegoized message. (Of course, the Khmer Rouge People's Enforcement Division might simply kill you anyway, but then they might kill you for merely having a computer. One would hope that Reno's Raiders would not do likewise, and that the existence of multiple images with "chaff" image planes would be sufficient to confuse things.) - the adversary may know you have an image with a chaff plane, but he doesn't know that you actually know how to decode that chaff, that that chaff is not chaff to you. [How is this any different from simply sending chaff messages conventionally, without using steganography? Why not use the full bandwidth? Answer: Stego provides some plausible deniability, more important in court cases in the U.S. than to the Khmer Rouge, of course. Having random messages filling up one's hard disk is suspicious, but having images of the Mona Lisa which _may_ contain stego bits and which _may_ be readable by the owner is considerably less suspicion-arousing.] This is my take on fixing the stego situation. Instead of worrying about a "stealth PGP version," which is likely to be only a slight speed bump (because of the statistics), think about flooding the detection channels. Longterm, however, I certainly think that cryptographic messages can be made virtually indistinguishable from low-order bit noise. (I have argued this since the late 1980s, so I'm not changing my views now.) --Tim May Boycott "Big Brother Inside" software! We got computers, we're tapping phone lines, we know that that ain't allowed. ---------:---------:---------:---------:---------:---------:---------:---- Timothy C. May | Crypto Anarchy: encryption, digital money, tcmay(a)got.net 408-728-0152 | anonymous networks, digital pseudonyms, zero W.A.S.T.E.: Corralitos, CA | knowledge, reputations, information markets, Higher Power: 2^756839 - 1 | black markets, collapse of governments. "National borders aren't even speed bumps on the information superhighway."

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InfoWar Epilogue 4 (Part III of 'The True Story of the InterNet')
by Matthew Ghio 17 Dec '03

17 Dec '03

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InfoWar Epilogue 4 / TEXT
by Matthew Ghio 17 Dec '03

