Sync, page 22
In search of alternatives, they consulted with Robert Newcomb, an electrical engineer at the University of Maryland who had designed his own brand of chaotic circuits. Newcomb had let his imagination run free. He hadn’t felt compelled to make circuits that mimicked Lorenzian waterwheels or lasers or any other physical system; he was just curious about chaos and wanted to explore it electronically. Carroll followed one of Newcomb’s recipes and confirmed that the resulting circuit produced wild fluctuations in voltage and current. Plotted on an oscilloscope, the variables traced out a strange attractor—not the same as Lorenz’s butterfly wings, but similar. The circuit was running at thousands of cycles per second and giving fast, beautiful chaos.
Now the synchronization scheme could be tested. Carroll built a second copy of the circuit, and wired it to the first one according to Pecora’s rules. The theory predicted that the two circuits should both oscillate spasmodically but in perfect lockstep. To test for synchrony, Carroll set the oscilloscope to plot the receiver voltage y versus its transmitter counterpart y’. If the two fluctuating variables were equal, they should line up on a 45-degree diagonal (because when y is graphed horizontally, and y’ is graphed vertically, the horizontal displacement y must equal the vertical displacement y’ if their values are always equal). And since y and y’ are always changing from moment to moment, they should race back and forth along that diagonal line but never depart from it.
Carroll flipped the switch to start the circuits. Within two milliseconds, the voltages leapt onto the diagonal and stayed there. “My hair still stands up when I think about it,” Pecora told me. “I don’t think I’ll ever have a moment like that again. It’s like seeing one of your kids being born.”
Last day of classes, MIT, December 1991. I’d just given the final lecture in my chaos course, and everyone had filed out except for one student. Beaming with pride, he handed me a piece of paper crammed with handwritten formulas and theorems, all enclosed in perfect rectangular boxes. To prepare for the upcoming final exam, he’d distilled the whole course to a single page. Looking at his minuscule, machinelike printing, I knew what I was dealing with. Sure enough, Kevin Cuomo turned out to be one of the best students in the class.
Cuomo was doing his Ph.D. research on synchronized chaos in electrical circuits and their possible uses in communications. At the time, I was vaguely aware of Pecora and Carroll’s 1990 paper, but had not studied it carefully. Cuomo wanted to tell me all about it—the words came tumbling out in a torrent—but then he jumped to his own work, and encouraged me to come see a circuit he’d built—the first electronic implementation of the Lorenz equations—and he also wanted me to check a mathematical proof he’d discovered, a demonstration of a new synchronization scheme that would always work for the Lorenz equations, no matter how the receiver and transmitter were started. He took a breath and continued: Pecora and Carroll had not offered any such proof, and that worried him—the reasoning wasn’t especially difficult, just a standard application of Lyapunov functions, like we’d done in class—so maybe he was missing something?
As it turned out, Cuomo had done everything right. His proof was sound, and his circuit did simulate the Lorenz equations (to this day, Pecora cheerfully admits that he has no idea how Cuomo got it to work). But none of this is what Cuomo is known for. Over the course of the next year, he and his adviser Al Oppenheim would be the first to demonstrate that chaotic encryption was possible: Synchronized chaos really could be used to enhance the privacy of communications.
Their method is based on masking, the same strategy used (unsuccessfully and unforgettably) by the secretive couple in Francis Ford Coppola’s movie The Conversation. Fearing that they are under surveillance, a man and woman walk around a busy town square and whisper to each other, trusting that the loud din of street musicians will hide their conversation. In Cuomo and Oppenheim’s version, the background noise is provided by the hiss of electrical chaos, generated by the variable x from a Lorenz circuit. Before any message is sent to the receiver, x is added on top of it, to mask it. For good coverage, x must be much louder than the message (just as the street music needs to be much louder than the whispered conversation) over its entire range of frequencies. Of course, if the receiver can’t disentangle the message from the mask, nothing has been accomplished. This is where synchronization comes in. Cuomo’s scheme ensures that the receiver, when driven by the hybrid signal (message plus mask), will synchronize to the mask, but not to the message. In effect the receiver regenerates a clean version of the mask. Subtracting it from the hybrid signal reveals the message. The method confers privacy because an eavesdropper has no easy way to perform the same decomposition; he wouldn’t know what to subtract, what part of the combined signal is mask and what part is message.
