Repost of the original message that started it all -- spread it far and wide:
Gentlemen,
A mathematical morsel for your entertainment and edification.
The number
6,857,599,914,349,403,977,654,744,967,
172,758,179,904,114,264,612,947,326,
127,169,976,133,296,980,951,450,542,
789,808,884,504,301,075,550,786,464,
802,304,019,795,402,754,670,660,318,
614,966,266,413,770,127
is the product of
5,174,413,344,875,007,990,519,123,187,
618,500,139,954,995,264,909,695,897,
020,209,972,309,881,454,541
and
1,325,290,319,363,741,258,636,842,042,
448,323,483,211,759,628,292,406,959,
481,461,131,759,210,884,908,747.
It highlights the absurdity of the whole thing, really. It seems hard for TI to argue that it's illegal to post a mathematical fact, doesn't it? Especially one that's so concise and straightforward.
I'd have to say the absurdity of the whole thing is that calculators are allowed on math tests at all, thus fueling an entire market of crippleware calculators.
(1) If you're trying to teach mathematical _concepts_ then make your problems come out to whole numbers that are easily done with pencil and paper.
(2) If (1) was true then maybe we'd see some real innovation in the calculator space, especially if the single-purpose electronic device costing $100 from TI had to compete with iphone apps costing $1. Or Google Calculator. Or Wolfram Alpha.
(3) I've anecdotally heard from my math-tutoring friends that there is a growing pandemic of high school kids in America that can't multiply single-digit numbers anymore. O_o
I teach mathematics at a college. Doing (1) for every type of problem is not feasible. In college algebra courses we now give problems that involve using the best linear, quadratic, or exponential fit. We definitely do not do these problems by hand. The calculator is a black box that spits out the correct approximation and the student is asked to answer questions based on what the calculator spits out for the correct fit.
Part of our jobs is to prepare students to use mathematics in engineering, physics, chemistry, etc. We don't want to always give problems where the numbers are nice.
Is (3) a problem? How many people can find interpolating polynomials? Very few. Interpolating polynomials are no longer needed by most students of mathematics because computing power is cheap and ubiquitous. The skill set for mastering mathematics has changed given this reality.
Think of it this way. Very few people can start fire without matches or flint. This used to be common knowledge years ago but is no longer needed. Fire making devices are cheap and plentiful.
Do you want your doctor to look at your symptoms and start leafing through reference books, or looking on Google to decide what you have? It doesn't matter how clever you are, or how good at reasoning, you have to have basic knowledge to use as the base.
AI researchers came to this conclusion a long time ago. Reasoning or experience is not enough - you must have codified, accessable knowledge on which to work.
Not knowing basic arithmetic cripples you when you need to work on the next stage. Not knowing that 7*8 is 56 means you can't work out the expansion of (x+1)^8. Working simple cases like that makes it clear what's happening when you differentiate, and gives insight that you can't obtain without doing the work.
The maxim from medicine is "See one, do one, teach one." The middle step is crucial. If you get the computer to do everything, you never gain the experience, and rarely have the insights.
And, personally, I can interpolate polynomials. It's an interesting example, because recently it allowed us to implement a new feature in our system. No one else to work out how to do it, and were amazed when I did.
In your terms, I can make fire in a world where most people can't. It just earned us real money.
Interpolating polynomials used to be taught in high schools. I learned it in high school and today almost no high school graduate knows how to do this. Is this a bad thing? I don't think so. With new technology and the advancement of ideas and knowledge the basic skill set that is necessary changes.
The division of labor has rendered many once necessary skills the province of specialists. This is a good thing. Is arithmetic one of those things? I begin to believe so. The expansion of (x+1)^8 does not require the ability to do 7*8. One can do the expansion using Pascal's Triangle.
Besides, the real question is ought one do the expansion of (x+1)^8 by hand? The answer is no. The expansion is a purely mechanical process. A computer does it faster and more accurately. In such a long problem I am likely to make a mistake and this mistake does not mean that I don't know what I am doing.