17 Dec '03

---------------------------------------------------------------------------- ---------------------------------------------------------------------------- The True Story of the InterNet Part III InfoWar Final Frontier of the Digital Revolution Behind the ElectroMagnetic Curtain by TruthMonger <tm(a)dev.null> Copyright 1997 Pearl Publishing ---------------------------------------------------------------------------- ---------------------------------------------------------------------------- InfoWar Table of Contents * Epilogue * gomez ---------------------------------------------------------------------------- Epilogue ---------------------------------------------------------------------------- Subject: Re: Just say "No" to key recovery concerns...keep OpenPGP pure From: Adam Back <aba(a)dcs.ex.ac.uk> To: tcmay(a)got.net CC: shamrock(a)cypherpunks.to, cypherpunks(a)cyberpass.net, eric(a)sac.net Tim May <tcmay(a)got.net> writes: > I'll try a different way of making my points... > At 9:12 PM -0700 10/14/97, Lucky Green wrote: > >I can't help but see a difference between enforcing to encrypt to a > >default key and storing the user's key outright. IMHO, the former entails > >less potential for abuse. > > All other things being equal, maybe the former is slightly less intrusive > than the latter. But maybe not even this, as the two give the same results. > After all, what's the real difference between "all mail, incoming and > outgoing, must also be encrypted to a CMR key" and "you must deposit a copy > of your key with us"? CMR keys are the root of all evil in pgp5.5. Without them almost any permutation of recovery care to construct would be less useful to the GAK'ers, for all the organizational, and inconvenience reasons Tim describes. Governments have problems handling complexity. So make their job complex. If you were one of the people writing the IRS tax software back in the 60s, and you were in deep cover, a proto-cypherpunk, and were bright enough to see the future possibilities you would have done all you could to fuck up the IRS system. You would have obfuscated the code. You would have put logic bombs in it. You would have destroyed the source code surreptitiously. (Destroying source code has analogies to destroying keys at earliest opportunity, you are destroying something which your enemy needs). Any bets as to if any of this actually happened on purpose? I reckon so. So, do you all reckon we can make task of fielding GAK impossibly complex for such a big disorganized government? Adam -- Now officially an EAR violation... Have *you* exported RSA today? --> http://www.dcs.ex.ac.uk/~aba/rsa/ print pack"C*",split/\D+/,`echo "16iII*o\U@{$/=$z;[(pop,pop,unpack"H*",<> )]}\EsMsKsN0[lN*1lK[d2%Sa2/d0<X+d*lMLa^*lN%0]dsXx++lMlN/dsM0<J]dsJxp"|dc` ---------------------------------------------------------------------------- ---------------------------------------------------------------------------- gomez I used to be a bright-eyed young computer guerrilla myself, many years ago, but I guess I have grown a little less radical as I have matured, since it now bothers me when youngsters such as Adam Back attempt to encourage others to sabotage the code of decent, hard-working programmers. This matter is of particular concern to me, since Basis, Inc. initially built the Purchasing and Contracting application in the IRS National Office, to run under UNIX on Zilogs, with a preSQL Informix database. Thus Back's intimations of "deep cover proto-cypherpunks" sabotaging the IRS could quite possibly cause serious harm to my professional reputation, in light of the wild-eyed claims made about me in 'The True Story of the InterNet' manuscripts. I have grown quite tired of being the target of some half-baked, delusional psychopath who is making me a character in the bizarre prevarications which drool from his mind, onto the printed page. I made the mistake of taking pity on a poor fellow one night in a Berkeley bar, buying him a few drinks in return for the atrocious tunes he played on a cheap, pawn shop guitar, and soon found him showing up on my doorstep every time he hit town, looking for a handout. Later, I found out that he had forged a letter of recommendation in my name, in order to help himself get established in the computer industry. It was very disturbing to discover some of the atrocities he had done under cover of affiliation with myself and Basis, Inc., causing me a great amount of legal problems. Now, after believing that his obsessive-compulsive fixation on me (because of his affection for my Jack Russell Terriers) was a thing of the past, I find that he is now pushing some crazy theory in regard to the Y2K problem, which is really a fairly insignificant matter that he has blown all out of proportion, and that he has drawn Adam Back into his madness to one degree or another. Mr. Back seems, from what little I know of him, to be a fairly reasonable individual, but lately he has been sinking into the same type of illogical rants and conspiracy rhetoric that the imbecile who calls himself the Author is notorious for. And now Back is making veiled references to persons connected to 'The True Story of the InterNet' obfuscating