Root of Unity

by SL Huang

  “Sonya,” said Arthur, his tone much more subdued than I was used to hearing from him, “you said you needed someone who understood the more technical aspects. Cas can do that. Fill her in.”

  Halliday turned to look down her nose at him, and the look was all wrong; even I could see that. Arthur had called them good friends, but Halliday was regarding him like he was a bug on a clean tablecloth. “Doubtful,” she said, turning to walk around her desk. She sat down in her office chair and started pulling up files on her computer. I tried to catch Arthur’s eye, but he was steadfastly not looking at me.

  “Sonya,” he tried again instead, “I just want to help you, okay?”

  She concentrated on her monitor. “I told you to leave me alone.”

  “You called me,” he pleaded.

  “In a moment of weakness. I believe I was very clear I do not want any help from you.”

  “Arthur?” I said.

  He made a “back off” motion at me with one hand. “You need help. You told me I wouldn’t understand, well, I brought someone who will. Just talk to us. Please.”

  “What is your area of specialization?” said Halliday.

  She hadn’t looked up, and it took me a second to realize she was talking to me. I wasn’t great at reading people’s tones, but it had never been more obvious someone was trying to set me up to fail.

  Oh, fuck you.

  I plopped down in one of the chairs across from Halliday’s desk, sprawling in an inelegant slouch. “You know. I do a little of everything.”

  “I do not mean to be rude,” said Halliday, “but Arthur does not grasp the level of depth and complexity in my field—”

  “Liar,” I said. “You do mean to be rude. Go on, say what you’re thinking.”

  She finally turned to regard me, folding her hands on the desk in front of her. “Miss…Miss Russell,” she said, the title only slightly questioning, “I know personally everyone in the same line of research as myself. You must understand how specialized areas of higher math are. Even doctorates in the same general area would require a great deal of study to understand what—”

  “I read all the papers you’ve published to date last night,” I said.

  She closed her mouth.

  “The Internet is a wonderful invention, isn’t it?” I said, deliberately misunderstanding her surprise. “Nice work on the new encryption algorithm using prime roots of unity to approximate randomness. That’s a clever trick.”

  Halliday’s voice tightened. “An elementary understanding of—”

  “For God’s sake, Sonya,” said Arthur from behind me, low and rough. “You told me you needed someone who understood your work. Let us help you.”

  And just like that, Professor Halliday crumpled. Not in a dramatic way—somehow I doubted she did anything dramatically—but her head bowed and her shoulders hunched and she took off her severe rimless specs to press shaking fingers to her face.

  Arthur swooped around the desk and put a supportive arm over her thin shoulders. “That’s okay. It’s okay. Just tell us what’s going on.”

  Her voice came croakily through her hands. “I think I understand now. What you said about getting in over your head, how easy it was—how you didn’t see what was happening until it was too late.”

  Arthur stiffened beside her.

  I perked up. “Wait, Arthur did something wrong?” Arthur and Halliday both froze, and the room got intensely uncomfortable in a way I tended to find perversely entertaining. “Do tell; I want to hear this.”

  “Don’t,” said Arthur, so quietly I almost couldn’t hear the word, his eyes fastened on nothing. He didn’t sound angry—he sounded like he was in pain.

  The tension in the room suddenly got a lot less entertaining.

  Shit. Way to go, Cas. I leaned forward and tried to go back to businesslike. “Professor. Arthur told me you had some work stolen. Why don’t you start there.”

  Halliday glanced back up at Arthur, who squeezed her shoulders in support and nodded her on. She leaned into him almost imperceptibly. “Not—not some work. All of it.” She’d reached up and was gripping one of Arthur’s hands so tightly the tendons stood out in her wrist. “All my current research. All my notebooks, at home and here at the office—gone.”

  Arthur sucked in a breath.

  “Okay,” I said. “I take it the police weren’t able to help?”

  “I didn’t—” She looked back up at Arthur, and then at me, hesitating.

  “I trust Cas with my life,” he said, surprising me.

