You might call
numerosity humans’ sixth sense. This number sense gives us the ability to visually discern quantity, such as how many peaches are resting in a bowl or how many marbles are scattered across the floor. Like seeing or hearing, it’s an evolutionary part of human and animal brains. Now researchers have shown that, like our other basic senses, numerosity depends on the organization and close proximity of very particular neurons.
Scientists have long known that the brain’s primary senses, such as sight and smell, lie in clusters of neurons in the sensory cortex. And these have been mapped to identify how and where the neurons for these sensory organs are organized in the brain. Such topographic maps have also been established for
numerosity in macaques. Scientists figured that human numerosity, too, relied on specific neuron organization. They just hadn’t been able to find it, until now.
Benjamin Harvey of Utrecht University and colleagues used functional magnetic resonance imaging (fMRI) to record the brain activity of eight adult participants as they looked at a series of dots that ranged in number and size. The researchers then turned this brain activity data into topographic neuron maps, and published their findings today in Science.
The researchers found that the posterior parietal cortex region of the brain, which plays a role in visual and spatial perception, was active in all eight adults during the study. They also found that specific neurons within that region of participants’ brains were encoded to respond to smaller quantities of dots and other neurons were encoded for the larger quantities. It turns out the area in the brain’s cortex that perceives numerosity is highly organized, similar to the areas of the brain that process our sense of sight or touch.
While number sense is not the same as recognizing the actual Arabic symbols that represent quantity, such as 2 and 10, the topographic map may help scientists learn more about how the brain processes numbers. “We believe this will lead to a much more complete understanding of humans’ unique numerical and mathematical skills,” says Harvey.
SAN FRANCISCO: On a recent afternoon, Homer Gaines hiked with girlfriend Tami Stillwell to the gusty peak of Angel Island in the San Francisco Bay, bent down on one knee and slipped a topaz and whitesapphire ring on her finger, capturing the entire marriage proposal on a computerized device that he was wearing like a pair of glasses. Gaines, a 41-year-old Web developer from Fort Myers, Fla, is one of 10,000 “explorers” testing Glass, the much talked about hands-free wearable computing device from Google that lets users take photos and videos, make phone calls, send and receive text messages, search the Internet and get turn-by-turn directions. “I would not have been able to pull off that level of spontaneity with any other device and instantly share it with the world,” Gaines said. “Glass gave me the ability to share with everyone that special moment from my point of view – the surprise on her face, the way she jumped around, the ring on her finger and the tears of joy in her eyes.” Glass won’t be widely available for purchase until early next year, but it’s one of the most anticipated new technologies in years.
The question many are asking: Can Google make digital goggles the world’s next must-have gadget? As Google sees it, Glass is a revolutionary new way to quickly and effortlessly connect people with information. Critics view Glass as an invasive new technology that – if it takes off – could rob people of what few shreds of privacy they have left. Lawmakers are alarmed by the privacy implications and have begun asking pointed questions of Google. And some commercial establishments – most notably casinos and bars – have already banned Glass. Google is downplaying the privacy and security risks, assuring the public that it will not permit facial recognition apps (or porn apps, for that matter). Google says it’s obvious when someone is taking pictures or recording a video on Glass. But some developers have already built a way to get around Google: an alternative operating system that runs on Glass but is not controlled by Google.
One developer is making a facial recognition app that will help users remember the hundreds of people they have met and should recognize but don’t. That in-your-face quality of Glass could wake up more people to their ever shrinking privacy in the rapidly advancing digital age, University of Washington law professor Ryan Calo said. Not only will people be more keenly aware that they have no reasonable expectation of privacy in public, Glass and devices like it could make it easier for government authorities to gain access to everything they see and record without a warrant, he said. And, with a warrant, the government might even be able to remotely turn on Glass’ video recording capability without the user’s knowledge, the way it has done with OnStar systems in cars, Calo said.