I do want my doctor to thumb through reference books particularly if my condition is rare. In such a case it is unlikely that the doctor will have encountered my condition and I want her to gain from the shared experience and collective wisdom of experts. That's what a reference book is for and it's a good thing. I want my doctor to be able to read and understand the book. That is what her training is for.
One can get insights into algebra using a computer. It's just that the insights are different than what one could get without a computer. Instead of teaching interpolating polynomials in high schools they are teaching basic statistics and in this present age the latter is much more beneficial than the former.
I'm sorry I can't convince you, but I've done both, and I know that insights are given in both realms, doing things by hand, and using the computer. To throw away one is to lose an advantage.
The point of doing the expansion of (x+1)^8 (to continue the example) is not to get the answer, but to see the patterns building. You say you can do it easily enough by using Pascal's triangle, but finding that insight alone is worth the time.
Just last week I saw someone differentiate x^(1/2) by using a calculator to convert 1/2 to a decimal, putting it at the front, then carefulyl subtracting 1 from it, and getting 0.5 x^(-0.5). Right answer. Then they did the same for x^(1/3) and got 0.334 x^(-0.667) and got full marks. Follow the process by rote, and don't think.
I know I'm fighting a losing battle, because "Get the computer to do it" is too easy. What is lost is hard to see, and once it's gone it's impossible to regain. I wish I'd learned a langauge when I was 15 or younger, instead of now when I'm nearly 50. I'm constantly surprised that I can get answers to questions far, far faster than my colleagues and employees using techniques I learned when I was 12. I'm not using experience or any great gift, I'm using what everyone was taught in school at that time.
It's still useful. Perhaps I should be grateful that these skills are disappearing because I'm certainly never out of work.
I agree with you on the first two points, but interpolating polynomials is not analogous to the ability to multiply single-digit numbers, which is indicative of a certain level of mathematical literacy which one would expect from even a third grader.
My experience is that the third graders know how to multiply without a calculator. But they use the calculator in 6th grade through 12th grade. By the time they get to college the connections in their brain used to multiply haven't been used in a long time and thus they have forgotten it. It isn't a travesty though.
It used to be the case that high school students were taught interpolating polynomials but they aren't anymore. It used to be taught to make estimates on what log(1.0356) is based on a table to values. When calculators became common the old timers would say things like,
"Wow, kids these days. They don't know anything. They have to rely on a calculator to find what log(1.0356) is. I can do that using interpolating polynomials and a table of values."
I suspect everyone on this website using a calculator to find log(1.0356).
I've interviewed senior candidates who can't do basic arithmetic in their head. I had one project management candidate, who was doing fairly well up until that point on the interview loop, who literally couldn't multiply 600 by 3 in their head, nor on the whiteboard.
My high school physics teacher did #1. The math was pretty simple if you knew the concept and the first week he showed us stuff like finding a square root on paper.
I did exactly the same thing when teaching physics in college for many years. I made the numbers easy, but made people memorize formulas. No calculators or "cheat sheets" allowed, but you could leave an answer as pi/sqrt(2).
I would tell the students: "The numbers should come out easy. If you get something weird, then one of us made a mistake!"
This simple change made a huge improvement in comprehension in the class. It wasn't about "number jamming" any more, but about learning.
This is where common sense comes into play. If you were in front of a judge, your argument that you are just posting a mathematical fact wouldn't stand for more than 10 seconds.
Tell the truth: why did you choose those two numbers, specifically? Oh, because they allow you to unlock a TI calculator. So (the judge would say), you are publishing a secret key. You are not publishing random number products for the fun of seeing how many digits your calculator can handle.
Taken to the extreme, this is the same argument as a hacker saying: I didn't break into the bank's computers, I just wrote a program, and the program is the one that broke into the bank. I was just typing on a keyboard, that's all I did.
It is an interesting fact, though, since there is a sequence of integers called "integers factored by computers". This integer is now a member of that set. The fact that a number this large was factored is interesting to the mathematical community. Should this information be suppressed because TI used this number to sign their OS?
