## A Mighty Number Falls 348

Posted
by
kdawson

from the time-to-generate-new-keys dept.

from the time-to-generate-new-keys dept.

space_in_your_face writes

*"An international team has broken a long-standing record in an impressive feat of calculation. On March 6, computer clusters from three institutions (the EPFL, the University of Bonn, and NTT in Japan) reached the end of eleven months of strenuous calculation, churning out the prime factors of a well-known, hard-to-factor number — 2^1039 - 1 — that is 307 digits long."*The lead researcher believes "the writing is on the wall" for 1024-bit encryption. "Last time, it took nine years for us to generalize from a special to a non-special hard-to factor number (155 digits). I won't make predictions, but let's just say it might be a good idea to stay tuned."
## What are they? (Score:5, Funny)

## Re:What are they? (Score:5, Funny)

## Re:What are they? (Score:5, Funny)

Hang on, I'm working on it. I'll get back to you.It's not going to take 11 months is it?

## Re:What are they? (Score:5, Funny)

On the plus side, the staff has quicker access to the nearest janitorial supply closet.

## Re:What are they? (Score:5, Informative)

1159420574 0725730643698071 48876894640753899791 70201772498686835353882248385

9966756608 0006095408005179 47205399326123020487 44028604353028619141014409345

3512334712 7396798885022630 75752809379166028555 10550042581077117617761009413

7970787973 8061870084377771 86828680889844712822 00293520180607475545154137071

1023817

factors:

5585366661 9936291260 7492046583 1594496864

6527018488 6376480100 5234631985 3288374753

×

2075818194 6442382764 5704813703 5946951629

3970800739 5209881208 3870379272 9090324679

3823431438 8414483488 2534053344 7691122230

2815832769 6525376091 4101891052 4199389933

4109711624 3589620659 7216748116 1749004803

6597355734 0925320542 5523689

(spaces added because of lameness filter)

## Re:What are they? (Score:5, Funny)

## Re:What are they? (Score:5, Funny)

## Wrong number, in both the GP and the summary! (Score:3, Informative)

I don't know where she/he got her/his number, but it's wrong.

Use Python, Luke:

## Re:Wrong number, in both the GP and the summary! (Score:4, Funny)

The factors are correct. Just checked.

And don't doubt me, I'm a 3 digits UID

## Re:Wrong number, in both the GP and the summary! (Score:4, Funny)

## Re: (Score:2, Informative)

Um, no it's not - that's somewhere between 2^1016 and 2^1017. Your factorisation is otherwise correct, but these aren't the numbers we're looking for.

## Re: (Score:2)

>>> 2**1039-1

58906 80864 31683 67664 47387 24917 74762 47119

38696 45981 50177 53575 68993 76584 32079 46555

59932 59138 49006 50140 34006 38916 15625 81754

37632 23144 51080 38858 45624 60719 42881 07610

69833 17459 92221 53387 11318 93632 01210 62386

22173 92146 90332 88521 55899 78237 00137 18480

62018 26907 36866 95341 12523 82072 65913 54912

10334 38768 44956 20912 65765 28293 887

however this is 313 not 307 digits as stated in the article... (8 rows w

## Re: (Score:3, Informative)

58906808643168367664473872491774762471193869645981 501775357568993765\

84320794655559932591384900650140340063891615625817 543763223144510803\

88584562460719428810761069833174599222153387113189 363201210623862217\

39214690332885215589978237001371848062018269073686 695341125238207265\

91354912103343876844956209126576528293887

## Re:What are they? (Score:5, Funny)

## Re: (Score:3, Funny)

## Damn, beaten, somewhat. (Score:5, Informative)

2^1039-1 = 5080711 * 5585366661993629126074920465831594496864652701848

is the correct factorization, as can be readily verified.

Also:

http://www.heise.de/english/newsticker/news/90031 [heise.de]

## Re: (Score:3, Informative)

## Re: (Score:3, Funny)

## Re:Damn, beaten, somewhat. (Score:5, Funny)

## Re: (Score:3, Informative)

With[{x = 2^1039 - 1}, Prime[Select[Range[1, 360000], (Mod[x, Prime[#]] == 0) &]]]

Finding the next factor like this (by trial division) should take a mere 10^70 or so times longer.