To counter that kind of growing apprehension, Google is trying to make the new technology seem as normal as possible. Google co-founder Sergey Brin constantly has a pair perched on his nose. He has worn Glass to the Oscars, to the TED conference, in the Hollywood film “The Internship,” and last year he stole the show at Google’s annual developers conference by wearing Glass. His cohort, Google Chief Executive and co-founder Larry Page, recently sported a pair as a groomsman in a wedding ceremony in Croatia. And he talked up Glass as the future of technology during Google’s second-quarter earnings conference call with analysts. Still, even inside the high-tech industry, some aren’t too keen on Glass. Los Angeles technology entrepreneur and investor Jason Calacanis has asked friends to remove Glass in his presence, banned Glass from poker games and coined a new term to describe what he feels like doing when he spots Glass wearers: “Glass-kicking.” And Glass hasn’t been able to ditch what could be its true Achilles’ heel: its dorky image. Labeled “Segway for your face,” it has become the butt of jokes on late-night television and on the Internet. Not only have Glass wearers been subjected to public ridicule for looking “glassed out,” they are referred to as a cross between Glass and a curse word. Google is the first to admit that Glass is not quite ready for prime time, with widely reported glitches. The battery drains quickly (but also charges quickly).
The capabilities are still very limited, with only a smattering of apps such as Twitter, Facebook and Tumblr. And some complain that it’s not easy to hear notifications or phone calls with the bone conduction speaker. Perhaps the most glaring omission: A way for the 64 percent of the US population who wears glasses to use Glass. Google has made a prototype of prescription frames designed to be compatible with Glass and said the company will release specifications for frames manufacturers. “We still have bumps in the road and obstacles,” Glass product director Steve Lee said. “Right now, you need to be an early adopter who is excited about the technology.” Glass is the first major product from Google X, the company’s super-secretive research laboratory for “moonshots,” big scientific bets such as self-driving cars.
The lab is located in two nondescript brick buildings about a half-mile from Google’s Mountain View, Calif., campus. A row of electric cars – including Teslas – is parked and charged out front. In a conference room inside Google X, Lee showed off an early prototype of Glass: safety glasses with a Nexus One smartphone attached to the right temple and the phone’s battery attached to the left. The device clearly didn’t win any style points and was not terribly comfortable (“guaranteed to give you a headache in 10 minutes,” Lee said). Over the last three years, Google has dramatically refined Glass. The latest incarnation weighs about the same as a pair of normal glasses and is more attractive and less obtrusive.
To keep careful watch over every detail, Google turned to Foxconn to manufacture Glass near its campus but won’t disclose the exact location. And Google chose a marketing strategy as novel as Glass. This summer, it opened three upscale pop-up stores that it calls “basecamps” – in a San Francisco office tower on the waterfront, in a penthouse in the Chelsea neighborhood of New York and in Google’s Venice Beach offices – to give thousands the white-glove treatment as they are outfitted with Glass. This shopping experience is even more exclusive than most. Explorers not only stretched their pocketbooks to be the first to test Google’s $1,500 device, they competed in an online contest for the privilege. “This is a big, new leap in tech, and I think it’s really important to get feedback on product and how people use it. Why would you do that in a vacuum, when you can do it with people across the United States?” said Ed Sanders, Google’s head of marketing for Glass. “When you are inventing something new and something that by its very nature is very intimate and personal, a device that someone wears on his or her face, then you have to listen to people.” In the San Francisco basecamp, an airy room with exposed ductwork, concrete floors and sweeping views of the bay, explorers first pick from five colors – demure hues of white, gray and black or bright pops of orange and blue – and then take a seat at wood tables. On each table sits a black shopping bag, a Chromebook Pixel laptop and a mirror.
One by one, guides deliver white boxes with the silver Glass logo. Inside the box under a single sheet of white filmy paper is Glass. The guides custom fit the lens-less glasses, bending the titanium and adjusting the nose guards. The explorers drink chilled champagne as they slide their fingers back and forth along the right side of the device and stare into a screen the size of a postage stamp above their right eye. Soon they are ordering Glass around with ease, dictating messages to family and friends and making plans to take Glass sightseeing, even hang-gliding. Erica Pang, 24, who works for a Palo Alto start-up, and her brother Aaron Pang, 20, a junior at Washington University recovering from spinal lymphoma, entered the #ifihadglass contest. She was picked. Their idea: to broadcast live virtual field trips to museums and zoos for children who are immobilized by illness or stuck in hospital beds.