What if there was a table of all numbers and their factorizations, 1 = 1, 2 = 2, 3 = 3, 4 = 2 * 2, ... If you noticed TI was using a number in this list as their signing key, you could just look it up. Would that be illegal?
Then why is it illegal to lazily generate this list as needed?
What other properties of integers are illegal to discuss?
(FWIW, I am almost certain that the DMCA does not apply to this case, anyway. TI is just sending letters Because They Can and because there is no penalty if they're wrong. I can send you a C&D telling you to take down the above comment. The appropriate response is to mock me, not to take the comment down.)
Basically, TI should have used a bigger integer. Why rely on something as flaky as the legal system to protect your business model when you can use mathematical fact instead? (Answer: because using a bigger number would have added 1 cent to the cost of every calculator sold. Sending C&Ds is free.)
You are still playing cute and don't understand how you'll be cut off by a judge (ie., common sense) in seconds.
Sir, is that the only large number that you published? Are you saying that your interest in this number has nothing to do with its presence as a key in TI's calculators? How come a search of your house found all those dumps of TI firmware? Seriously?
Don't get me wrong, I'm on your side, but this defense is called "playing stupid" and it is known not to work.
The good news is that this will never see the inside of a courtroom, as publishing factorizations of numbers is legal. So is doing whatever you want to your own hardware.
But anyway, how hard does the math problem have to be before it becomes illegal to solve? If my signing key is 4, would it be illegal for someone to say "Gentlemen, it has come to my attention that the number 4 is the product of 2 and 2."? Why?
How is 4 different from
68575999143494039
77654744967172758
17990411426461294
73261271699761332
96980951450542789
80888450430107555
07864648023040197
95402754670660318
614966266413770127 ? (I need to know what the law is before I can try to avoid breaking it.)
Also, is the mere act of looking up the factorization of this number in a table illegal? If not, what about using a program that has the number stored in a table? If not, what if that program lazily computes the factorization. Is strict evaluation legal, but not lazy evaluation?
How about if I have a friend that randomly wants to know the factorization of a number and I compute it in my head? What if it's for breaking a cryptosystem, but I don't know that, I just think it's a problem he's challenging me with? Who broke the law? What law was broken?
Anyway, I don't think you've thought this through. You used the word "cute" dismissively, though, which was very cute...
> But anyway, how hard does the math problem have to be before it becomes illegal to solve?
Well, that is the point, isn't it. It is about intent, not the difficulty of the problem. Picking up a wallet is easy, taking someone's wallet is illegal.
But, the intent here: "use my own software on my own hardware", is (or should be) legal - all the rest is fluff.
While I agree with the general principle you're espousing, from a legal point of view, it's clearly possible to decide that the probability that someone "just happened" to factor a sufficiently large number is low enough to determine beyond reasonable doubt that they had some ulterior motive in doing so.
> How is 4 different from 68575999143494039 77654744967172758 17990411426461294 73261271699761332 96980951450542789 80888450430107555 07864648023040197 95402754670660318 614966266413770127 ? (I need to know what the law is before I can try to avoid breaking it.)
How is "Limp" different from "Limp, the body of Gorrister hung from the pink palette; unsupported—hanging high above us in the computer chamber; and it did not shiver in the chill, oily breeze that blew eternally through the main cavern. The body hung head down, attached to the underside of the palette by the sole of its right foot. It had been drained of blood through a precise incision made from ear to ear under the lantern jaw. There was no blood on the reflective surface of the metal floor. [...]"?
There might be legal precedent on this issue since there used to be books of tables of logarithms published in the days before calculators and computers. The publishers would place subtle errors in a few random points many digits out to prove that someone else had copied the numbers without computing the logarithms themselves. Of course IANAL.
Forgive me my lack of insight, but I'm not entirely certain what that has to do with this particular case. What you are listing is a mechanism for copyright protection - while the numbers themselves cannot be copyrighted, the book containing them might qualify for copyright (depending on the local legal system).