## Re: (Score:2)

You missed one:

5080711x

5585366661 9936291260 7492046583 1594496864

6527018488 6376480100 5234631985 3288374753

×

2075818194 6442382764 5704813703 5946951629

3970800739 5209881208 3870379272 9090324679

3823431438 8414483488 2534053344 7691122230

2815832769 6525376091 4101891052 4199389933

4109711624 3589620659 7216748116 1749004803

6597355734 0925320542 5523689

= 2^1039-1

## Re: (Score:3, Funny)

## Re: (Score:2)

## Re: (Score:3, Informative)

## Re: (Score:3, Funny)

## Re:What are they? (Score:5, Funny)

## Re: (Score:3, Informative)

## How many people have the computing power ... (Score:2)

## Re:How many people have the computing power ... (Score:5, Informative)

That said, not all 1024-bit numbers are hard to factor, in fact you have about a 1 in 300 chance of pulling 1024-bit prime out of your ass. The trick here is that RSA numbers are random and have less algebraic structure than Mersenne numbers.

Of course, with all that said, people should be using ECC anyways.

Tom

## Re:How many people have the computing power ... (Score:5, Funny)

in fact you have about a 1 in 300 chance of pulling 1024-bit prime out of your assWow, now *that* is a cool trick!

## Re:How many people have the computing power ... (Score:5, Funny)

Originally at http://www.primenumbershittingbear.com/ [primenumbe...ngbear.com] but that's long dead, so I dug it out of the Wayback Machine and put it up at http://rpresser.googlepages.com/primenumbershitti

## Re: (Score:2)

## Re:How many people have the computing power ... (Score:5, Insightful)

## Re: (Score:3, Insightful)

## Re: (Score:3, Insightful)

## distributed network computing? (Score:2)

## Re:distributed network computing? (Score:5, Interesting)

## Re: (Score:3, Funny)

## Re:distributed network computing? (Score:5, Interesting)

## Re:distributed network computing? (Score:5, Interesting)

This has already been done as early as 10 years ago.

I was working in Eastern Europe on a now unclassified project, working against a low budget illegal foreign intelligence agency. They were selling and distributing porn CD's and DVD's with thousands of pictures, one or more of which would contain an encrypted stenographic message. Their contact would purchase the DVD at one of hundreds of little markets, and decrypt the proper image(s).

It was really quite a good plan. Not only were there many possible valid messages to one or more agents, but there were also an unknown number of false messages, they even may have even been all false messages that could only be put together by inference. However, since they were encrypted with PGP, we never were able to break that particular system before I left the project.

The real genius of the plan was that it brought them in some much needed cash as well.

## Re:distributed network computing? (Score:4, Funny)

## Next step: FPGA cracking (Score:4, Interesting)

## Re:Next step: FPGA cracking (Score:5, Informative)

## Re:Next step: FPGA cracking (Score:5, Funny)

## Re:Next step: FPGA cracking (Score:5, Funny)

(forgive me, I love quantum-related jokes... ^_~)

## Re:Next step: FPGA cracking (Score:5, Funny)

(forgive me, I love quantum-related jokes... ^_~)

(forgive me, I love logic-related jokes

## Re:Next step: FPGA cracking (Score:5, Funny)

Quantum computers have that one nagging flaw: they actually exist.

## Security (Score:3, Insightful)

## They better hurry and copyright that number (Score:2, Funny)

## Still would take a while... (Score:2)

## Three years isn't a whole lot. (Score:5, Insightful)

I understand that they'll be able to crack 1024, but still, 3 years to see my e-mails. It's not worth it for them. Now when they got it down to 3 hours I'll be worried, but by then we'll probably be using 4096.True, but what you need to think about is forward secrecy.

There are lots of things being transmitted today that are still going to be in use three years from now. For example, think of financial information: if you use an encryption standard that's acceptable right now, but can be broken in three years (or, is trivially breakable in three years due to increases in computer power or techniques), then you're in trouble, because some of that information is still going to be sensitive/valuable in three years. The fact that you'll be using 4096 bits

thendoesn't matter, if someone grabs it now and crunches on it for a while. Same with identification numbers (SSNs, etc); if I grab a batch of numbers today, most of them will probably still be good in ten or fifteen years, and some of them will still be good in 30 or 40. That's how far out you need to be thinking when choosing an encryption standard for that data.There are some things where only immediate security matters (transmitting big session keys that get thrown away a few hours or minutes later), but many other things -- and I think general file encryption falls into this category -- where it's hard to predict for how long the encrypted information might be sensitive or valuable.