They got the idea when Aaron spent three months in the hospital. “We want to let kids see the world without any of their limitations getting in the way,” Erica Pang said. Michael Kendle, an app developer from Springfield, Miss, in flip-flops and jeans and lounging on a gray couch at the Google basecamp, is a selfprofessed gadget junkie and says he always has to be the first to get the new, new thing. Now he has Glass, and he says he plans never to take it off except to shower and sleep. “I love being connected to everything, to have all this information at my fingertips,” Kendle said. —MCT
When Alfonso Cuarón’s astronauts-adrift epic Gravity arrives in theaters Oct. 4, it will be grounded in science, thanks to Kevin Grazier. The planetary physicist is well-known to fans of Syfy’s Defiance and Battlestar Galactica, and TNT’s Falling Skies, as the science adviser who helps give authenticity to sci-fi adventure.
Grazier spent 15 years at NASA’s Jet Propulsion Laboratory as investigation scientist and science planning engineer for the Cassini-Huygens Mission to Saturn and Titan. He’s currently penning Hollyweird Science with co-author Stephen Cass. The book, due out in 2014, will cover studios that got the science in their sci-fi right. Still an active researcher focused on computer simulations of solar system dynamics, Grazier told Discover Associate Editor Gemma Tarlach that Hollywood is less a shark tank than a sandbox full of smart kids.
Discover: How did you start working as a science adviser in Hollywood?
Kevin Grazier: I’m a blend of right brain and left brain, and I got into this with my right brain. When I was at grad school at UCLA, Paramount would still take unsolicited scripts. My buddy and I sent one in for Star Trek: Voyager, and seven months later I got a call from the executive producer’s assistant to come pitch stories. Through that opportunity, I met with people like Bryan Fuller (Dead Like Me) and Michael Taylor (Battlestar Galactica).
D: Do you find yourself often at odds with writers who want to put scientifically implausible things in the script?
KG: The writers come in having done their research, but sometimes it’s hard for non-scientists to grasp all the implications. Science can get in the way of a good story if you let it, but good writers don’t let it. There’s a saying in Hollywood about playing in your sandbox. When you give constraints, some writers look at it as stifling, as suffocating: You’re building walls around their sandbox and trapping them. Other writers write on the walls you’ve erected around them and find ways to work with the story. Science can be enabling. Sometimes science is more out there and cooler than what the writers might have thought. It can be liberating for them when you show them, “Actually, it would do this.”
D: Oh, come on, no screaming matches? No stereotypical nasty Hollywood egos to contend with?
KG: The science adviser is not a copy editor. There’s no guarantee they’ll listen, but good writers are very keen to listen to what you have to say.
D: Which do you prefer, working on TV series or movies?
KG: I prefer TV to movies because movies are a one-off. Your work is often done early on in production, on your own. On a TV series, you’re part of a team, you get to know people and make friends.
D: Was Alfonso Cuarón one of the writers who did his research?
NB: Absolutely. He’d done a lot of work already and wanted to get everything right. He wanted to know details down to which direction the toggle switches would flip.
The Samsung Galaxy S4 is the Korean brand’s most powerful device yet. It boasts specs that would make most smartphones question the reason for their existence and is successor to Samsung’s best selling smartphone to date, the Galaxy S3.
Granted, the much awaited and hotly anticipated S4 has big shoes to fill, but after spending a week with it, I feel this device has the capability needed, to be able to step up to the challenge. However, this particular phone comes with too many unnecessary whistles and bells of sorts.
It feels as if the gimmickry is in place to compensate for the lack of something else. In reality, this phone can only show its efficiency and capability once you get rid of all the excess baggage that you will never need.
The S4 is a robust device with a beautiful display, but is it worth the price tag?
The S4 is similar to the S3, among other Galaxy products in design, but better. It has the same round-edge rectangular look with a chrome trim wrapped around the edges, housing the power button to the right, 35mm audio jack on top, the volume rocker to the left and micro-USB port at the bottom.
It’s remarkable, that Samsung has managed to make the phone lighter, faster and narrower while increasing the screen size from 4.8 inches to a full 5 inches on the S4. It also deserves to be said that although it’s lighter and narrower, it feels like more of a premium phone, build wise, than its predecessor.
Unlike its nemesis, the S4 has a removable battery, under the detachable plastic cover, a feature applauded by a surprising amount of people. I personally don’t think much of it; I’d rather have a fixed metal finished back cover that represents the feel of the price tag it carries like say, the HTC One.
Micro-SD storage and Micro-SIM ports are placed adjacent to each other, above the battery and beneath the LED Flash. Other manufacturers use a nifty tray that houses the SIM or external storage and slides into the profile of the phone.