The question here is not about copyright, however. (Probably. Maybe.) The question here is whether it is legal to factor an arbitrary number and inform others of the results.
England and the United States have decided differently (england: yes, us: no) whether to allow copyright of lists of otherwise uncopyrightable information. The long term outcome has been a thought provoking natural experiment: the United States, with no such copyright allowance, has a thriving industry of competing databases of various kinds of publicly available information. The UK has no such thriving ecosystem. Or at least this was all according to a newspaper article a few years back. For all I know they may have since seen the light and banned copyright on databases.
Copying the table may be illegal (although probably not; you can copy the phonebook), but referring to it while figuring out how to make a flat screwdriver unscrew a phillips-head screw is probably not. How is using a flat screwdriver to unscrew a phillips-head screw different from using your own OS on your own calculator instead of using TI's?
[Source of MS Windows in binary] + 1 = [Source of MS Windows in binary + 1]
However, I while I feel that breaking into systems should be punishable, I think that breaking cryptography should not be, as long as you inform the owner a six months before going public, and do not use the result for any personal gain.
No cryptography was broken. Given a number, he found a number that divided it. It's a property of the universe, and you can't copyright properties of the universe. (That gravity is mine!!!)
I think that breaking cryptography should not be, as long as you inform the owner a six months before going public, and do not use the result for any personal gain.
It worries me that anyone would say something like this. I notice that 2 * 2 is 4, and I have to wait 6 months before telling anyone? Or what? What would TI do with this information? Why does TI have a right to tell me what I can do with chunks of plastic with TI's logo on it? (Remember, he's not copying their software; he wants to delete their software and use his own.) What legal framework would this fit into? Factoring numbers has nothing to do with copyright law, and a new law suppressing the publication of scientific research is a clear violation of the Constitution. Even if it is illegal, people can't be stopped; you can't police the entire Internet, and you can't stop people from installing software on their calculator. There is no way a government or corporation could monitor this.
So you basically want to make a law that is a violation of hundereds of years of American principles, that is completely impossible to enforce, and that will drastically slow down scientific and mathematical progress. All so a piece of plastic that says "TI" on it can only run TI's buggy code?
Wow. I ... just don't know what to say.
You also use the word "owner". Who owns a number? Does TI own the number because they used it to sign some software? Can I "own" the number 4 because I say I do? If you are in a situation where you have four pieces of candy, and want to split them between you and your friend, should you have to inform me of your intentions and wait 6 months before giving yourself two pieces of candy and your friend the other two pieces? Why not? I'm the owner of the number 4, and anything you do with it needs my approval.
The thing about "personal gain" is also insane. Everyone uses facts for personal gain. Big investment banks analyze market trends to make themselves billions of dollars. TI assumes you can't factor a number, and they use that to prevent you from cheating on math tests. All "personal gain" of the same variety. Should all enterprise be made illegal?
Now that I think about it, I guess your post is sarcasm... but you had me worried there for a minute.
> No cryptography was broken. Given a number, he found a number that divided it. It's a property of the universe, and you can't copyright properties of the universe. (That gravity is mine!!!)
Breaks of cryptography systems in general can also be expressed as numbers. Quite naturally. And so can all binary representations. The American law system certainly does view numbers as being able to be placed under copyright (or similar restrictions).
And numbers aren't a property of the universe. They are made up by humans.
I disagree. Integers are a property of the Universe. If you have 4 sticks, it's intrinsic in the way the Universe works that there are two groups of two.
Yes. For a silly example, one cloud and another cloud do not make two clouds, but one bigger one.
But more philosophical, it's the question of `oneness' that problematic. How can you define objects? All edges are arbitrary and if you look close enough they even blur.
(Of all the numbers the `natural' numbers are the most unnatural. Though they do describe some features of nature quite well, most scientiest would agree that e.g. complex numbers and vector fields are much more closely aligned to nature.)
Leopold Kronecker thought, "God created the integers; all the rest is the work of Man." Today we now better. The integers are no exception.
