Joseph still looked as if the shock had not worn off. Back at his lodging, he pulled out his laptop with a stiff expression and opened an encrypted communications network.
Tap, tap—
[Report on Yu Seoha’s Movements and the Development Status of Next-Generation PQC]
—Recipient: David, Senior Researcher, NSA
David. I thought your assessment of him was exaggerated. But even you may have underestimated Seoha. His capacity for thought is far beyond the human category. What I saw was...
(omitted)
No cryptographic system humanity has built is safe before him. If Seoha slips beyond our control, I am afraid even to imagine what kind of disaster may befall national security.
While fully cooperating with the new PQC development he is leading, I urgently request that the surveillance and protection level for Seoha be raised to Tier Zero.
Joseph’s hands trembled.
Tak—
When he pressed the Enter key, a message appeared saying the email had been sent successfully. In his eyes, reflected by the monitor’s light, fear was unmistakable.
“This is the best I can do.”
Joseph muttered to himself.
That overwhelming intellectual violence, dismantling encryption with an innocent expression as if he were taking out the trash. And yet Seoha seemed entirely unaware that he himself was a bomb capable of destroying, in an instant, the digital civilization humanity had built.
***
[Apex Lab KOREA — Director Yu Seoha]
“I heard Yu Seoha is your student, Professor. You must have mixed feelings.”
Minister of Science and ICT Lee Hyeon-seok asked, glancing at Professor Kim Ji-yun, who had accompanied him. Ji-yun smiled faintly and shook her head.
“In mathematics, the word student is not used so lightly. Seoha is no one’s student.”
“That is complicated.”
“It is. It is also a sensitive matter. But Seoha will have plenty of students of his own in the future.”
Rumors were rampant in academia that Seoha would soon be appointed to a faculty position.
Probably after his discharge from the military.
She had heard that Ivy League universities were racking their brains because of him.
Knock, knock!
They had been told he was inside, but there was no answer.
One more knock.
Knock, knock!
Creeak—
Ji-yun carefully opened the door.
Seoha stood by the window. He was wearing headphones and staring at the blackboard. Ji-yun glanced over the formulas.
Lattices, probability, complexity, and curves whose meaning she could not discern, along with a few short lines of text.
[Is being hard to solve incompatible with being beautiful?]
It was a question every mathematician who studied cryptography had pondered at least once.
Ji-yun’s expression turned wistful. Five years ago, she had found her own answer to that question.
Yes, they were incompatible.
All the beautiful cryptosystems she knew were not outstanding in terms of performance and efficiency.
Elliptic curve cryptography (ECC), used in blockchain and secure communications, was mathematically quite elegant, but it had almost no quantum resistance.
‘What conclusion will you reach?’
Her thoughts were complicated.
She expected that he might give her a different answer, and yet, at the same time, a petty part of her believed that even Seoha would have no choice.
After waiting a few seconds, the minister was about to call out to Seoha, but Ji-yun raised a hand to stop him.
Time passed.
One minute.
Two minutes.
Ssssk.
As if he had realized something, Seoha suddenly drew a graph on the blackboard. The line repeatedly expanded and contracted, forming a strange pattern. Ji-yun tried to understand it, but she could not grasp its meaning.
“Huh? Professor!”
He was stretching when he noticed the two people standing by the door.
Ji-yun smiled and greeted him.
“Hello, Seoha. This is Minister Lee Hyeon-seok of the Ministry of Science and ICT.”
Seoha’s face stiffened for a moment. Then, after thinking for about two seconds, he hurriedly yanked off his headphones.
“Loyalty!”
His posture was sharp as a blade, still bearing traces of training camp. Ji-yun made an impressed “oh” expression.
“Oh my! Director Yu! There’s no need for that. I’m not a soldier.”
The minister laughed heartily and took Seoha’s hand.
“Be at ease. I heard you went through training very diligently.”
Pour, pour.
Ben, who had now taken on secretarial duties as well, came in and served black tea. It was the highest-grade tea Morris had flown in from England.
“Ahem! I heard about what happened a few days ago. They said you broke the encryption the researchers made.”
Perhaps unable to contain himself, the minister brought up the main subject first.
Joseph’s report had arrived at the NSA at two in the morning.
After reading the report, Director Murphy convened an emergency meeting that same morning. It was a vice-ministerial-level meeting attended by officials from NIST, finance, and the Department of Defense.
The United States planned to apply three PQC algorithms, including Kyber, to financial institutions and military communications within the next ten years. The Department of Defense alone had already allocated hundreds of billions of dollars in budget... and they could not ignore the possibility that the foundation of all that might be shaken.
“It was possible because it was in the early stages of development. LWE isn’t that weak.”
Seoha’s expression said it was nothing special.
As if his insides were burning, Minister Lee picked up his cup and gulped down the hot black tea.
‘It would have been nice if he’d gone into engineering...’
From the perspective of Minister Lee, who had once been an engineer, Seoha was clearly wasting his talent.
If that brilliant mind were used to develop cutting-edge technology, just how much economic value could it create?
“So what are you planning to do from now on?”
Ji-yun pulled a chair over in front of him and sat down as she asked.
“Professor.”
“Yes?”
“I don’t think I’m particularly suited to cryptography.”
A vein stood out at the minister’s temple.
‘What kind of nonsense is this!’
After smashing a cryptosystem dozens of experts had agonized over, this was what he had to say...
But he exercised superhuman patience and kept his mouth shut. He had been warned several times that Seoha greatly disliked being forced into anything.
“What made you think that?”
Ji-yun asked, looking at him as if coaxing him.
“Cryptography is basically like giving an unspecified number of people a math problem, right?”