The Super AMOLED capacitive touchscreen boasts a 1080 x 1920 resolution, coughing up a mind blowing 441 pixels per inch (ppi). There is absolutely no problem under direct sunlight, or any other surrounding.
The display on the S4 has to be without a doubt the best screen Samsung has ever put into production. Adjusting the brightness is as easy as pulling the notification menu down and making your choice; alternatively you could select the auto-mode that adapts according to your surroundings.
The incredible pixel density can really be put to the test while reading long documents. Generally when you view an entire page worth of text on a phone, the words tend to blur, but not here: the text is sharp and very readable even from a distance without having to zoom in.
Samsung has a whole array of features lined up to enhance user experience with the S4.
‘Air view’ allows you to preview before selecting while hovering over different areas of the screen, you can essentially preview information or numbers before you choose to click on them. We saw this feature in the Galaxy Note 2 as well, only difference is that previously it was dependant on the S pen.
‘Air gesture’ enables users to scroll web pages in screen size jumps through air jump. The user can also move between pictures, pages or music tracks via air browse. You could wave your hand over the screen to accept incoming calls. All of these tricks sound great but are not as functional as I would have hoped.
‘Smart stay’ ensures that the screen stays on as long as you are looking at it. Smart rotation adjusts the display direction to the angle of your sight and Smart pause makes sure that the video will pause when it detects that you are facing away from the screen. Again, this sounds great but if more than one person is watching the video or you happen to be a multi-tasking while doing something else this feature falls flat since you do not want your eye movement to interrupt the process.
‘Smart scroll’ is meant to detect your eye movement with the front camera so you can scroll pages by tilting your head or the device. It may not work when the front camera fails to detect your eyes, when the source of light is behind you or in dark situations. It will also not work when the device is shaking, or you are interacting with the device in other ways such as tapping the screen or using palm motions.
S beam can be used to transfer files through Wi-Fi as oppose to Bluetooth on Android beam resulting in quicker transfer speeds. The S4 can also be set up as a Wi-Fi hotspot for others to share internet usage. In Pakistan this would be very expensive since most of us do not have unlimited mobile data plans.
WatchON and S Health
Samsung has developed applications that will integrate your lifestyle with your S4. Through an infra-red blaster located on the top of your phone WatchON can act as your universal remote with separate devices setup in different rooms of the house.
Those of you like me who spend entirely too much time looking for the TV remote will be very satisfied with this setup. For once you can actually call the remote to see where it rings and additionally, you can customize it with as many setups as one may have in their house. The bedroom, TV lounge etcetera can all be saved as presets and with the touch of your fingertip you’re in control.
WatchON takes it up a notch by allowing you to control not just TV but the set-top box, dvd player, AV receiver, Streaming media player and blu-ray player.
Furthermore, you can have programs chosen or recommended for you. This feature again does not work so well in Pakistan because of the lack of digitization with respect to cable operators; most of us do not have access to on demand television.
HELSINKI: Microsoft Corp on Tuesday said it will buy Nokia's mobile phone business for 5.44 billion euros ($7.2 billion).
In a statement, Microsoft CEO Steve Ballmer says the deal will bring Nokia's capability and talent in hardware design, engineering, manufacturing, sales, marketing and distribution to Microsoft.
The companies say that when the deal closes in early 2014, about 32,000 Nokia employees are expected to transfer to Microsoft, including approximately 4,700 people in Finland.
The operations affected by the transfer generated approximately 14.9 billion euros in 2012, or almost 50 per cent of Nokia's net sales, the company said.
Of the total purchase price of 5.44 billion euros, 3.79 billion relates to the purchase of Nokia's Devices & Services business, and 1.65 billion relates to the mutual patent agreement and future option.
"It's a bold step into the future - a win-win for employees, shareholders and consumers of both companies,” Microsoft's outgoing CEO, Steve Ballmer, said in a statement.
“Bringing these great teams together will accelerate Microsoft's share and profits in phones, and strengthen the overall opportunities for both Microsoft and our partners across our entire family of devices and services.”
The deal is subject to approval by Nokia's shareholders and regulatory approvals.
Nokia plans to hold a news conference in Espoo, Finland, on Tuesday morning.
Nokia partnered in 2011 with Microsoft and uses Microsoft's Windows software to run its mobile phones.
Stepehen Elop, a Canadian hired by Nokia in 2010 from Microsoft, has been one of the favourites to take over as Microsoft chief when Ballmer steps down.