the IRS software code and planting logic bombs in it. Of particular concern to me is the fact that I personally had a hand in a rather innocent affair that resulted in our company making some coding changes which could possibly be misconstrued to indicate that we were involved in some type of testing of a Year2000 bomb in the IRS software code. One of the requirements for the application we built for the IRS was that no PO (Purchase Order) could be issued without a signed Requisition being referenced. Also, no PO could reference a Requisition issued outside the current fiscal year. Thus, no PO could be written for a Requisition from the previous year. Butthe BIG Butthe IRS staff had long been doing thisviolating the strict letter of the IRS rules-especially around the beginning of a new fiscal year. Lacking a current fiscal year requisition, they referenced one dated the previous (just closed) fiscal year. Suddenly, however, our application stopped their usual practice dead in the water, once certain PO's could not be processed because we required reference to valid (current year) Requisitions. We later discovered that the staff had been "cheating" by defeating our system's logic, after we were called in to tighten up all the design's enforcement of logic. Next thing you know, we were asked to relax the logic. I would like to stress that it is a matter of record that the calendar year related problems were a result of the government specifications and requirements for the application, and not the result of some devious plot on the part of myself or Basis, Inc., to surreptitiously investigate the potential for designing code that would take advantage of the unique problems surrounding the advance from the 1999 calendar year to the year 2000 in order to bring the IRS system to its knees. Neither was the fact that the garnisheeing of my paycheck by the IRS stopped shortly after our work on the IRS system started in any way related to my work on their system. It was merely a coincidence of timing. Conspiracy theorists might also make much of the fact that several Basis applications programmers shortly thereafter took jobs with the IRS for half of their former pay, but this can be explained by their desire for the long-term security and benefits that come with working for the federal government. The fact that most of them left the IRS and returned to Basis shortly after completing a major reworking of the calendar year updating system which locked the IRS irrevocably into two digit year entries indicates nothing more than the fact that they preferred living on the Left Coast. People often offer raised eyebrows when discovering that I went from having my paycheck garnisheed to receiving a $150,000 tax refund on a $100,000 salary, each year, but this is because they fail to take into account the fact that my work as a Research Faculty member for the Center for Policy Alternatives at MIT involved an interdisciplinary research center with links to Sloan School, which gave me access to superior financial management advice. The Center's close relationship to MIT's engineering departments was in no way connected to any sort of covert government activity, nor to any secret work on the development of the forerunners of today's InterNet in a manner that would make it possible for powerful non-elected Committees and secret Commissions to take control of a large segment of government and society. The Center for Policy Alternatives ceased to exist years ago, and the rumors that the Center was merely moved underground to secretly work toward establishing organizations such as FEMA, and the like, are preposterous. I am really growing sick and tired of the lies and slander that surround myself and Basis, Inc., as a result of the jealousy of other major league computer companies over our success at the peak of the Open Systems computer consulting industry. The constant insinuations that we cannot be trusted with highly sensitive systems such as those at the IRS and PacBell, simply because of our early history as Deadheads, is an insult to our integrity. To suggest that Acidhead Grateful Dead fans all have radical political agendas is no more true than suggesting that all Malcolm X followers are black. Well, OK, maybe that's a bad example Regardless, I would like to put to rest, once and for all, the preposterous rumors that have been circulating which link myself and other top computer industry executives to some mythical secret organization whose aim is to subvert the legitimate machinations of authority and the established power structure. I totally deny any connection to 'The True Story of the InterNet' or to anyone involved in the writing and/or dissemination of the manuscripts, and I would advise others to do the same. Copyright TruthMonger <gomez(a)BASISINC.com> "Last one seen fixing it, gets the blame." ---------------------------------------------------------------------------- "The Xenix Chainsaw Massacre" "WebWorld & the Mythical Circle of Eunuchs" "InfoWar (Part III of 'The True Story of the InterNet') Soviet Union Sickle of Eunuchs Secret WebSite ----------------------------------------------------------------------------