(Anyway, in practice every mathematician behaves like a Platonist no matter what his philosophical viewpoints are.)
Why should I worry about "a clear violation of the [US American] Constitution"? I never signed it, and do not even live under its jurisdiction. Why should I care about care about "hundereds of years of American principles"? Do you care about the Abstraction principle (http://en.wikipedia.org/wiki/Abstraction_principle_%28law%29)? Neither do I.
If you ask me in private, I think that copyright is no longer enforceable, and thus should be done away with. And philosophical, so called intellectual property restricts what I can do with my physical property.
My grandparent post expressed a more guarded opinion. And telling somebody that you broke his cryptosystems (or his key) before going public with it, is at least good manners.
I don't think it is so simple as that. In this thread many people are arguing something to the effect of "factoring numbers and posting it online is not illegal".
I tend to disagree. Not with the argument itself, but with the deception in covering the original issue with a much toned down one. The hacker who did this was not trying to innocently post the results of some factoring problem. He posted the results to a specific factoring problem. Then he posted an article titled "83+ OS signing key cracked!". Clearly he was not trying to just factor those numbers.
How Texas Instruments should have ideally reacted to these results is debatable, the fact remains that what the hacker did was not pure white.
How would you feel if somebody figures out your SSN and post it online? Is it just another mathematical truth that is allowed to flow freely through the universe?
If you use a 512-bit RSA modulus then you are asking for it to be factored. I am in the process of factorizing the 512-bit RSA modulus of a very famous company and will disclose it (responsibly) once it is done. Factorizing keys of that size is doable on high-end consumer hardware with a little patience.
The more fundamental question here: can the DMCA really be used to prevent a person from factoring a number and disclosing the results? This is insanity.
I'd guess that it can, if that number is being used to protect a copyrighted work. That sounds like exactly a situation that the DMCA was designed to make illegal.
Actually the DMCA has an exemption for allowing software interoperability. This sounds like exactly the situation that it was designed not to protect against.
It's not that the number is being used to protect copyright - it's a signing key that prevents you from installing different software on your calculator.
But yes. The DMCA has the provision about publishing DRM circumvention techniques and tools. This seems to be one.
Stupid and unfair, BTW. TI is hurting itself more than its user base.
Tom Cross, a security technology researcher in Atlanta, received a cease-and-desist letter from TI after merely posting about the hackers on his blog, ... [and] took down his link.
Excuse me, taking down a link to a discussion of the key? Does the DMCA really extend that far? If it does, it really sounds like it could be challenged on First Amendment grounds. Otherwise, either this guy has no backbone, or there is more to the story.
My comment is largely reiterative, but important enough to emphasize.
It's not the law as written, but as it plays out in the larger socio-economic sphere, that is often determinant. It doesn't matter what your rights are, if the other guy can spend you into the ground with impunity.
Taken to extreme, it's one reason we regulate monopolies (in the U.S.). You may come up with a better product, but if the other guy has a lock on the market...
If you don't have the resources, the wherewithal yourself to execute the fight, how do you make it? On your own, your one option is the shop for a lawyer who believe you have enough of a chance that their willing to invest in the effort (in return for a hefty share of any judgment in your favor). If you are fortunate, the EFF or another organization may take up your case and use the pooled contributions of its members to fund the fight.
Even then, what will the publicity do to your career and other aspects of your life, depending on who you work for, etc. Fighting such things, one way or another, is important, but it is not easy and not everyone has the means, financial or otherwise. The means to fight, and/or the means to survive if the judgment (and penalties) goes against them.
The original forum thread was entertaining to read after the fact (parts have since been redacted).
"Oh my good lord...
You need to explain how you did this. Now. To somebody.
EDIT: I just confirmed this is what I think it is. This needs to go to ticalc.org. Are you being intentionally cryptic about this and want to stay unknown?"
What could TI's motivation possibly be? How could they lose any business by hackers modifying firmware? Instead, it seems it would be more likely to increase hardware sales.
I think this is down to the educational market. For many exams you can only used an approved make/model calculator.
