Re: Almost-Everywhere Superiority for Quantum Computing
Russell Nelson [EMAIL PROTECTED] wrote: If quantum computers make brute-force cryptanalysis tasks easier, don't they also make brute-force cryptographic tasks easier as well? At 01:12 AM 10/18/1999 -0400, Vin McLellan wrote: The problem to worry about, of course, is that maybe not everyone is going to have access to the same oracle. Consider what was involved when the NIST lab at Boulder created a qubit a couple of years ago. As I recall, to get their qubit they had to trap a single atom with missing electrons (an ion) and two energy levels by nailing it down with an array magnetic and electric fields at minus 273 degrees C. For instance, will it fit in your palmtop or smartcard? Probably not. It's not clear that, in practice, it will be possible to get high enough resolution out of quantum computers to affect crypto - a resolution of 20 bits is enough to annoy smartcards by forcing the encryptor to use more key bits, but doesn't bother other computers. A resolution of h-bar is ~10**47 or 150 bits, but by the time we get that much resolution, it probably won't bother palmtops much, except maybe for RSA key generation. Quantum devices with resolutions like that probably aren't small or portable However, if you can get much bigger resolution improvements out of quantum devices without some way to use lower-resolution devices in parallel while still collapsing one big waveform in some kind of Quantum Black Magic. Hard to say if that will be portable or not, or affordable except by large organizations (particularly, even with Moore's Law constantly making things cheaper, will the cost be some large multiple of the cost of small portables, Pocket Area Networks, Pencil Area Networks, RF grocery tags, you-are-here broadcasters on store shelves or license plates, etc. Thanks! Bill Bill Stewart, [EMAIL PROTECTED] PGP Fingerprint D454 E202 CBC8 40BF 3C85 B884 0ABE 4639
Digital Contracts: Lie in X.509, Go to Jail
-BEGIN PGP SIGNED MESSAGE- Aside from noting the vicious hypocrisy of the Clinton administration saying they support the 11th Ammendment, I've also decided that the bill mentioned in the New York Times Story excerpted below, like most current state digital signature legislation, could more properly be called the "Lie on an X.509 'Certificate' and Go to Jail Act of 1999". "Crypto in a crime", indeed. The solution to this madness, is, of course, bearer credentials, as Stephan Brands points out in his recently published doctoral dissertation "Rethinking Public Key Infrastructures and Digital Certificates -- Building in Privacy", now published by Ponsen and Looijen in the Netherlands, ISBN 90-901-3059-4. The resulting book is in very well-written English, it's about 300 pages, and, in it, Brands, the best of his generation of financial cryptographers, completely demolishes, all the way down to the level of cryptographic protocol, most of the received wisdom about "certification" - -- and the current cult-like mystification of identity which underlies it. Even better, Stefan unveils a whole class of bearer-credential cryptographic protocols which get the job done with infinitely more privacy. More important, I'm personally convinced that Brands' bearer credentials are significantly lower in cost than current book-entry methods of "certification", especially after the costs of repudiation and enforcement are taken into account. However, given my current business, my biases on this subject are rather plain here, so don't take my word for it: get the book and see for yourself. Stefan's thesis committee was Ron Rivest, Adi Shamir (yes, the R and S in RSA), and Claus Schnorr (yes, *that* Claus Schnorr). Three men who could be easily said to be the fathers of digital "certification", if its patrimony was ever in dispute. I would highly recommend that *everyone* who's serious in the study of digital commerce -- and I mean legal professionals in particular -- order this book immediately and go read it. It goes without saying that anyone who calls himself a financial cryptographer, much less a cryptographic or digital commerce software engineer, should have this book in his library as well. Cheers, RAH - --- begin forwarded text Date: Tue, 19 Oct 1999 08:20:23 +0200 (CEST) From: Anonymous [EMAIL PROTECTED] Comments: This message did not originate from the Sender address above. It was remailed automatically by anonymizing remailer software. Please report problems or inappropriate use to the remailer administrator at [EMAIL PROTECTED]. Old-Subject: NYT Story: Digital Contracts To: [EMAIL PROTECTED] Subject: NYT Story: Digital Contracts Sender: [EMAIL PROTECTED] Reply-To: Anonymous [EMAIL PROTECTED] Fight Over Electronic Contracts Heads to House Also: U.S. Shut Out in First Round of Internet Board Elections ASHINGTON -- With the clock ticking toward adjournment for the year, Congressional leaders and the Clinton Administration are working to eliminate political infighting and pass legislation that would give electronic contracts the same legal weight as their traditional paper