## Re: (Score:3, Informative)

We have in germany some really brain-fucked law about the requirement of digital signatures (s/mime based) on electronic invoices, but one idea they actually got right: You will get an invoice which is signed by the vendor. If you are required to keep incoming invoices (businesses) every once in a while you need to take the current file and sig

## Re: (Score:2)

## Why Does Encryption Need to "Scramble" Information (Score:2, Interesting)

Code Talkers.

The Navajo language basically served as a one time pad in WWII - why not use programs which generate their own method of communication (their own "language") for use in transmitting inform

## Re:Why Does Encryption Need to "Scramble" Informat (Score:2)

## Re:Why Does Encryption Need to "Scramble" Informat (Score:5, Informative)

You mean, like generating a analogous OTP out of a pseudo-random number generator? Not only has that been done before [wikipedia.org], but you're still left with a key: The seed which produced the pseudo-random sequence.

The Navajo code-talkers worked because the encoding was extremely obscure (security through obscurity at its finest!) and cryptography was still in its infancy. I sincerely doubt that the Navajo codes would stand up to a modern cryptographic analysis.

http://en.wikipedia.org/wiki/Navajo_Code_Talkers [wikipedia.org]

## Re:Why Does Encryption Need to "Scramble" Informat (Score:5, Funny)

Navajo code is pretty easy to crack.

## Re: (Score:2)

Incidentally, I suspect security through obscurity can serve a purpose in encryption. As I understand it, with public key encryption, it may be computationally intensive, but at least it can be automated and the problem is well-defined. If the eavesdropper had to first figure out, from the set of all possible encryption methods, which encryption method you were using, it m

## Re:Why Does Encryption Need to "Scramble" Informat (Score:2)

## Re:Why Does Encryption Need to "Scramble" Informat (Score:2)

Perfectly secure methods (one time pad) are perfectly secure because even if you have the cryptotext and the plaintext, the probably that the cryptotext is the plaintext is the same for all plaintexts if you don't know the key (e.g. if you knew the cryptotext is one of two plaintexts, the probability that it was one or the other is 0.5 regardless of what you know about the algorithm).

The Navajo language is an example of sec [wikipedia.org]

## Re:Why Does Encryption Need to "Scramble" Informat (Score:2)

## Re:Why Does Encryption Need to "Scramble" Informat (Score:5, Interesting)

The Navajo language basically served as a one time pad in WWIINo, they served as code-talkers. A one-time pad is a system whereby every bit of the encryption key is independent of the others (never reused, unlike codewords) and entropy is maximal. Simply translating stuff from one word to another is simple substitution, a simple code.

The reason Navajo Code Talkers were succesful wasn't because the scheme was particularly advanced. In fact, it would have been computationally trivial to break. However the messages relayed were only ever "tactical" in nature; i.e. communications in the field, of use during a fight, but old news in about 10 minutes. Had Navajo code talking been used to relay top-secret messages, it would have been broken fairly quickly. The reason for its success was that is was extremely cheap to implement for the US, and the secrets protected weren't valuable enough to spend huge effort on breaking. Economics, rather than mathematics.

Navajo wasn't used in Europe, because Germany had sent anthropologists to the US to learn native languages, anticipating precisely this scheme.

## An NSA spokesperson disagrees (Score:4, Funny)

## this too (Score:5, Funny)

## Re:this too (Score:4, Funny)

Why yes, I am a big hit at parties.

## Re: (Score:2)

Knowing this, too, will not help you pick up chicks in a bar....But I'm fluent in javascript as well as klingon! [youtube.com]

## The real sticky point... (Score:4, Interesting)

This artificial limitation is going to become more and more glaringly obvious as time goes on.