If you can hack the OS you can use it to store answers, notes or similar aids in cheating for an exam. If the platform isn't locked down they'll lose their exam board approval which would have a big impact on that models sales.
Notice on the page it also says -
"Count on TI calculators at exam time. You can use this TI graphing calculator on the PSAT, SAT, and ACT college entrance exams, IB, and AP tests."
I'm thinking it goes something like this. The TI calculators have something called "press-to-test", where a teacher can hit a combination of keys to disable functionality on the calculator. When in this mode, a LED flashes on the calculator so a teacher can tell at a glance that the student isn't "cheating".
Now imagine a homebrew OS where you hit the "press-to-test" keys, and the light blinks, but all the calculator's functionality is still there. If this became prevalent, then schools could outright ban the calculators on tests, so students would be less likely to buy them. Lost sales from this would dwarf any pickup from home hobbyists.
We weren't allowed calculators on most tests. (And when we were, most of the time I forgot to bring one. I've never been so fast with mental and written arithmetic since.)
I'm not sure about the details, but perhaps they're afraid that if the ROM key is broken, people can more easily read out the ROM code, and use it in a cheap clone of the hardware?
There might be a larger monetary value associated with the actual OS code, than with the hardware design itself.
As I said though, I'm not sure about the details here; you'd of course expect the CPU on the calculator to be able to access the ROM code in order to run it, but perhaps it's somehow obfuscated when that happens.
My understanding (which is incomplete) is that this is only a signing key.
Even if they are somehow relying on this key to prevent cloning, are they seriously under the impression that a 512-bit key is sufficient to keep a bunch of guys in Shenzhen from making a firmware-compatible device? Any effort to clone has long since come to fruition, and it's extremely likely they know this.
My guess: lawyers justifying their existence by picking fights with people they expect can't fight back. I'm glad the EFF has stepped in.
It's not even keeping anyone from cloning, because a Chinese manufacturer could just copy the ROM wholesale, signing key and all. Or patch out the signing key checks, or replace the key with one of their own... quite literally, the only thing this key is doing is keeping device owners from installing unauthorized applications and replacement OSes.
Didn't this hacker make a replacement OS better than TI's? I don't think TI will face serious competition from clones, except possibly in Asia and third-world countries. This hack doesn't even help in making clones, except maybe by giving engineers a chance to practice programming calculator OS's.
This would be plausible except AFAIK all of their models have different hardware capabilities. The 83 and 84 even un the same exact software; they just have different specs.
In the last 10 years, computers have advanced tremendously. Why do I still see the same TI-whatever calculators in on the store shelves? They don't need to play video or browse the web, but you'd think they'd at least be thinner.
Does TI have any competitors? Do they have an exclusive agreement to supply calculators that can be used on standardized tests?
Do they have an exclusive agreement to supply calculators that can be used on standardized tests?
In a great many cases, yes. That's why they never seen to change or innovate, and also why they're coming down like a load of bricks on anyone who hacks their firmware.
umm...amidst of all these no-doubt informative discussion, I found no one here appreciated how cool Hardware Hacking can be. And when it is done with something as minimalist as a Calculator(while these have more power than the first PDPs at MIT labs),there is just one word for it MIND BLOWING. I mean...look at them, they are making fcking GAMES for these. DUDE even FPS?A FPS on a calculator,in your face X-Box.
Okeyyy...If I getting down voted these many times,hardware hacking may not be cool.Or did something else ticked someone off?Please any comment might help me understand what is so offensive about this post.
I'd guess that your tone, combined with the borderline off-topic-ness and the fact that you haven't really added anything to the conversation ticked people off. I wouldn't and didn't downvote you based on that comment, but certainly wouldn't upvote. I suspect if you'd phrased it differently and maybe made it a little more useful, people would have responded differently, e.g. "I really admire these calculator hackers, it's amazing what can be done on these simple machines, e.g. these cool games: [link], [link] and [link]."
The HN userbase appreciates a high signal/noise ratio. Merely calling something "cool" is considered noise.