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The reason terrorists carry fly-swatters
by nobody＠REPLAY.COM 17 Dec '03

17 Dec '03

>From News of the Weird: * The San Jose Mercury News reported in March on some working models of the Defense Department's tiny flying machines ("micro air vehicles"), no larger than birds, including one helicopter that could fit inside a peanut shell, that are suited for tasks such as locating hostages in occupied buildings, sniffing out poisonous chemicals, and finding enemy snipers. Each micro air vehicle carries cameras, sensors, transmitters, and antennas.

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Re: Is PGP still private?
by Anonymous 17 Dec '03

17 Dec '03

> Your reencryption scheme fails because of the management of the short > term encryption keys, among other things. Wait a minute. Didn't Adam abandon the reencryption idea and switch to key escrow? Or did that one turn out to be a non-starter too?

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Re: Security flaws introduced by "other readers" in CMR
by Jean-Francois Avon 17 Dec '03

17 Dec '03

From: Fabrice Planchon <fabrice(a)math.Princeton.EDU> To: Adam Back <aba(a)dcs.ex.ac.uk> Cc: cypherpunks(a)cyberpass.net Subject: Re: Security flaws introduced by "other readers" in CMR >They do: the title is "Le programme PGP se range", which I unfortunatly > have know idea of how to translate. It can be translated by "PGP gets into the groove" it "behave itself". It finally obtemperate (wether prompted or not). It grow's up and become reasonable. "se ranger" is what you do with your car when a cop car pulls you on the side. it gets in line There is an ambiguity in this expression. "Se ranger" is usually used for a transition to an unusual or not approved behaviour to a accepted or acceptable behaviour. It entails a "higher" authority or standard. This is why I use a much more americanized language when I speak in french about civil liberties and related subjects. The french language almost equate government intentions with fatherly love... :-) Over the years, I found that most French peoples have a feodal attitude with a tendency to almost blind veneration of hierarchy. (flames to dev(a)nul.con) I suppose it comes from their History. In general, if a govt said it, it *has* to be good. It is as if they were trained since young age to surrender their capacity for independent judgment and reason. And quite distressingly, it happens even with extremely intelligent individuals. OTOH, the uncommon ones that do not exhibit such psycho-epistemological trait tend to be real hot shots, independent type. I am not French, but Canadian (french) and we often have scrapes with French on this topic. They tend to trust their govt, unless somethings warrant otherwise. Here in french Canada, we are much more American in that sense. Notwithstanding what trumped-up polls say, we have a much more american-styled opinion. Ciao JFA

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index.html
by Jim Choate 17 Dec '03

17 Dec '03

CNN logo Navigation Infoseek/Big Yellow Pathfinder/Warner Bros Tech banner IBM AS/400 rule WILL HEADLESS HUMAN CLONES GROW ORGANS IN 10 YEARS? 'Human Cloning' graphic October 19, 1997 Web posted at: 8:39 p.m. EDT (0039 GMT) LONDON (AP) -- Headless human clones will be used to grow organs and tissues for transplant surgery in the next five to 10 years, a leading authority on the ethics of human cloning predicted Sunday. Dr. Patrick Dixon, author of "The Genetic Revolution," which forecast the cloning of animals, made the prediction after The Sunday Times reported that British scientists have created a frog embryo without a head. Scientists believe the technique used to create the headless frogs could be adapted to grow human organs such as hearts, kidneys, livers and pancreases in an embryonic sac living in an artificial womb, the paper said. Many scientists believe human cloning is inevitable following the July 1996 birth of Dolly, the world's first cloned mammal. She was created at the Roslin Institute in Edinburgh using cells from a sheep's udder. "I predict that there will be great pressure to combine cloning technology with the creation of partial fetuses, missing heads, arms or legs as organ factories for tomorrow's people," Dixon told the British news agency, Press Association. "These will be developed on an experimental level somewhere in the