  “I need you to mean that.”

  “I promise,” said Arthur. “You can trust her.” Some sort of fuzzy tingly feeling crinkled in my chest at his quiet confidence; I tried not to let it show.

  Halliday’s eyes flicked to me, to Arthur, and back. “I might be in—I’m in some trouble. Maybe—a lot of trouble. Arthur…I’m sorry. I’m so sorry.”

  “It’s okay,” Arthur said again, almost too firmly. “We’re here to help you. Just tell us what happened.”

  She closed her eyes for a long moment, then opened them and focused on me. “I found an efficient integer factorization algorithm.”

  I stared at her, the implications crashing through my brain. “You what?”

  “I was only thinking of the math!” She raised a hand as if to excuse or defend herself. “Truthfully, I never let myself believe I would actually solve it. Dwelling on the consequences of a pipe dream—it felt so arrogant, and now…”

  “How efficient is it?” I asked. “Even if we’re talking polynomial time, if there’s a large enough constant in there—”

  “It’s fast enough,” she said. “I think. The programming part isn’t my area. But it’s fast enough.”

  “Shit,” I said, though I couldn’t help the word coming out half-admiring. “Wow.” And then the purpose of our meeting came thundering back. “Wait, someone stole it from you? Why the hell are you calling us instead of the police or the FBI or, I don’t know, anyone? I’m usually in favor of going outside the law, but when it comes to wrecking the entire global economy—”

  “I know!” cried Halliday. “I know I should have. I tried talking to a friend of mine who works for the NSA—just hypothetically, as if I were considering working on a problem like this, and he told me—” She took a breath, swallowed. “The amount of oversight they wanted if I began work on the problem, the care he told me to take if I was getting close…if they found out I—”

  “Yeah,” I said. “Yeah, okay. I get it.” Arthur’s face was a study in confusion; I turned to him and tried to explain. “Pretend you built a nuclear bomb for fun and someone stole it. I don’t know if there’s a law on the books for this, but if they decided they wanted Professor Halliday to go down, they could probably find one. Hell, cryptographic algorithms used to be classified as munitions under U.S. law—a few years ago you could go to jail for sending someone three lines of Perl script. They take this shit seriously.”

  “I should have gone to the authorities,” Halliday said. “It was selfish. I was—I confess I was frightened. I have a handful of friends who consult with the NSA, mathematicians, and I could have, I should have talked to them right away. And now…now I fear it might be too late.”

  “Why do you say that?” said Arthur.

  Still gripping his hand with one of hers, Halliday reached down with the other and unlocked a drawer in her desk. She drew out a blue file folder and handed it up to him. “I found this in my office the next day.”

  He opened the folder, and his expression twitched, the muscles in his face tightening. He passed the folder to me.

  It had one sheet of paper inside. Plain white paper, with plain black lettering printed on it:

  We aren’t planning to wreak destruction. But pretend this never happened, or else.

  “Huh,” I said. “I suppose it could be worse.”

  “How?” said Halliday incredulously, some of her control slipping.

  “Whoever stole your
proof isn’t planning on destroying the world, only being selfish with it—probably getting rich. Unless they feel threatened, apparently. This gives us some time.”

  “Time for what?” Halliday demanded. “Whether or not we find who stole it, whether or not we get it back—the information is out there now!”

  “Stop panicking,” I said. “Or at least go somewhere else to do it after you give us the rest of the information.” I snapped the file folder closed. “We’ll keep this. Now, who else knows anything about the proof?”

  “No one,” she said, steadying her voice with an obvious effort. “It was my pet project. My Fermat’s Last Theorem. I was embarrassed even to tell anyone else I was working on it; it seemed too fantastic.”

  “You didn’t have any collaborators?”

  “No, not on this. Or—only Rita. I talked to her about it sometimes, but I swore her to secrecy. And she didn’t know I had finished.”

  “Who’s Rita?” I said.

  “You talking about Dr. Martinez?” asked Arthur. “Your doctoral advisor?”