## Re:The real sticky point... (Score:4, Interesting)

A few weeks ago I was reading Steven Levy's

Crypto(not a bad book, although a little out-of-date now, but it brings back the dot-com nostalgia), in which he spends a lot of time describing the NSA's objections to strong civilian crypto in the U.S. in the 80s and early 90s. They went from absolutely opposing civilian crypto (particularly public-key stuff) tooth and nail, to suddenly just throwing in the towel. While I'm sure that much of that was just political pragmatism -- with the Cold War over, they were having a harder and harder time maintaining their objections in the face of 'progress' (in the form of a lot of pressure on Congress from business and the tech sector) -- but I can't help but wondering if they didn't figure something out that made them withdraw their objections to bigger key sizes.Particularly since it's now known that some people on the government side knew about public-key crypto before it became public (the early-70s GCHQ paper, and I find it hard to believe that at its peak during the Cold War, someone at the NSA didn't find the same thing), they've had a long time to work on the problem -- though it's possible that they just totally squandered whatever lead they had, and are now at the same point that the unclassified world is, that just seems unlikely to me.

## Re:NSA computing power vs. EPFL+UofB+NTT? (Score:4, Interesting)

I'm not even sure that it's really raw 'computing power' that you'd want to try and assess, anyway; I was thinking about something like a novel way of factoring general numbers very quickly, something that could be implemented in specialized hardware. That doesn't seem too outside the NSA's traditional forte -- they have some good mathematicians and probably have relationships with hardware companies that would let them source a lot of (odd) stuff without anyone noticing.

I do think it's interesting to note that of the algorithms listed as part of the NSA's "Suite B" [wikipedia.org] Good-Housekeeping-seal-of-approval list, all the PK systems are based on elliptic curves, and not prime factorization, for the trapdoor function.

## I don't expect much of an AC, but really. (Score:4, Informative)

Um, AES is elliptic curve? News to me...For christsakes, at least read the list; I linked to it. And I did say only the public key algorithms, so AES isn't even relevant.

NSA Suite B:

What I said:

"...all the PK [public key] systems are based on elliptic curves, and not prime factorization, for the trapdoor function."

Of the algorithms in Suite B, AES and SHA aren't public-key algorithms; they're a symmetric block cipher and a hash function. The three relevant PK algorithms are ECMQV, ECDH, and ECDSA, and all of those are specifically noted as being "elliptic curve" variants, rather than the more common RSA-style prime-factorization-based algorithms.

PK algorithms which use elliptic curves use an entirely different set of mathematical functions as the basis for their 'trapdoor' or 'puzzle' (the function that's easy to compute but difficult to run backwards) from RSA. They're based on a variation of the discrete logarithm problem [wikipedia.org]. (From what I understand, the purest form of the discrete logarithm problem isn't reversible at all -- you can run it in one direction, but from the output you can't figure out all of the input parameters were with certainty -- so specific variations of the general problem are used, of which elliptic curves are one.)

Given the popularity of RSA, I think its absence from the list is notable at the very least, and it's furthermore interesting that the NSA seems to really like elliptic curve systems as a basis for PK crypto. At least according to Wikipedia, nobody has ever published a proof of the mathematical hardness of elliptic curve systems...maybe they're even better than is currently realized. (Although, the real tinfoil hat response is, 'maybe they're really flawed somehow, and

that'swhy the NSA wants you to use them!' However, I think this is doubtful for any number of reasons.)## -1 author stupidity (Score:5, Informative)

That they pulled off a SNFS on a 1024 bit number is cool, but not the same amount of work for a GNFS against an 1024-bit RSA key.

Tom

## on the wall, eh? (Score:4, Insightful)

## One down, (Score:4, Funny)

## RSA uses primes. (Score:2)

## "the writing is on the wall" for 1024-bit (Score:3, Interesting)

I'm having a hard time making the coorelation.

## Re:"the writing is on the wall" for 1024-bit (Score:5, Informative)

RSA 101.

## That's nothing! (Score:3, Funny)

two large primes, not just some numbers with lots of small factors? Bummer!## ummm.... (Score:2, Offtopic)

Obviously, that's a long and involved argument. But in this case - factoring a very large prime number - just by using methods we *knew* would work - but had never dedicated the resources to - what kind of real progress is that?

## Re: (Score:2)

Lenstra was even quoted in the article to counter your argument of no progress/learning -

"We have more powerful computers, we have come up with better ways to map the algorithm onto the architecture, and we take better advantage of cache behavior."This is an incremental improvement that gives us benefits outside of just factorization, so I would say that the greater good may have## Re: (Score:2)

---

won't shut up! [douginadress.com]

## Some details of the computation size. (Score:3, Informative)

## Long distance before 1024 bits (Score:2)

The RSA Factoring Challenge [wikipedia.org] has been suspended, i.e. they are no longer giving out prize money, but the numbers still stand as a good reference for where we are in comparison to 1024. There's a lot of mileage between here and there.