counterparts. The legislation is considered crucial for the future of electronic commerce, and it is part of an effort by the Commerce Department both domestically and internationally to make the standards for such contracts, with their "digital signatures," as simple as possible. But Republicans and Democrats in the House are still battling over how far the legislation should go, a fight that could play out on the House floor this week. The House is scheduled to take up digital signature legislation as early as Tuesday, but first leaders must decide how to proceed. At issue are states rights, and whether individual states should have the power to make their own rules for recognizing digital signatures. The White House and most Democrats are pushing for a bill that would make digital signatures legal only in those states that don't already have laws recognizing the validity of electronic contracts. But Republican leaders in the House have been pushing for more sweeping legislation that would not only pre-empt state digital signature laws but would also eliminate some of the paper-record keeping and notification requirements that some states impose on financial institutions and insurance companies. The House Judiciary Committee last week narrowly approved a version of the bill backed by Democrats that would recognize current state laws on both electronic signatures and record-keeping. The bill is similar to a White House-endorsed Senate proposal by Senator Spencer Abraham, a Michigan Republican, that is awaiting passage in that chamber. The House Commerce Committee, meanwhile, in August approved a bill by Chairman Thomas J. Bliley Jr., a Virginia Republican, that would establish a uniform national standard for authenticating electronic signatures, and require that states pass laws
Re: Digital Contracts: Lie in X.509, Go to Jail
In message v0421012db4321dc2f55c@[204.167.101.62], Robert Hettinga writes: The solution to this madness, is, of course, bearer credentials, as Stephan Brands points out in his recently published doctoral dissertation "Rethinking Public Key Infrastructures and Digital Certificates -- Building in Privacy", now published by Ponsen and Looijen in the Netherlands, ISBN 90-901-3059-4. Do you know where to order this? None of the amazon.com sites has it, nor doe s barnesandnoble.com. --Steve Bellovin
Re: Digital Contracts: Lie in X.509, Go to Jail
Evidently, there are only 500 in the first printing, but I bet Stefan didn't give them *all* away. :-). I bet that if you put in a special order to Amazon with the ISBN and the publisher in it, they'll manage to sell one to you on order. Upon receiving a bunch of orders for the book from some place like Amazon, if and when the publisher sells out, they'll probably print some more, or at least make a deal to print it on this side of the pond. Cheers, RAH At 11:56 AM -0400 on 10/19/99, Steven M. Bellovin wrote: In message v0421012db4321dc2f55c@[204.167.101.62], Robert Hettinga writes: The solution to this madness, is, of course, bearer credentials, as Stephan Brands points out in his recently published doctoral dissertation "Rethinking Public Key Infrastructures and Digital Certificates -- Building in Privacy", now published by Ponsen and Looijen in the Netherlands, ISBN 90-901-3059-4. Do you know where to order this? None of the amazon.com sites has it, nor doe s barnesandnoble.com. --Steve Bellovin - Robert A. Hettinga mailto: [EMAIL PROTECTED] The Internet Bearer Underwriting Corporation http://www.ibuc.com/ 44 Farquhar Street, Boston, MA 02131 USA "... however it may deserve respect for its usefulness and antiquity, [predicting the end of the world] has not been found agreeable to experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'
The Privacy Snatchers
Here's something I wrote two years ago that may be timely when evaluating
whether or not to trust the government. At least, that is, when police say
they'll not abuse wiretaps and backdoors inserted into Internet protocols.
-Declan
http://www.pathfinder.com/time/digital/daily/0,2822,12609,00.html
The Privacy Snatchers
By Declan McCullagh
History reveals that time and again, the FBI,
the military and other law enforcement
organizations have ignored the law and spied on
Americans illegally, without court authorization.
Government agencies have subjected hundreds of
thousands of law-abiding Americans to unjust
surveillance, illegal wiretaps and warrantless
searches. Eleanor Roosevelt, Martin Luther King
Jr., feminists, gay rights leaders and Catholic
priests were spied on. The FBI used secret files
and hidden microphones to blackmail the
Kennedy brothers, sway the Supreme Court and
influence presidential elections.
In these cases, police violated the law by
eavesdropping without a judge's approval, which
the Constitution requires. Now the FBI wants to
require Americans to use only computers and
telephones with a secret backdoor. Such easy
access is the fantasy of every unethical policeman
and corrupt bureaucrat. Of course, they pledge
never to use it without court authorization. Can we
trust them?