world in countries where there is little or no gene legislation within the next 5-10 years because of the overwhelming demand," he said. "The demand is there, the technology is almost there as well," he said. Dixon called for an urgent global biotechnology summit to examine every aspect of genetics, and provide the foundations for international agreements. International inconsistencies on various aspects of genetic engineering, including human cloning, urgently need to be ironed out, he said. "Genetic engineering is a very exciting technology, it has the power to feed the world, cure disease but you do have to ask fundamental questions before it is too late," Dixon said. "The headless frog embryo is another example of the way the technology is racing far ahead of public understanding," he said. "We must get the thinking in place which looks over the horizon beyond today's headlines to what tomorrow will bring," Dixon said. "Scientists have been making up the rules as they go along." Genes manipulated to suppress frog's head Jonathan Slack, professor of developmental biology at Bath University, told The Sunday Times that he created headless frog embryos by manipulating certain genes -- and used the same technique to suppress development of a tadpole's trunk and tail as well. The headless frog embryos have not been allowed to live longer than a week because under British government rules, embryos are not considered animals until they are a week old, the paper said. Slack believes the breakthrough could be applied to human embryos because the same genes perform similar functions in both frogs and humans, The Sunday Times said. The newspaper said the lamb and frog techniques could be combined so that people needing transplants could have organs "grown to order" from their own cloned cells. These organs would exactly match the patient and there would therefore be no danger of rejection. It would also ease the shortage of organs for transplant, The Sunday Times said. Slack told the paper that using intact human embryos would not be possible because they would have to be killed. "It occurred to me a half-way house could be reached," he was quoted as saying. "Instead of growing an intact embryo, you could genetically reprogram the embryo to suppress growth in all the parts of the body except the bits you want, plus a heart and blood circulation." Growing parts of human embryos to cultivate organs could bypass many legal restrictions and ethical concerns, because without a brain or central nervous system the organisms may not technically qualify as embryos. Professor: 'No ethical issues' Lewis Wolpert, professor of biology as applied to medicine at University College London, called Slack's suggestions sensible and feasible. "There are no ethical issues because you are not doing any harm to anyone," The Sunday Times quoted him as saying. But Oxford University animal ethicist Professor Andrew Linzey told the newspaper: "It is morally regressive to create a mutant form of life." "It's scientific fascism because we would be creating other beings whose very existence would be to serve the dominant group," he was quoted as saying. Copyright 1997 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed. rule CNN Plus * Message Board: Cloning Health Special: Cloning Related stories: * Report: Cloned sheep has human gene - July 24, 1997 * Embryo splitting caught in cloning controversy - June 25, 1997 * Scientists grow monkeys from cloned embryos - March 2, 1997 * Firms team up to make cloned cattle with special milk - October 7, 1997 Related sites: Note: Pages will open in a new browser window * Genzyme Transgenics * Genetics and Ethics - Biomedical Ethics Unit, McGill University * Institute for Philosophy and Public Policy - Animals as Inventions: Biotechnology and Intellectual Property Rights * Genetic Engineering Home Page External sites are not endorsed by CNN Interactive. _________________________________________________________________ Infoseek search ____________________ ____ ____ _________________________________________________________________ rule Watch these shows on CNN for more sci-tech stories: CNN Computer Connection | Future Watch | Science & Technology Week rule Message Boards Sound off on our message boards You said it... [INLINE] IBM AS/400 rule To the top © 1997 Cable News Network, Inc. All Rights Reserved. Terms under which this service is provided to you.