Finland's Nokia, once the undisputed leader in mobile phones, has been struggling to respond to the challenge from smartphone makers such as Apple and Samsung.
Nokia said in a statement it expected that Elop, along with senior executives Jo Harlow, Juha Putkiranta, Timo Toikkanen, and Chris Weber, would transfer to Microsoft when the deal was concluded. It did not say what roles they would take at Microsoft.
Nokia board chairman Risto Siilasmaa would take over CEO duties while the Finnish firm looked for a new CEO, it said.
Analysts say Elop's bold bet in 2011 to adopt Microsoft's untested Windows Phone software has yet to pay off.
Last month, Nokia finalised the purchase of German engineering giant Siemens' 50 percent stake in Nokia Siemens Networks for 1.7 billion euros.
NSN, which is specialised in high-speed mobile broadband, was set up as a joint venture between the two companies in 2007, a partnership that expired in April. The unit has posted stronger earnings than Nokia's mobile phone business.
NSN posted a net profit of 8.0 million euros in the second quarter of this year, compared to Nokia's net loss of 227 million euros in the same period.
A trip to the beach may require an extra layer of sunscreen—and not just for you landlubbers.
Marine biologists in Canada and Mexico have shown that increased exposure to the sun darkens certain whale species’ skin. As in humans, UV rays trigger an increase of the pigment in whales’ skin, creating a tan of sorts. In some cases they even burn and blister. And like people, whales accumulate such sun-caused damage to the DNA in their skin as they age.
For three years, researchers took skin samples from blue, sperm and fin whales at various points during their sun-seeking annual migrations, and they published their results in Scientific Reports today.
Blue whales are pretty pale, so during their sunny spring migration, the researchers found increases in pigment and the same kinds of DNA damage in sunburned human skin. Fin whales were the darkest; their skin was already heavily pigmented, so the researchers saw less color change and fewer blisters. Sperm whales tend to be in the middle in terms of pigment, but they also spend more time at the surface in the sun, so their skin reacted more strongly in response to the UV rays. Newcastle University researcher Amy Bowman explains in Science Daily:
“We saw for the first time evidence of genotoxic pathways being activated in the cells of the whales — this is similar to the damage response caused by free radicals in human skin which is our protective mechanism against sun damage.”
So despite living underwater, perhaps these thick-skinned mammals should consider slathering on some sunblock, too.
In a previous post I described mathematicians’ ongoing search for key properties of prime numbers. That effort may seem to belong entirely within the realm of pure mathematics; but surprisingly, the importance of primes goes far beyond the abstruse obsessions of ivory-tower mathematicians. In fact, the use of prime numbers underlies some of the most dramatic events in the news these past weeks: the story behind Edward Snowden’s revelations that the National Security Agency (NSA) is snooping on the communications of both American citizens and European diplomats.
While the Europeans have protested about their internal communications being intercepted by the NSA—ironically—the tools that one can use for protection from spying by anyone are readily accessible online, in the professional literature, and in publicly-available manuals and textbooks. These methods all rely on clever uses of prime numbers.
The essentials of these techniques are far from new. The foundations of a program to create codes so powerful that they could not be broken even if an eavesdropper were to use the entire available worldwide computing power were laid more than 35 years ago. The year 1976 saw the development of theDiffie-Hellman key exchange method (named after Whitfield Diffie and Martin Hellman; the names Ralph Merkle, James Ellis, Clifford Cocks, and Malcolm Williamson are often also associated with it); and the following, 1977, witnessed the appearance of the RSA algorithm. Both methods have advanced over the past three and a half decades, but information about their extensions is also readily available to anyone.
How do these techniques work? I will explain both methods here—necessarily in a simplified way. (Those interested in learning more can read some of the articles in the links that appear throughout this post.)
Alice sends Bob a secret message
The Diffie-Hellman key exchange idea has been described in a clear and concise way using an analogy by Terence Tao, whose work on prime numbers I mentioned in my previous post. The idea is as follows. Alice wants to send Bob a secret message (cryptographers prefer to use “from Alice to Bob” instead of the mundane “from A to B”) and she wants to prevent Eve (the “eavesdropper”) from reading it. So Alice places the message in a box, puts a good lock on it, keeps the key, and sends the package to Bob. (If Alice were to separately send Bob the key, there would be a chance that Eve could intercept both the package and the key.)