Martin Luther King
The FBI's campaign to destroy Dr. Martin Luther
King began in December 1963, soon after the
famous civil rights March on Washington. It
started with an extensive -- and illegal -- electronic
surveillance of King that probed into every corner
of his personal life.
Two weeks after the march, the same week King
appeared on the cover of Time magazine as "Man
of the Year," FBI agents inserted a microphone in
King's bedroom. ("They had to dig deep in the
garbage to come up with that one," FBI director J.
Edgar Hoover said of the Time cover story.) Hoover
wiretapped King's phone and fed the information to
the Defense Department and to friendly
newspapermen.
When King travelled to Europe to receive the
Nobel Peace Prize, Hoover tried to derail meetings
between King and foreign officials, including the
Pope. Hoover even sent King an anonymous
letter, using information gathered through illegal
surveillance, to encourage the depressed civil
rights leader to commit suicide.
"The actions taken against Dr. King are
indefensible. They represent a sad episode in the
dark history of covert actions directed against
law-abiding citizens by a law enforcement
agency," a Senate committee concluded in 1976.
Hoover's legacy? The FBI headquarters proudly
bears his name today.
Mail Monitoring
Opening mail may be an imprecise form of
surveillance, but that didn't stop the FBI and CIA
from surreptitiously reading hundreds of thousands
of letters from 1940 to 1973. Government
employees (who took special classes to learn this
skill) would stealthily open the envelope and
photograph whatever was inside.
The CIA did it randomly. One agent testified before
Congress, "You never know what you would hit."
Included in the agency's dragnet were three U.S.
senators, a congressman, a presidential
candidate and many business and civil rights
leaders.
Under federal law, opening mail not addressed to
you results in fines of up to $2,000 and five years
in jail. But not one agent appears to have been
prosecuted.
Eleanor Roosevelt
Even the personal life of the First Lady of the
United States is fair game to the eavesdrop
Is there an anonymous contribution protocol?
A couple of months ago, someone (unfortunately, I dont recall the name or date) wrote to the New York Times, suggesting that all political contributions be made anonymously. Given the continuous contention that the issue of political contributions causes in the US, I was intrigued by the idea and have thought about it, off-and-on, since even though connected people snort derisively at the idea and I assume it has the political viability of the Caesar cipher at the NSA. Nevertheless, I wonder whether a suitable protocol has already been invented. Typically, a contributor would like to be able to confirm his donative status with a political party or candidate, as would the party or candidate for, their targeted mailings and such. Thus, any anonymous political contribution protocol would involve proofs of membership and share much with secure election protocols. However, these are not sufficient. It should be difficult for the recipient to discover the amount of the claimed contribution. The contributor might say to the recipient, I will contribute $100,000 to your campaign at exactly 2:03 PM USA PST. While a third party, that anonymizes the transfer of funds, would be involved, immediate tracking of the balance could reveal the identity of the contributor. Limited reporting of the current balance or, using MixMaster-like techniques to obfuscate fund transfers, may do for high traffic ($$$) recipients. Further, it ought to be be difficult for a contributor to collect proofs of contribution from other people that he has illicitly funded in the classical "Gore-Buddhist-Temple" attack (http://www.realchange.org/gore.htm#buddhist) to later present to the recipient. This seems hard and may justify a very low minimum contribution. While, "I am not a lawyer", I am certain that there are labyrinthian free speech issues involved in mandated anonymous political contributions. While the fact that we voted is recorded in the US, the side that we voted for is concealed (I hope so, having made no detailed examination of the internals of voting machines. Such is Trust.). Even though anonymous voluntary free speech is protected here, it is a thick slice between "I contributed" and "I contributed $100,000. Here is my receipt! Get the bedroom ready." Michael
Bernstein Delay Motion
Thanks to Cindy Cohn we offer the USG's motion yesterday to delay en banc reargument in Bernstein: http://cryptome.org/bernstein-mot.htm A quote: "The revisions being implemented by the Department of Commerce entail extensive changes in the existing terms of the encryption export regulations. At this time, the details of the revised regulations are under review. One of the subjects currently under review in connection with the policy update is the regulatory treatment of encryption source code. It is possible that the revised regulations will not materially change the treatment of source code. But it is also possible that the revised regulations will alter the treatment of source code in ways that could have a bearing on the constitutional issues before this Court.(1) (1) In connection with the announcement of the Administration's encryption policy update on September 16, the Department of Commerce issued a "question and answer" document regarding the update that indicated, inter alia, that existing controls on the export of encryption source code would not be changed. That document does not reflect the review that is currently taking place." End quote.