A year after he took my course, Cuomo returned to give a live demonstration of his encryption scheme to my latest crop of chaos students. First he showed us his transmitter circuit: a small board loaded with resistors, capacitors, operational amplifiers, and analog multiplier chips. The voltages x, y, z at three different points in the circuit were proportional to Lorenz’s variables of the same names. When Cuomo graphed x against y on an oscilloscope, the familiar butterfly wings of the strange attractor appeared as a glowing, ghostly image on the screen. Then, by hooking the transmitter up to a loudspeaker, Cuomo enabled us to hear the chaos. It crackled like static on the radio. Next he grabbed another circuit board, a receiver built to match the transmitter, and connected them with an alligator clip in a strategic place. Using the oscilloscope again, he demonstrated that both circuits were now running in sync, offering the usual 45-degree diagonal test as evidence.
Cuomo brought the house down when he used the circuits to mask a message, which he chose to be a recording of the hit song “Emotions,” by Mariah Carey. (One student, apparently with different taste in music, asked, “Is that the signal or the noise?”) After playing the original version of the song, Cuomo played the masked version. Listening to the hiss, one had absolutely no sense that there was a song buried underneath. Yet when this masked message was sent to the receiver, its output synchronized almost perfectly to the original chaos, and after instant electronic subtraction we heard Mariah Carey again. The song sounded fuzzy but was easily understandable.
When Cuomo and Oppenheim’s paper was published in 1993, their dramatic results came as no surprise to Lou Pecora. He and Tom Carroll had been toiling along the same lines for three years already, but they weren’t allowed to say anything or publish what they’d found.
As early as the fall of 1989, once their chaotic circuits were successfully synchronizing, Pecora and Carroll had begun considering the problem of chaotic encryption. Lacking even a rudimentary background in communications or coding theory, they came up with a clumsy method, one that required sending two signals. One signal was used to establish synchrony between the receiver and the transmitter. The second was a hybrid, a mask with a message added to it at very low power. It’s essentially the same strategy that Cuomo and Oppenheim proposed a few years later, though less elegant in the sense that Cuomo’s method uses only one signal (x plus message) for double duty—it both establishes sync and carries the message. But the general idea is the same.
The Space Warfare group at the Naval Research Laboratory became interested in Pecora and Carroll’s work, because of the potential it offered for new ways of encoding and encrypting satellite communications. They had been funding Carroll for the preceding year, and now wanted a closer look at what the physicists were up to. A senior officer told Pecora to keep quiet about the work until the Space Warfare people had a chance to evaluate it; they were going to send an outside expert to assess the circuit. Pecora was given strict instructions about how to behave. He and Carroll were not allowed to ask the expert anything: not who he worked for, not even his name. What should we call him? Pecora asked. “Call him Bill,” said his superior. In private, Pecora and Carroll referred to him as Dr. X.
Dr. X turned out to be a young man, serious and competent, carrying a computer loaded with software for simulating analog circuits. He seemed unfamiliar with chaos theory, but he clearly understood the communications ideas, and managed to get his own simulations of the circuit running very quickly. Pecora and Carroll were later informed that Dr. X had concluded that their circuit performed as described, though he had doubts about whether it could be made digital and secure.
Other visitors from the Space Warfare group soon followed. Pecora, in his naïveté, bet one of them a beer that he could hide a sine wave in the chaos, and challenged the visitor to extract it. The visitor ran the circuits for a minute, measured the voltage waveforms, then did a computation called a fast Fourier transform to measure the strengths of all the component frequencies being transmitted. The sine wave stood out nakedly as a spike in the spectrum. Pecora realized then that he had a lot to learn about encryption.