## Quadruple AES? (Score:3, Interesting)

Now for my actual question. There isn't a symmetric crypto algorithm that I know of that can use 1024 bit keys (except for stream ciphers, maybe RC4?); the best block cipher is AES (Rijndael) which supports 256 bit keys. If one would "invent" QAES, i e quadruple QAES, for a total key length of 1024 bits, what would the "effective" key length be?

## Re:Quadruple AES? (Score:4, Informative)

The standard 3DES process is 3DES-EDE which uses 2 keys, thus giving you 112 bits.

ENCRYPT data with key1DECRYPT output with key2ENCRYPT output with key1Since DES is symmetric, any paired combination of encrypt and decrypt will give you the same result. You can do E(D(data)) to get your result, or you can use D(E(data)) for the same thing. If you used the same key for key1 and key2, this would be the same as doing regular DES, and would just take 3x as long.

If you used three different keys for your 3DES instead, you would have the 168-bit key length. Thus, you can apply the same concept to 4AES, and depending on which way you do it you will end up with 256-, 512-, 768- or 1024-bit key strength.

## Quadruple AES would be effectively 512 bits (Score:5, Informative)

The reason 3DES provides an effective key length of 112 bits, not 168, isn't because only two keys are used instead of three. We only bother using two keys because the effective length of three-key 3DES is still only 112 bits, so there's little reason to bother storing and managing a third.

The reason the effective length is only 112 bits is something called the "Meet in The Middle" attack. Suppose three keys were used and that the attacker has plaintext and ciphertext to mount a known-plaintext attack. An attacker can apply the first encryption step to the plaintext message using all possible 56-bit keys and then store the results in a big dictionary. Then, the attacker picks a 112-bit key and performs the first two decryption steps on the ciphertext. If the result is in the dictionary, then the attacker has probably found all three keys. If not, he picks another 112-bit key and tries again. So the attacker's work is (a) the effort required to create the dictionary plus (b) the effort required to brute force search a 112-bit keyspace. Since (b) completely dominates (a) we can ignore (a) and use (b) as our estimate of the attack complexity.

In the case of any quadruple encryption, then, the Meet in the Middle attack would require building a dictionary of all possible encryptions using the first two keys, then brute forcing the space of the last two keys. So, the effective strength is equivalent to the size of two of the four keys. Quintuple encryption is equivalent to three keys. Double encryption is equivalent to one key, which is why 2DES was never used.

What does all of this have to do with 1024-bit RSA keys? Not a thing. 1024-bit RSA keys consist of numbers that are the product of two 512-bit prime numbers. That means they're pretty sparse among the set of all possible 1024-bit numbers, and it means they have a particular mathematical structure that can be exploited.

Symmetric ciphers, like AES, are different. Unless there's something wrong with them, their keyspaces are flat, meaning that if they use n-bit keys, every possible n-bit value is a legitimate key. They have no particular mathematical properties, and there is no way to identify the right one except by trying them all.

So, assuming that AES doesn't have some weakness that allows attacks faster than brute force to succeed, how long until we need to use keys bigger than 256 bits?

Forever, basically. Barring weaknesses in the algorithm, 256-bit symmetric keys are safe until, as Bruce Schneier put it "computers are built from something other than matter and occupy something other than space."

In Applied Cryptography he outlines an interesting little computation to demonstrate why this is. Suppose you had a computer that contained a 256-bit register that was maximally efficient, meaning that toggling a bit required exactly one quantum of energy. Since smaller units of energy don't exist, you can't do better than that[*]. With that assumption, you can calculate how much energy it would take to cycle your 256-bit counter through all possible states. Schneier calculates that if you could capture all of the energy from a typical supernova and run your counter on that, you could count from 0 all the way up through about 2^219. So you'd need about 130 billion supernovas to run your counter through all of its 2^256 possible states.

That completely ignores the energy you'd need to perform a trial encryption with each of those values, and it also completely ignores how long it would take to perform all of these operations.

Quantum computers that can somehow handle the complex structures of symmetric ciphers, or some other radical change in computing technology would be required to make 256-bit keys accessible to brute force. A flaw in AES is far more likely, IMO.