Re: BXA
On Wed, Sep 29, 1999 at 07:41:34PM -0700, I wrote: At 04:49 AM 9/29/99 , Donald Ramsbottom wrote: What really intrigues me is the end of your post relating to the distinction between object code and source code. So if I understand you correctly, you will still require the old style regime and restrictions on source code. If so does that not mean that there is effectively no liberalisation? [...] Specifically of interest are general question #18, which indicates that technical assistance, APIs, and source code will continue to be controlled under the old regime; technical question #7, illustrating a new detailed approach to API regulation; and technical question #8, reiterating that only object code will be subject to the new policies. It appears that this may no longer be correct. John Young has made available on his website a document http://cryptome.org/bernstein-mot.htm filed by the US Government with respect to the en banc rehearing of the Ninth Circuit's decision in the _Bernstein_ case. In short, the US Government is asking the court to postpone oral argument in the case until the US Government has revealed the new regulations, promised for release on December 15 1999. As the filing states - "It is possible that the revised regulations will not materially change the treatment of source code. But it is also possible that the revised regulations will alter the treatment of source code in ways that could have a bearing on the constitutional issues before this Court.[1]" where footnote 1 says that the BXA's question and answer document "does not reflect the review that is taking place." Thus, reliance on that document may no longer be appropriate. BXA's website does not reflect that change in status. -- Greg Broiles [EMAIL PROTECTED]
Re: BXA (fwd)
- Forwarded message from Greg Broiles - Date: Tue, 19 Oct 1999 15:13:53 -0700 From: Greg Broiles [EMAIL PROTECTED] Subject: Re: BXA It appears that this may no longer be correct. John Young has made available on his website a document http://cryptome.org/bernstein-mot.htm filed by the US Government with respect to the en banc rehearing of the Ninth Circuit's decision in the _Bernstein_ case. In short, the US Government is asking the court to postpone oral argument in the case until the US Government has revealed the new regulations, promised for release on December 15 1999. - End of forwarded message from Greg Broiles - Which shouldn't be relevant since his rights were impacted under the *old* law. Even if the new regulations do permit unlimited export of crypto then he'd still have a reason to push the case. The best lack all conviction, while the worst are full of passionate intensity. W.B. Yeats The Armadillo Group ,::;::-. James Choate Austin, Tx /:'/ ``::/|/ [EMAIL PROTECTED] www.ssz.com.', `/( e\ 512-451-7087 -~~mm-'`-```-mm --'-
Re: Is there an anonymous contribution protocol?
At 3:48 PM -0400 on 10/19/99, Reusch wrote: "I contributed $100,000. Here is my receipt! Get the bedroom ready." Right. See http://www.xs4all.nl/~brands/order.txt There's an echo in the room, isn't there? :-). Cheers, RAH - Robert A. Hettinga mailto: [EMAIL PROTECTED] The Internet Bearer Underwriting Corporation http://www.ibuc.com/ 44 Farquhar Street, Boston, MA 02131 USA "... however it may deserve respect for its usefulness and antiquity, [predicting the end of the world] has not been found agreeable to experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'
Re: Is there an anonymous contribution protocol?
Michael Reusch, [EMAIL PROTECTED], writes: A couple of months ago, someone (unfortunately, I don't recall the name or date) wrote to the New York Times, suggesting that all political contributions be made anonymously. Given the continuous contention that the issue of political contributions causes in the US, I was intrigued by the idea and have thought about it, off-and-on, since even though connected people snort derisively at the idea and I assume it has the political viability of the Caesar cipher at the NSA. Nevertheless, I wonder whether a suitable protocol has already been invented. One way to approach protocol design for such problems is to first specify how the system would be done if there were a trusted third party (TTP). The TTP is perfectly trustworthy, honest, and reliable. Define the behavior he would have which would solve your problem, then look for a crypto protocol which accomplishes the same thing. Generally, any TTP protocol can be turned into a crypto protocol by straightforward techniques, but the resulting crypto protocols will generally not be efficient. You can then look for shortcuts or approximations to produce efficient crypto protocols to accomplish the same thing. Maybe you could try to clarify how the TTP would behave in order to accomplish what you desire with regard to anonymous contributions. What information would the TTP have? What kinds of interactions would it have to have with the participants? Expressing this in detail would be a good first step in looking to see whether a crypto protocol could be designed to do the same thing.