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Re: "First do no harm"
by Jean-Francois Avon 17 Dec '03

17 Dec '03

On Sun, 19 Oct 1997 16:31:10 -0700, Tim May wrote: >At 5:23 PM -0700 10/19/97, Jean-Francois Avon wrote: >First, are you reading Cypherpunks? No. I don't have time to even read all of e$ and e$pam. I am not yet retired... :-) >If so, post there. If not, please >forward my reply to the "e$" Done. >If I operated a small company, I would have no policy on this at all. Just >as I wouldn't routinely tape-record employees, I wouldn't monitor their >mail. >Do most small businesses steam open all of the incoming and outgoing paper >mail? Let me describe a scenario: one key employee isn't there because... ...he is somewhere else [ :-) ] Then, comes an urgent message forcing the company to make a split-second decision before the employee gets back. It happens _all_the_time_ in small businesses where they do not have the ressource of the big companies. You, the boss, want to re-read some critical documents before making a move. Of course, your employee was the most consciencious and trustworthy employee on this side of Andromeda, only, your memory fails as what was the *exact* wording of the critical documents. It furthermore happens that the said faithfull employee is, unknown to you and in his mind not-company-detrimental, getting intronized to the mile-high club between L.A and Tokyo in the plane lavatory, therefore not reacheable by any modern means. Wouldn't you wish you had his corporate private key? In the old days, you used to scramble through his mountain of paper on his desk. In more recent days, you frantically scanned his exemplary organized data structure and found the documents. Today, the said data structure is still exceptionally orderly, but only to reveal several very describing filenames with a dot-asc or dot-pgp extension. Any comments? >If a small business owner doesn't trust his employees, all is lost anyway. >(Employees can post stuff from home, can take diskettes out of the >building, etc., so what is gained by monitoring their mail?) I thought that this was so self-evident that I did not even thought usefull to explain. I was not talking about having employees giving copies of their personnal private key but of their corporate private key. The one they use to transact business over unsecured lines, not the one they use to send raunchy messages to their lover. >And if an employer is known to monitor employee mail, this actually >increases his liability for certain things. (If an employer does not, then >he has a better defense of claiming that some employee was acting on his >own, etc.) I never talked of monitoring e-mail, only of making sure the business retains access to employee-generated documents. I used to have a private key for the company at a previous work location. The first thing I did was to give my boss my passphrase for being afraid of what could happens if ever they needed a critical document during my absence. He did not even know how to use PGP, but I made sure that somebody did. This is why I said that a "secret" copy would be given to the boss. He could encrypt the employees keys, print it on a micro-dot and hide it under his gold tooth cap. And that way, there is always deniability since there is official company policy. Or better, all the boss has to do is to generate the employee's keys himself. He could forever deny having made a copy and nobody could prove that he did. Personnally, I would make sure that all keys to be used to transact business be generated by the boss, to be handed on a diskette in person. As far as trying to prevent malicious action, I agree that there is no way an employer can do that. Anybody really intent on screwing up things can. Ciao jfa -- Jean-Francois Avon, Pierrefonds(Montreal) QC Canada DePompadour, Societe d'Importation Ltee Finest of Limoges porcelain and crystal JFA Technologies, R&D consultants physicists and engineers, LabView programing. PGP encryption keys at: http://w3.citenet.net/users/jf_avon http://bs.mit.edu:8001/pks-toplev.html ID# C58ADD0D : 529645E8205A8A5E F87CC86FAEFEF891 ID# 5B51964D : 152ACCBCD4A481B0 254011193237822C