  Halliday nodded. “Collaborator now, and we’re very close friends. But she couldn’t be involved.”

  “I don’t know,” I said. “Sounds like motive to me, stealing a colleague’s secret proof to publish yourself.”

  Halliday snorted. “I would sooner believe Rita capable of murder.”

  Arthur and I exchanged a look over her head, but Halliday’s gaze had unfocused into the distance, and she missed it.

  “How did the robbery happen?” Arthur prompted gently, after a moment.

  “Wednesday I came home and—there was no sign someone had broken in, nothing,” Halliday answered. “But all of my notebooks were gone. Just gone.”

  “What about your computer?” I asked.

  She shook her head. “I work in longhand. I was only just now going back through the proof to rewrite it for publication—so many years, so many dead ends and notes, and…they took it all. I went back to my office immediately and found my work here cleaned out, too. Even though none of it was relevant to the factorization problem, they still took everything. And the next morning I found the note.”

  “I’ll go by your house in a few hours with some equipment,” said Arthur. “And then we’ll come back and look at your office again. Don’t handle nothing you ain’t touched already.”

  Halliday made an abortive gesture at the books and papers surrounding her. “I have to…my work—”

  “Can wait,” Arthur said.

  “He’s right,” I put in. “Take the day off, go have a stiff drink or three. We’ll call you.”

  “I don’t drink.”

  Of course she didn’t. “Then sit in a park and read some combinatorics papers or something. What else do people do to relax?” I asked Arthur.

  He gave me a funny look, but addressed Professor Halliday instead. “Sonya, she’s right. Go get some coffee; try to stay calm. We’ll figure this out.”

  “Things don’t always work out, Arthur. You should know that better than anyone.”

  Arthur didn’t reply, though his movements hitched for a second before he became the supportive friend once more, nudging Halliday gently to her feet. “Give me your keys, okay, hon? We’ll call in a bit.”

  She obeyed, and Arthur guided us out of her office and locked the door.

  “You going to be okay?” Arthur asked.

  She hesitated. “My biggest fear is—I don’t know if I can recreate it. My greatest achievement, and I don’t even know…what if it’s gone?”

  Arthur took her by the shoulders. “Ain’t gonna make you no promises I can’t keep. But Russell here is the best there is, and I ain’t too shabby myself. Take this one day at a time, okay? We’ll call.”

  She nodded.

  “Come on. We’ve got a lot of work to do,” I prodded Arthur.

  He squeezed Halliday’s shoulders one last time. As we headed off at a trot, he glanced back several times to where she stood thin and bereft in the hallway.

  Well, this sucked for Arthur. Of course, that didn’t mean I wasn’t going to take his head off the moment we were out of sight.

  Chapter 3

  In the end, I was very well-behaved. I waited until we were in the car. Arthur was pulling out of the visitor’s parking lot and had the gall to say to me grimly, “So, I get this is a big deal. Can you give me the layman’s rundown?”

  “You first,” I said. Penguins could have gotten frostbite from me.

  He hesitated. “Me first what?”

  “Fuck you,” I said, though I couldn’t force as much vitriol into it as I wanted. “The client will pay my rates?”

  “You’ll be paid—”

  “She didn’t want me there. She didn’t even want you there.”

  “She came around, though, right? I knew she’d let us help if—”

  “You lied to me.”

  “Okay, yeah, but I didn’t know if—”

  “If what?” I bit out. “If I’d come along if you weren’t paying me to?”

  “You got to understand—she’s too important to me. I didn’t mean—I needed you; I ain’t thought—”

  “You thought if you said, ‘hey, Cas, help me out,’ that—what, I’d say no?” Voicing the words stung. I bit my lip.

  “Well, to be fair, money’s what you always—and you can’t be too hard on me, Russell, if this ain’t no official job for you, can you take it anyway?”

  It was a fair concern. After all, Arthur knew what happened when I wasn’t working—he was one of the few. That didn’t mean I wanted to concede. “You could have asked me. For the record, I’ll be fine.”