“You could see it that way.”
“But you have to give them a problem they can’t solve under the given conditions.”
Seoha frowned.
RSA was based on the idea that factoring very large numbers was difficult. Difficult to a realistically unbearable degree. With a home computer, it would take thousands of years, perhaps hundreds of millions of years or more.
Seoha’s paper had drastically reduced the computation time for that problem by narrowing the range in which primes could be arranged.
“That’s right. That’s the core concept of modern cryptography.”
“LWE is ultimately the same idea. Mixing in noise to make the calculations bothersome. Making the problem complex enough that people won’t want to solve it. To be honest, I think it’s a fairly reasonable method. Lattice problems will probably remain safe for decades to come.”
Ji-yun nodded.
“Cryptographers, myself included, also think that’s the best option for now.”
“But lattice problems will certainly become easy to solve someday. If a groundbreaking algorithm appears, or if far more powerful computers emerge. In the end, you could say it’s a method with a predetermined lifespan.”
Why worry now about a future decades away!
The minister’s face flushed red.
Ji-yun set down her black tea and stared intently at Seoha. That was because she sensed in his attitude the particular hesitation of a mathematician.
Seoha could probably create a cryptosystem that satisfied NIST’s requirements right this moment. But he was not doing so.
Because he valued mathematical integrity. And then, suddenly, a thought flashed through Ji-yun’s mind.
‘Integrity?’
Calming her trembling heart, she asked.
“Are you, by any chance, thinking about the P≠NP problem?”
P≠NP, or P=NP, one of the Millennium Problems.
P is the set of problems computers can solve quickly. NP is the set of problems whose answers can be checked quickly, but are difficult to solve.
In other words, the P≠NP problem can be defined in a single sentence like this.
‘This problem has an answer that is easy to verify, but you will not be able to solve it easily.’
And most existing cryptographic systems are built on the assumption that this proposition is true.
The existence of problems that cannot be solved efficiently. In a way, this is also a question about the essential limits of humanity.
Seoha smiled awkwardly and nodded.
“Ah!”
Ji-yun finally understood.
The reason Seoha had shown little interest in cryptography. It was because cryptography’s major premise had not been proven.
For Seoha, who had a kind of mathematical mysophobia, it must have felt like building a house on sand.
Ji-yun asked in an urgent voice.
“Don’t tell me you’re planning to solve P≠NP first?”
How long had it even been since he solved the Riemann Hypothesis...? Ji-yun was horrified, thinking Seoha might be about to throw himself into another difficult problem. Seoha laughed and waved his hands.
“No. Of course not. I’m not that unrealistic.”
If the Riemann Hypothesis was the holy grail of number theory, then “P not equal NP” could be called cryptography itself. More than ten years ago, there had been a large-scale survey of mathematicians.
Mathematicians overwhelmingly believed P≠NP to be true.
Eighty-three percent agreed with the conjecture, while only nine percent answered that P=NP was likely correct. It could hardly be otherwise, because if P=NP were to hold true, the world would become completely different from what it was now in every field, including pharmaceuticals, biotechnology, mathematics, computer science, and nuclear fusion.
Ji-yun wanted confirmation. In a trembling voice, she asked Seoha.
“Seoha. Do you think P equals NP?”
Tick, tock. Tick, tock.
Silence flowed.
It was only a few seconds, but to Ji-yun, that time felt dozens of times longer.
“No.”
Seoha spoke firmly.
“P is not NP.”
Ji-yun let out a sigh of relief.
If Seoha had answered the opposite, it would have meant the world might change on a scale comparable to the overturning of heaven and earth. That was how much Ji-yun trusted Seoha.
“Then there shouldn’t be any problem with creating encryption, right?”
Seoha believed that essential difficulty existed. If so, it should be possible to create cryptography using it.
“Yes! Actually, I have something in mind. Would you like to see it?”
Rustle.
This was the purpose of his visit to the research institute today. A tense light crossed the minister’s face.
What Seoha handed over was a single sheet of paper. Ji-yun received it and began to read.
“Noncommutative algebraic cryptography...”
“As expected, Professor. You know it.”
This was a matter that cryptographers had already reached a conclusion on long ago.
About twenty years earlier, when braid group cryptography appeared, academia had briefly been stirred up. A noncommutative structure, where the result changed depending on the order, like the way hair was braided.
From an attacker’s perspective, the number of cases increased explosively, so in theory, it had seemed impregnable.
But in less than four years, that cryptosystem collapsed. The attacker had transformed the noncommutative structure into a commutative one.
Seoha knew all this, and yet he had chosen a noncommutative structure. With an innocent smile, he said,
“Don’t worry. I’m going to build the structure using ‘Yu-Riemann geometry.’”
Ji-yun lifted her head and looked at the blackboard again.
The graceful curves were tangled in a geometrically dizzying form. She thought it was truly beautiful.
[Is being hard to solve incompatible with being beautiful?]
Seoha was saying firmly:
Absolutely not.
Creeak— tak.
The door closed, and Seoha was alone again.
His expression turned cold.
‘Why did you lie? You’re planning to solve P≠NP.’
Seoha rose from his seat and looked at the duck mask on his desk.
“Because I think it’s a problem that can be solved. But I’m not ready yet. It might take years.”
‘Tch!’
“How about being satisfied with cryptography? They seem to like it quite a lot.”
‘Ah, yes. The truth is, you want someone to challenge you too. You’re hoping there will be plenty of fools who walk fearlessly into the abyss.’
Ducky laughed with a bizarre sound.
Seoha put the headphones he had taken off back over his ears. From the player flowed Gyeo-ul’s performance.