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One-Way Functions
by semprini＠theschool.com 17 Dec '03

17 Dec '03

I've read tons about how DES, PGP, and all of those programs use a one-way function involves the difficulty of factoring out numbers. Are there other one-way functions? --Dylan

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Re: A Math problem related to cryptography
by phelix＠vallnet.com 17 Dec '03

17 Dec '03

On 19 Oct 1997 18:36:26 -0500, Jeffery Walker <jeff(a)beol.net> wrote: > > I have been working on some math and C++ code to implement it. The >problem relates to a possible new means of factoring. However, I have run >into a problem which I cannot solve. Indeed it maybe impossible. The >problem is this: > > Find and equation using only multiplication, division, GCD, LCM and >powers (not addition or subtraction) in terms of a and b which is equal to >a+b for all Integers. If so desired it can be assumed that a and b are >relative primes with a>b. Note that equivalent problems include but aren't >limited to finding a similar equation for a+1. If other functions such as >logs are required they may be allowed but I would have to look into it. > > To encourage people to answer I am offering $20 for a solution and >$10 for a proof that it is impossible. I will judge all answers as to the >validity both in terms of actually being correct (or in the case of a proof >a valid argument) and in that they do not use functions such as addition or >subtraction which are not allowed. > (why do I get the feeling that I'm doing somebody's homework assignment) Claim: In the realm of Integers, There does not exist a formula using only multplication,division,GCD,LCM,and powers such that f(a,b)=a+b for all integers a,b Proof (by contradiction): I assume that a,b are primes > 2. Lets also define x,y,z,w,m,n and any integer >=0 and C,D,E as any integer > 0. Now let's look at the operations available to us: multiplication and powers: C=C*(a^0)*(b^0) CD=E(a^0)*(b^0) where E=CD a=1*(a^1)*(b^0) a^2=1(a^2)*(b^0) Cab^2=C*(a^1)*(b^2) well , all results can be expressed in the following form: C*(a^x)*(b*y) division: a !| b and b !| a (here !| means "does not divide") (From here on out, lets assume that a !| C,D,E and b !| C,D,E. it makes things simpler) Ca/a=C*(a^0)*(b*0) Ca/D=E*(a^1)*(b^0) assuming that E | C I won't go through all this. It should be obvious that for all allowed divisions, the result can be expressed as C*(a^x)*(b*y) GCD: gcd(a,b)=1 gcd(a,C)=1 gcd(Ca,Da)=max(a,gcd(C,D)) gcd(Ca,Db)=gcd(C,D) gcd(C(a^x),C(a^y))=C(a^(min(x,y))) gcd(C(a^x)*(b^y),D(a^z)*(b^w))=gcd(C,D)*(a^(min(x,y)))*(b^(min(z,w)))) =E(a^m)*(b^n) for E=gcd(C,D) m=min(x,z) and n=min(y,w) so, all results of GCD can be expressed in the form: C(a^x)(b^y) LCM: LCM(C,D)=CD/gcd(C,D) LCM(C(a^x)*(b^y),D(a^z)*(b^w)) =CD*(a^(x+z))*(b^(y+w))/gcd(C(a^x)*(b^y),D(a^z)*(b^w)) =( CD/gcd(C,D) * (a^(|x+z-min(xz)|)) * (b^(y+w-min(yw)) = E(a^m)*(b^n) where E=CD/gcd(C,D)=lcm(C,D), m=x+z-min(x,z) , n=y+w-min(y,w) so, all results of LCM can be expressed in the form: C(a^x)(b^y) Now, as you can probably tell by now, I don't remember a damn thing about groups, rings, and fields. If i did, I could have done all of the above in one paragraph. SO, what I'm trying to say is that all the numbers you can operate on here will be of the form C(a^x)(b^y). SO you really have to prove at least the following C(a^x)(b^y)=a+b for some C>0, x,y>=0 integers Now, you could probably say that C=a+b, x,y=0, but that's a cop out and is probably not allowed. Now, watch while I cop out and provide an easy counterexample: assume some formula exists such that f(a,b)=a+b with f only using *,/,gcd,lcm,^ operators let a=5, b=7 Since I know that ultimately anything I do with constants and the operators 8,/,gcd,lcm,^ will result in something in the form of C(a^x)(b^y), I won't even mess with the formula there must exist some C>0, x,y>=0 that makes: C(5^x)(7^y)=5+7=12 of x>=1 and y>=1 then C(5^x)(7^y)>12 for all C>0 if x=0 and y=0, C=12 (this works) if x=1,y=0 then 5C=12. there is no C which satisfies this. if x=0,y=1 then 7C=12. there is no C which satisfies this. so, the only solution is C=12, or more generally, C=a+b oops, that can't be the formula since we're not allowed to do that. QED. Well, that was pretty sloppy, so lets be more general about this: C(a^x)(b^y)=a+b Now, I'm sure that somewhere in the history of mathematics somebody has proven that for primes a and b where a,b>2 that a*b>a+b. So, for any x,y,C>0, C(a^x)(b^y)>a+b. C, of course, cannot be negative, or zero, so all were left with is x or y being zero. Let y=0. we have: C*a^x=a+b C*a^x - a = b a*(C*a^(x-1) - 1 ) = b a*E=b Now, a and b are primes and a!=b. From the fundamental theorem of algebra, every number can be represented by a series of primes such as: p1^a1 * p2^a2 * p3^a3 * ... In a*E=b, if we factor each side down to its primes, we find that the right hand side does not contain the prime 'a' which is on the left hand side, so a*E != b. A similar argument can be used for the case x=0. This leaves the only solution to C(a^x)(b^y)=a+b being x=0,y=0 C(a^0)(b^0)=C=a+b which isn't allowed. QED Now, If you're like me, you've probably read the above and don't have a f*cking clue what I meant (I sure wouldn't if I read all that crap). So here's the executive summary. With the operators you've defined on the number set you want to work with, no matter how complicated or crazy your formula is, I can express the result at C(a^x)(b^y). And hopefully, I've shown that C(a^x)(b^y) cannot equal a+b except in the trivial case of C=a+b (with x,y=0) which is what you don't want. >Thanks, >Jeff >P.S. I am very interested in joining the coderpunks list as I do program >and am interested in cryptography as related to programming. If someone >would be kind enough to recommend me I thank you. > I wasn't aware that you had to be invited to the coderpunks list. Just send a message to majordomo(a)toad.com with the following in the body: subscribe coderpunks -- Phelix, who really should have been able to give a better proof (especially since one of my degrees has the word 'Mathematics' on it)

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