[*] Just because someone will call me on it, I should point out that reversible computing means that in theory you might be able to do better than the theorized maximally-efficient computer. In practice, that probably isn't going to make your energy budget small enough to be reachable, and it certainly isn't going to help with the time factor.

## in related news (Score:3, Funny)

there is absolutely no such thing as 100% security

and there never will be

for most of us, 99.9999999999999999999999% security will do

for the rest, sweaty heart palpitations and paranoid schizophrenia will do

## What about dynamic encryption algortithms? (Score:5, Interesting)

Decryption itself only makes sense once we know what method was used, ie RSA, DES, Blowfish etc. However what if that algorithm itself was dynamic and formed part of the encryption? Sort of like a more generalised version of onion encryption, ie encrpyting the same content a number of times using different algorithms. So that the algorithms used and the sequence in which they are used form a sort of "meta-key"

## Re: (Score:3, Insightful)

## How long is long-enough? (Score:3, Interesting)

From TFA:

Reading Lestra's comments, I get the feeling that he has a fairly high degree of confidence that they will succeed in making the leap to a mathematical generalization within a modest time frame.

Can any security researchers tell me what GPG key length I should be using in the real world to give me a good trade-off between computational simplicity and future security please? I'm only using crypto for email and secure file storage.

-P## Before someone starts panicking (Score:3, Informative)

The lead researcher believes "the writing is on the wall" for 1024-bit encryption.It should be noted (before people start panicking) that the 1024-bit key size refers to

asymmetricor public-key cryptography (software like PGP, GPG, algorithms like DH/DSS, RSA), not tosymmetriccryptography (software like TrueCrypt, algorithms like AES, Blowfish, Twofish, etc.).A 256-bit symmetric key is completely infeasible to brute force and even if quantum computers become available, the complexity of brute force attack will still be 2^128 (which is again infeasible to brute force).

## You may also be interested to know (Score:3, Funny)

## Re: (Score:2)

## Re: (Score:3, Insightful)

By and large, "we" don't even use *mild* crypto, even in places where we really should be using *hard* crypto.

Do we actually *want* privacy? Seems not.

## Better than a slide rule (Score:5, Insightful)

It's simply insane to use general purpose computer clusters to factor prime numbers when specialized devices built for factoring prime numbers can do the job thousands of times faster per node. These stunts are meaningless. All money funds for those waste of times should be put into developing better purpose built devices and more clever algorithms.

here's an example pdf [arg4] of one such device. It's a tin can with single chip that has LED's integrated onto a shift register and a light detector at one end. costs about the same as one super computer node and is faster than a large cluster. Note that it's designed by the S in RSA so this is not baloney. it's not perfect and it needs technology refinement to scale to numbers larger than about 512 bits. That's where money wasted on this stunt should have been spent.

What's even stupider is that the calculations themselves serve no purpose. Anyone with an napkin and a pencil can tell you whether or not the calculation is feasible on a given size computer cluster. The expected time to crack in a brute force application of a seive is entirely predictable. So what does cracking one prove?

People who do this are more than harmless idiots. They waste money.

## fixed link (Score:3, Informative)

## another good link (Score:3, Informative)

## Exactly...it proves nothing.... (Score:3, Insightful)

With climate change looming, pointless waste of electricity like this should be discouraged.

PS: It's well known that RSA will fall. Number factoring is one of the half-dozen-or-so tasks a quantum computer can actually do. It's just a matter of time before a working quantum computer renders the whole public-key system unsafe.

## The god question and quantum computing (Score:3, Interesting)

But for quantum computers the question has a resonance. Can a quantum computer create prime numbers so large that another quantum computer could not factor the composite?

I realize that there's always quantum crypto. But for most folks we need to be able to use RSA not some new scheme for the privileged.

## Re: (Score:3, Insightful)

## Re: (Score:3, Interesting)

What's even stupider is that the calculations themselves serve no purpose. Anyone with an napkin and a pencil can tell you whether or not the calculation is feasible on a given size computer cluster. The expected time to crack in a brute force application of a seive is entirely predictable. So what does cracking one prove?

People who do this are more than harmless idiots. They waste money.

I thought it was for the entertainment value. It reminds me of the slashdot stories of DES getting broken "quickly."

http: [wikipedia.org]