Bob has no key to Alice’s lock. So what he does instead is to put his ownlock on the box. And he now sends the package back to Alice, locked twice: using both her lock and his. Alice gets the package, removes her own lock using her key, and then sends the box, still safe because it bears Bob’s lock, back to Bob. Now Bob uses his key, opens the box, and gets the message! Each person here used his or her own lock and key—and yet a message was passed perfectly safely from Alice to Bob.
This idea is implemented digitally in the Diffie-Hellman key exchange. The message to be sent from Alice to Bob is a secret number, call it n. Alice’s “key” is an exponent, a, which she chooses, and then uses it to raise n to. So the “locked box with the message” that Alice sends Bob is na. Bob has his own “key,” which is a number of his own choosing, b, that he uses as an exponent. He doesn’t know n or a, but he has na, which he got from Alice, so he raises this number to the power b. He thus sends Alice the “box with the two locks”: nab. Alice’s using her own key to open her own lock means her taking the ath root of nab, which, from the simple math of exponents, we know gives her nb, which she now sends back to Bob. Using his “key,” his exponent b, Bob takes the bth root of nb, and he thus obtains the secret number n that Alice wanted to convey to him.
It is possible to send a secret number from Alice to Bob as I just described, and if the numbers are large enough, one would have a reasonable probability that the number might not be deduced by Eve. In actuality, however, modern implementations of the Diffie-Hellman key exchange use more sophisticated elements to make it more difficult to break the code. And the secret number is not sent from Alice to Bob, but rather deduced by both of them using the formula nab (which, of course, is also equal to nba).
Alice and Bob choose a prime number, which they assume can be known to Eve, or to anyone in the world. Let’s say that this number is 11. They then do all calculations using the mathematical multiplicative group of integers modulo 11 (like a clock going around to 12 and then starting from 1, this group starts to count again after reaching 11). They also choose a base, and let’s suppose it is the number 5. Alice then chooses her secret number, say 3. Independently, Bob chooses his secret number, 4.
Alice raises the commonly-agreed-on base of 5 to the power of her secret number 3, and does the calculation modulo 11. She gets: 53 = 125, but 125 modulo 11 is 4 (it’s the remainder of dividing 125 by 11, which gives 11 and a remainder of 4—it acts like 16 hours in a clock, but this clock is based on 11 rather than 12). She sends Bob the answer, the number 4. Recall that Bob had chosen a secret number of 4, so he raises the 4 he got from Alice to the 4th power, modulo 11, and this gives him 44 = 256, but 256 modulo 11 is 3 (because 11×23 = 253, leaving the remainder 3), which is his final answer.
Alice gets from Bob the original 5 they had both agreed on, but now raised to the power of his secret number, 4, modulo 11, which is 625 modulo 11, which is 9 (as 11×56 = 616, leaving a remainder of 9). She then raises this number to the power of her secret number of 3, again doing this calculation modulo 11. She gets the same number that Bob got, 3 (because 93 = 729, but modulo 11 it is 3, since 11×66 = 726, which leaves a remainder of 3).
Using this complicated modular arithmetic based on a prime number, but essentially raising a number to hidden powers as in the previous section, Alice and Bob establish a common secret number, in this example, 3. Modular arithmetic using prime numbers helps make the algorithm much more difficult to decipher by an eavesdropper.* In reality, the prime number is large, and so are the other numbers. When Alice and Bob use secret numbers 100 digits long, the common number jointly deduced by Alice and Bob cannot be learned by Eve even if she has access to all the world’s available computing power.
Once Alice and Bob have established a common secret number, they can use it as a key to encrypt messages from one to the other and should have a high probability that their communication will not be deciphered by an outsider.
The year after the Diffie-Hellman algorithm was published, three academics then working at MIT—Ron Rivest, Adi Shamir, and Leonard Adelman—came up with a brilliant idea for encrypting messages. What they tried to do was to avoid the stage in which Alice and Bob must create a common secret number, since this stage slows down the communication between them.
The three MIT scientists developed the notion of a pair of keys: a public key and a private key, which are then jointly used for communicating secret messages. The public key can be published and known to all. Its use saves time. The private key is a secret that Bob keeps, allowing him to decipher coded messages from Alice (or from anyone who knows his public key). Bob publishes his public key, which is a large number. This number is obtained when he multiplies together two very large prime numbers, known only to him (they constitute his private key). When Alice wants to send Bob a secret message, she encrypts it using his known public key. But in order to decrypt the message, one would need to know Bob’s private key, which is the two prime numbers he had used to create his publicly-known key. Supposedly, only Bob can do this.