Re: Digital Contracts: Lie in X.509, Go to Jail
Robert Hettinga [EMAIL PROTECTED] writes: Evidently, there are only 500 in the first printing, but I bet Stefan didn't give them *all* away. :-). I bet that if you put in a special order to Amazon with the ISBN and the publisher in it, they'll manage to sell one to you on order. Upon receiving a bunch of orders for the book from some place like Amazon, if and when the publisher sells out, they'll probably print some more, or at least make a deal to print it on this side of the pond. Cheers, RAH Amazon only sells books distributed in the United States. Cheers, Julian.
Re: BXA
Unless, of course, this quiet announcement (in the Bernstein court papers filed by the US Govt) that the source code issue is currently being reviewed within the Executive Branch -- despite White House assurances to the contrary to leading Congressional figures -- was a purposely misleading representation, intended only to further stall and delay the Berstein hearing before the full appelate court. Of course, it is probably just because I'm a cynical Child of the SiXties that I view the DoJ posture with a skeptical and jaundiced eye. sigh Suerte, _Vin At 03:13 PM 10/19/99 -0700, Greg Broiles wrote: On Wed, Sep 29, 1999 at 07:41:34PM -0700, I wrote: At 04:49 AM 9/29/99 , Donald Ramsbottom wrote: What really intrigues me is the end of your post relating to the distinction between object code and source code. So if I understand you correctly, you will still require the old style regime and restrictions on source code. If so does that not mean that there is effectively no liberalisation? [...] Specifically of interest are general question #18, which indicates that technical assistance, APIs, and source code will continue to be controlled under the old regime; technical question #7, illustrating a new detailed approach to API regulation; and technical question #8, reiterating that only object code will be subject to the new policies. It appears that this may no longer be correct. John Young has made available on his website a document http://cryptome.org/bernstein-mot.htm filed by the US Government with respect to the en banc rehearing of the Ninth Circuit's decision in the _Bernstein_ case. In short, the US Government is asking the court to postpone oral argument in the case until the US Government has revealed the new regulations, promised for release on December 15 1999. As the filing states - "It is possible that the revised regulations will not materially change the treatment of source code. But it is also possible that the revised regulations will alter the treatment of source code in ways that could have a bearing on the constitutional issues before this Court.[1]" where footnote 1 says that the BXA's question and answer document "does not reflect the review that is taking place." Thus, reliance on that document may no longer be appropriate. BXA's website does not reflect that change in status. -- Greg Broiles [EMAIL PROTECTED]
Re: Digital Contracts: Lie in X.509, Go to Jail
For details of how to order, see www.xs4all.nl/~brands/order.txt What is it about wanting to change the instantaneous electronic world that generates this sort of time paper hazing ritual? Yours in irreverent confusion, Lightning Rod
Re: Digital Contracts: Lie in X.509, Go to Jail
At 9:20 AM +1000 10/20/99, Julian Assange wrote: Robert Hettinga [EMAIL PROTECTED] writes: Evidently, there are only 500 in the first printing, but I bet Stefan didn't give them *all* away. :-). I bet that if you put in a special order to Amazon with the ISBN and the publisher in it, they'll manage to sell one to you on order. Upon receiving a bunch of orders for the book from some place like Amazon, if and when the publisher sells out, they'll probably print some more, or at least make a deal to print it on this side of the pond. Cheers, RAH Amazon only sells books distributed in the United States. That's not quite true, as the US publishers of Harry Potter found out. You can easily order from England http://amazon.co.uk and Germany http://amazon.de Unfortunately, neither list 'Rethinking Public Key Infrastructures..." Ponsen and Looijen have a web site http://www.p-l.nl/ That's as far as I got. Arnold Reinhold
Re: Digital Contracts: Lie in X.509, Go to Jail
$94 for the $50 book from the US, ($19 shipping and bank costs on the publisher's side, $25 for an "international money order"). Robert Hettinga wrote: At 2:27 PM -0400 on 10/19/99, Somebody, at the head end of a long line of other Sombodies, wrote: Where can I get this book by Brands? To which, at 8:16 PM +0200 10/19/99, Stefan Brands wrote: For details of how to order, see www.xs4all.nl/~brands/order.txt :-). Cheers, RAH - Robert A. Hettinga mailto: [EMAIL PROTECTED] The Internet Bearer Underwriting Corporation http://www.ibuc.com/ 44 Farquhar Street, Boston, MA 02131 USA "... however it may deserve respect for its usefulness and antiquity, [predicting the end of the world] has not been found agreeable to experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'