The Space Warfare scientists concluded that this new scheme was interesting but hardly something the navy should depend on. Pecora and Carroll were finally given permission to disclose their results, but because they wanted to apply for a patent, their lawyer advised them to extend their silence about what they were doing. So they still didn’t publish anything.
Space Warfare also put them in touch with a contact at the National Security Agency, the ultrasecretive arm of the government concerned with the making and breaking of codes. Pecora visited the agency headquarters and presented his results to an audience of cryptographers who listened attentively, but wouldn’t respond to any of his questions. “It was like talking in a black hole,” said Pecora. “Information goes in and none comes out.” After the meeting, Pecora realized he’d forgotten something and needed to get back in touch with his contact at NSA. Having lost the phone number, he looked in the phone book, and was surprised to find a listing for this most clandestine of organizations. He dialed the number and reached an information desk. The conversation was reminiscent of a Monty Python sketch:
“May I have the phone number for Colonel Y?”
“I cannot confirm or deny that anyone named Colonel Y works here.”
“OK, how about if I give you my number and you tell him to call me back?”
“I cannot confirm or deny that he works here.”
“This is the information desk, isn’t it?”
“Yes. What information would you like?”
The early work on synchronized chaos led to a jubilant sense of optimism about the prospects for chaotic encryption, especially among physicists with no background in cryptography. It was common in the early 1990s to see papers in physics journals with hopeful titles about “secure” communications. But the experts knew better. From the beginning, Al Oppenheim cautioned Cuomo and me about hyping the results. “You must never call this method secure,” he warned. “Secure means secure—unbreakable. We don’t know if it’s secure. It may give some low level of privacy, but that’s all. Masking schemes are usually pretty easy to break.”
For people using cellular phones, even a minimal level of privacy would be welcome. Princess Diana needed it when reporters intercepted her conversations with her lover James Gilbey, later publicized as the embarrassing “Squidgy” tapes. Prince Charles was caught speaking even more intimately to Camilla Parker Bowles in 1989. When Newt Gingrich and his lawyers were discussing the ethics case against him, their cell-phone conversation was taped by Democratic loyalists using a police scanner. Cell-phone scramblers do exist today, but they tend to cost several hundreds of dollars. Chaotic masking might turn out to be a cheaper alternative for defeating casual eavesdroppers.
For military and financial applications, on the other hand, much stronger encryption is required. So far, chaos-based methods have proved disappointingly weak. Kevin Short, a mathematician at the University of New Hampshire, has shown how to break nearly every chaotic code proposed to date. When he unmasked the Lorenzian chaos of Cuomo and Oppenheim, his results set off a mini–arms race among nonlinear scientists, as researchers tried to develop ever more sophisticated schemes. But so far the codebreakers are winning.
One of the most promising developments comes from the 1998 work of Gregory VanWiggeren and Rajarshi Roy, physicists then working at the Georgia Institute of Technology. They gave the first experimental demonstration of chaotic communications using lasers and fiber optics, instead of electrical generators and wires. In their optical system, chaotic waves of light carried hidden messages from one laser to another at speeds of 150 million bits per second, thousands of times faster than the rates achieved electronically. And there’s no theoretical barrier to even higher speeds.
Another advantage of communicating with chaotic lasers is that the chaos is much more complex, making it tougher to crack. The complexity is quantified by a number called the dimension of the strange attractor, which is a natural generalization of the ordinary concept of dimension. But unlike a straight line (which is one-dimensional) or a flat plane (which is two-dimensional), strange attractors typically have dimensions that are fractions. The Lorenz attractor, for example, is made of infinitely many two-dimensional sheets, which implies that it has an infinite surface area but no volume. Arcane as it may sound, it’s more than a surface but less than a solid, and its dimension, accordingly, is greater than 2 but less than 3. For VanWiggeren and Roy’s erbium-doped fiber lasers, the dimension of the strange attractor is unknown but it is almost certainly a fraction and, more important, it is huge. It seems to be at least 50, corresponding to an extremely wild form of chaos. It remains to be seen whether this new form of encoding will be more secure than its predecessors.