Encrypting and decrypting messages using the RSA algorithm is a complicated mathematical procedure that relies on modular arithmetic and prime numbers similarly to the way they are used in the description of the Diffie-Hellman system above. But it is more sophisticated so that it can allow deciphering using only the private key. The public key alone is useless for deciphering the RSA code.
The essential element of RSA is the fact that the public key is composed of the product of two very large unknown prime numbers. It so happens thatfactoring a number into its prime components is very difficult when the primes are large. (35 = 7×5, a product of two primes, is easy; but 46,324,637 = 5,881 × 7,877 is harder, and primes used in RSA encryption are much larger still.) It is this fact alone that keeps Eve in the dark. She knows the product of the two prime numbers—but she can’t easily (and hopefully not at all) deduce what the two primes are!
Right after the RSA system was invented, Martin Gardner published in Scientific American an encrypted message and a large RSA number, with 129 digits, that was the product of two primes. He challenged his readers to break the code, offering a $100 prize. It took 17 years for the number to be factored and the message deciphered. This was a relatively short period of time—many had expected that it would take an exceedingly long time, and Rivest, Shamir, and Adelman had jested that it could take several “quadrillion years.” The complex operation was achieved using distributed computing with thousands of computers around the world performing parts of the general calculation—thus demonstrating the power of such an approach.
RSA Security, founded by the academics, has since published several similar numbers, and for a time there was a cash prize offered for their factoring into pairs of primes, which the company subsequently withdrew. By now, some of these challenges have been met by mathematicians using distributed computing. Here is one problem that is still outstanding, an RSA number with 210 digits, that has never yet been factored into two primes:
RSA-210 = 245246644900278211976517663573088018467026787678332759743414451715061600830038587216952208399332071549103626827191679864079776723243005600592035631246561218465817904100131859299619933817012149335034875870551067
Obviously, the larger the number to be factored, the longer the time needed to break it into a pair of primes. Beyond a certain length (in decimal digits), the RSA code becomes impregnable and therefore any message based on it undecipherable (in a reasonably finite length of time) by an eavesdropper. The RSA algorithm is widely used today in Internet security.
NSA’s uses and abuses of encryption
In adopting standards for encryption in the United States, and for exporting encryption products, the NSA has pushed for, and succeeded in implementing, legal limits on the size of the numbers used in RSA coding, so that—with its supercomputers—it would be able to decipher any message based on it. Presumably, the Europeans are not bound by these restrictions, and their cryptanalysts should have been able to easily devise an unbreakable RSA code (by choosing primes that are large enough) for use in routine European diplomatic communications as well as protecting their computers from hacking.
And as history has shown, supercomputers are less effective than wide-ranging worldwide distributed computing for breaking advanced codes—but by its very nature, the NSA could never employ the latter. On the other hand, the most recent revelations seem to indicate that one of the purposes of NSA searches is in fact to identify people or entities that use encryption in their communications. If so, all the more reason for the European governments to use established, Western, advanced codes, so as to set themselves apart from terrorist entities, whose codes would necessarily look different. This would actually help the NSA concentrate on identifying real threats rather than wasting resources on intercepting Brussels messages such as: “Pierre, Italian or Chinese for lunch today? Yours, Hans.”
Thus we find ourselves where we do now, in an arms race of encryption and decryption, a world in which pure mathematics plays the key role in helping invent better and better codes. As the codes become more sophisticated, so do the code-breakers, and the cycle perpetuates itself. What is so amazing is that codes that were considered absolutely unbreakable a few decades ago do become breached as the technology improves—but then again, those designing new encryption methods, on all sides, use ever more complicated math to keep a step ahead of their pursuers.
*There are two good reasons for using modular arithmetic. The first is that it acts as a many-to-one function, in the sense that many numbers, when divided by a prime, will give the same remainder—thus making Eve’s life much more complicated (she can’t uniquely reconstruct Alice and Bob’s secret numbers). Using the clock example, if she should overhear that a meeting is to take place at 1 o’clock, she couldn’t tell if it’s a.m. or p.m., or which day. The second reason is that it puts a cap on the size of numbers involved when using exponentials, since (by definition!) without modular arithmetic these numbers grow “exponentially,” and could make computations intractable.