  “I’m sorry,” he said, though I didn’t hear much repentance in his tone. “It was too important; can you understand? Please? But I’m sorry. I am.”

  “So who is she?”

  He took a long breath. “Sonya and I—we grew up together. Childhood friends.”

  “And then?”

  “And then what? Life happened. We grew apart. Ain’t mean I don’t still care about her.” He kept his eyes glued to the road in front of him, like someone who wasn’t telling me anything close to the whole story. “So, uh. This math stuff. Help an old guy out—why is the world ending?”

  “This isn’t over,” I grumped, but I let him change the subject. For now. I slumped in the passenger seat, sticking my boots up on the dash. “Do you know anything about encryption?”

  “Not a thing.”

  “Okay. Well, a whole hell of a lot of our current crypto depends on the idea that factoring large integers is a really hard problem. In simple terms, we encrypt information by multiplying large prime numbers together, and the fact that no one can un-multiply them easily is what keeps everything secure. And ‘everything’ means everything—from your credit cards to the Department of Defense.”

  Arthur let out a low whistle.

  “Yeah,” I said.

  “So Sonya cracked the crypto?”

  “Sort of,” I said. “The ticket is, we’ve always thought factorization was a hard problem, but we’ve never actually known it was hard. Nobody’d ever proven it was.”

  Arthur frowned. “Why’s everyone use it, then? Seems kind of unwise.”

  “Not that unwise. A lot of really smart people had been working on the problem of integer factorization for a very long time, and nobody’d come up with a fast way of doing it. Key word being ‘fast’—we can do it; it just takes years, far too long to be useful in code-breaking. So building an encryption algorithm based on the fact that nobody’d ever discovered a way to do this quickly, well, it was actually pretty genius.”

  “Except Sonya found a way,” said Arthur.

  “Yeah.” I still couldn’t believe it. As grave as the situation was, part of me was ravenous just to read her proof. “Yeah, she thinks she did.”

  “And you say everything runs on this math.”

  “Yeah. Checker might know better than I would where all it’s being used, but I’m pretty sure i
t’s across the board. Every financial transaction people send electronically. Our whole economy, national security, all of it.”

  “Jesus, Mary, and Joseph,” whispered Arthur. “So if whoever got her proof decides they’re bored just making themselves rich…”

  “Modern apocalypse,” I said. “It’s possible. I think we’ve got a little breathing room, though. Professor Halliday said she was in the midst of going back through decades of notes and rewriting the proof for publication—it’ll take them time to organize and absorb all her work. And they’ll probably need someone in the field to help them with it. Plus they’ll have to write whatever actual computer code they want to use—I’ll have to talk to Checker and see if he can estimate how long that’ll take—”

  “Wait,” said Arthur. “Did you just say they’d need a mathematician even if they have her notes?”

  “Yeah,” I said. “Probably more than one.”

  “Shit,” said Arthur, yanking the wheel to slue the car toward the next exit, “Sonya—she ain’t safe—”

  A car slammed into us from behind.

  Metal shrieked and the seatbelt wrenched me across the chest. The car spun more than 180 degrees and slid into a skid across four lanes of freeway, traffic screeching by us the wrong way around—

  I reached for the wheel and yanked it over, Newtonian mechanics erupting in my brain like a fountain. “Accelerate!” I bellowed in Arthur’s ear; he immediately let go with his hands and slammed his foot down on the pedal.

  “Switch with me!” I shouted, diving for my seatbelt release with my other hand and cursing Arthur’s insistence that I wear it. Horns deafened the air in a cacophony around us, and a screeching crash blasted through the noise as if it were right next to my ear—two cars avoiding us had smashed into each other and one had flipped over the median. I swung the wheel the other way with a solid wallop of inertia, sending us barreling between a semi and a minivan as I brought us out of the skid. The minivan’s driver jerked away, and it turned directly into the path of a bright blue sports car. I could have screamed—not that you could have heard it over the deafening explosion of metal and kinetic energy. “I wasn’t going to hit you!” I yelled in pure frustration.