Re: BXA v. Bernstein
_Bernstein_ case. In short, the US Government is asking the court to postpone oral argument in the case until the US Government has revealed the new regulations, promised for release on December 15 1999. Which shouldn't be relevant since his rights were impacted under the *old* law. Even if the new regulations do permit unlimited export of crypto then he'd still have a reason to push the case. Since Bernstein is primarily asking that the law regs be changed (struck down as unconstitutional), if the government changes them itself to be constitutional, then a large part of the case would be won. However, the chance of that is vanishingly slim. First, a big problem is in the statute, which BXA can't change, only Congress the Pres can. It eliminates judicial review of BXA decisions, yet all permanent free-expression suppression decisions MUST be made by a court. Second, a fundamental constitutional problem with the export controls is that they require that you ask the government for a license *before* you publish something, rather than merely trying to punish you later if you err. (A "prior restraint"). The so-called liberalization we're promised did not alter this scheme; they want you to go through a "one-time review" before they'll issue you a license to publish your software. Tinkering with the source code and technical assistance rules will not change this. It would be a fundamental restructuring, eliminating all that lovely administrative discretion so near and dear to NSA's heart. The discretion to deny licenses arbitrarily (or extend your "one-time review" into taking infinite time), in unpublished decisions, means that the regs don't actually have to say what the rules are; they can make them up as they go along. This is why we seek to end this discretion (and why courts back us up, in cases involving free expression). I encourage everyone to jump up and down and tell BXA to reform all the unconstitutional parts while they're messing around in there. Who knows, it might work. But I don't see their motion to put off the appeal hearing as anything more than their usual delaying-the- inevitable tactic. You can call me a cynic; I think of it as years of experience at NSA-watching. Sure, they'll revise the regs by Dec 15, change the source code rules, even make them constitutional. And pigs will learn to sing. We'd better wait til next year just to be sure, eh? The government is always free to revise the regulations at any time. It has known in gory detail what constitutional problems exist, since 1984 (http://www.eff.org/bernstein/Legal/950922_tien.exhibits/). It is always free to consider making them constitutional. But it has never yet done so. John
Re: Bernstein Delay Motion
John Young [EMAIL PROTECTED] writes: It is possible that the revised regulations will not materially change the treatment of source code. But it is also possible that the revised regulations will alter the treatment of source code in ways that could have a bearing on the constitutional issues before this Court.(1) Someone quote this next to "doublespeak" in the dictionary. On a more serious note, when Patel issued Bernstein III, I seem to recall a quote where she admonished the government for changing the regulations out from under her while she was working on a decision. Unfortunately, I can't find this quote. Does someone else recall this, and have a pointer? If I'm remembering this right, and if the en banc review remands (again), I can see her not looking positively on another request for a stay. Marc
Re: size of linear function space
[EMAIL PROTECTED] wrote:
Consider functions of one variable whose domain and range are both
{0,1,2,...,n-1}. There are n^n possible functions.
n!, I'd say, since the range of any function that isn't one-to-one is
_not_ {0..n-1}. Did you mean that the range was a subset of {0..n-1}? Or
perhaps (equivalently) you meant to say "codomain" instead of "range"?
How many of these
are linear [i.e. F(a+b) = F(a) + F(b) + c, where c is the same for all
a,b (if it were different, that would be trivial)]? For any one
definition of +, there will be some number;
This strikes me as completely false. Can't be bothered to prove it,
though. Especially since the problem is currently not well-defined :-)
I'm interested in the sum
over all definitions of + that satisfy the usual requirements of
associativity, commutativity, additive identity, etc.
Hmm. This is horribly inexact. Do you mean the usual requirements for a
group? A field? What?
And like anonymous says, if you are going to ask these weird questions
(some of which are quite entertaining), you could at least say why.
Cheers,
Ben.
--
http://www.apache-ssl.org/ben.html
"My grandfather once told me that there are two kinds of people: those
who work and those who take the credit. He told me to try to be in the
first group; there was less competition there."
- Indira Gandhi