Leaving encryption aside, the more lasting legacy of synchronized chaos may be the way it has deepened our understanding of synchrony itself. From now on, sync will no longer be associated with rhythmicity alone, with loops and cycles and repetition. Synchronized chaos brings us face-to-face with a dazzling new kind of order in the universe, or at least one never recognized before: a form of temporal artistry that we once thought uniquely human. It exposes sync as even more pervasive, and even more subtle, than we ever suspected.
• Eight •
Sync in Three Dimensions
MY FIRST ENCOUNTER WITH SYNC OCCURRED BY chance on a dismal day in Cambridge, England, in 1981. I was studying math there on a Marshall Scholarship after graduating from college, and feeling entirely displaced. The English girls never got my jokes, the brussels sprouts were gray, the drizzle was relentless, and the toilet paper was waxy. Even my coursework was drab: old-fashioned topics in classical physics, like the rotational dynamics of spinning tops. It was complicated stuff, and not inspiring.
Hoping to rekindle my academic passion, I walked across the street to Heffer’s Bookstore to browse the books on biomathematics. (As a senior in college, I had written a thesis about the geometry of DNA, and that whole experience—doing original research with a world-class biochemist, using some of the math I was learning and applying it to an unsolved problem about chromosome structure—had been so thrilling that I was convinced I wanted to become a mathematical biologist.) As I scanned the shelves, with my head tilting sideways, one title popped out at me: The Geometry of Biological Time. Now that was a weird coincidence. My senior thesis on DNA had been subtitled “An Essay in Geometric Biology.” I thought I had invented that odd juxtaposition, geometry next to biology. But the book’s author, someone named Arthur T. Winfree, from the biology department at Purdue University, had obviously connected them first.
The blurb on the back flap looked promising: “From cell division to heartbeat, clocklike rhythms pervade the activities of every living organism. The cycles of life are ultimately biochemical in mechanism but many of the principles that dominate their orchestration are essentially mathematical.” I dipped into the table of contents. Right away I could see that this was the work of an unusual scientist. No, not just unusual. Arthur T. Winfree was breaking all the rules. Above all, he was playful. In a chapter about the mathematics of the menstrual cycle, he used data from his own mother. Other chapters had equally quirky elements in them: puns in their titles, personal stories from the author (“Nixon chose that week to invade Cambodia”), and I started to wonder if Winfree was for real. So I slid the book back into its place on the shelf, and left the store.
A few days later I felt myself being tugged back to Heffer’s. Winfree’s book was beckoning, and I had to look at it again. To check his credentials, I turned to the bibliography: 36 papers between 1967 and 1979, with several in the most prestigious journals, such as Science, Nature, and Scientific American. That should have been convincing enough, but for some reason I put the book back again, only to revisit it a few days later. Eventually it occurred to me that this was getting ridiculous—and God forbid that someone else might snatch the store’s only copy. I surrendered and bought it.
Every day of reading the book was a new delight. Winfree’s synthesis was brilliant and utterly original. Chapter by chapter, he built a mathematical framework that exposed an underlying unity in how various biological oscillations work. He applied his ideas to heart rhythms, brain waves, menstrual cycles, circadian rhythms, the cell division cycle, even waves in the gut. But his ideas went far beyond that. They made startling predictions that had kept turning out right in experiments. Some of them dealt with matters of life and death.
For the first time, I could sense my career path beginning to unfold. Excitedly I wrote to Winfree to ask for ideas about where to go to graduate school for mathematical biology. (I hadn’t heard of any formal programs in it. The subject was too new, too much on the fringe.) Two weeks later, my pulse quickened when I picked up the mail and spotted the Purdue return address. Inside, scrawled in red Magic Marker on blue-lined school paper, with a few phrases connected by swooping arrows, was a reply from Winfree himself:
