The top student must be diligent.

Chapter 111 Millionaire Problems
The main principle used by the RSA encryption algorithm is the difficulty of decomposing large numbers into prime factors.

For example, we all know that 17*13=221, but when we see the number 221, it is not certain whether we can immediately see that it is equal to 17×13.

The larger these two numbers are, the harder it is to crack.

Of course, the sieve method, which is specifically used to factorize large numbers, has become an important method to crack this kind of encryption.

After all, the principle of the sieving method itself is to eliminate various factors by continuously multiplying upwards.

Therefore, in the RSA encryption system, there is an attack method called general number field sieve, which is recognized as the most effective method to crack RSA encryption.

Of course, the same problem is that the parity check problem in the sieve method makes it more troublesome when dealing with particularly large numbers. The modern RSA encryption algorithm uses particularly large numbers. Therefore, when using the sieve method, it is inevitable that there will be huge deviations in the cracking process.

But now...

"Yes... In the past, it was quite difficult to crack the RSA cipher using the sieve method. After all, the parity check problem is a big problem."

Maynard patted Xiao Yi on the shoulder and said, "But now, the impact of the parity check problem has been directly suppressed by your classification screen. Those cryptographers are going to have a headache."

Tao also smiled and said, "Don't people outside always think that our math has no practical application? Now, let's give them a practical cryptographic attack."

Seeing the two mathematicians gloating over the misfortune, Professor Kleinlock, a computer scientist standing beside them, said without any curiosity: "Do you think this way? If there is a problem with the bank's password system, our social stability will also be in trouble."

"Don't worry, we all know that there is only a risk of being cracked. It is still a long way to go to actually crack it. After all, even if we use the sieve method to crack it, it will take a lot of time." Tao Zhexuan was not scared. He waved his hand and said, "But we are still very happy to give them a headache."

People like them who study pure mathematics are often asked what applications their research has, which often leads to friction between the pure mathematics community and other fields.

For example, after Zhang Yitang made a breakthrough in the twin qualities conjecture, Google invited him to give a speech, but he refused because he was afraid that after the speech, others would ask him what the use of his achievements was.

Similarly, after Perelman proved the Poincare conjecture, he was invited to major universities to give lectures and explain his proof.

As a result, someone asked him what applications this method had after he figured it out. Upon hearing this question, Perelman became furious and asked how someone could ask such a stupid question. Finally, he couldn't stand it anymore and went back to Russia directly.

In short, those who work in pure mathematics and others who work in applied mathematics basically cannot pee in the same pot.

Of course, this has also formed a temperament of some pure mathematicians. The less applicable their work is, the more satisfied they are with it, and they think that this is true "pure mathematics."

Hearing the discussion of the professors in front of him, Xiao Yi shook his head helplessly.

This was not his intention. He never thought that if he invented the classification screen, cryptography would be at risk of being cracked.

"Okay, don't gloat over my misfortune. I just want to ask you how to solve this problem. This kind of thing involving pure mathematics will eventually fall on the shoulders of pure mathematicians like you."

Professor Kleinrock said.

The three mathematicians then calmed down and waited for Kleinlock's explanation.

"Hmm... To explain what a multi-party security algorithm is, let's start with a classic problem."

“That’s the millionaire problem. Do you know this problem?”

Terence Tao nodded. He also had some research on computers. More than ten years ago, he had come up with something called Information Acquisition Guidance Theory. Simply put, this was a digital compression imaging technology. Eventually, this technology was widely used in the information field and other fields, fully demonstrating that he also had very strong abilities in applied mathematics.

However, Xiao Yi and James Maynard seemed a little embarrassed.

The latter was fine, saying that he had heard of this, "I remember the person who asked this question was a Turing Award winner."

"Haha, yes." Kleinlock nodded and said, "Come to think of it, this Turing Award winner is also Chinese, just like Xiao Yi. His English name is Andrew Yao, and his Chinese name seems to be Yao Qizhi."

"Simply put, the millionaire problem is, suppose there are two millionaires named Alice and Bob, and now they want to compare who is richer, but they don't want to reveal how much money they have to each other. In this case, how should they compare their wealth?"

As he spoke, Kleinlock also wrote the description on the blackboard.

[Assume that Alice and Bob have assets i and j respectively, and the values ​​of i and j are both between 1 million and 10 million. How can we make the other party not know the specific value of i or j and compare the values ​​of i and j? ]

Looking at this problem, Terence Tao knew how to solve it, but Maynard and Xiao Yi began to think about it. This problem also seemed quite interesting.

After a while, the two of them had some time to think, but Kleinlock did not want them to answer on the spot, and then he said: "Okay, we can answer this question later..."

"and many more."

However, at this moment, Xiao Yi spoke up, "I think this problem may be solved in this way."

Then he walked to the blackboard, picked up a pen and started writing.

[Let M be a set whose elements are all N-bit non-negative integers, and let QN be the permutation group from M to M. ]

“If my design is correct, as long as they follow the following protocol, they can complete the wealth comparison while keeping their assets confidential.”

“First, Alice randomly selects an element Ea from QN as the public key and tells it to Bob, while keeping the inverse Da of Ea as her own private key.”

"Then, Bob randomly selects an N-bit integer x and uses the public key given by Alice to calculate k=Ea(x)..."

As Xiao Yi began to speak, the three people next to him fell silent.

They looked at each other, somewhat caught off guard by the scene before them.

Especially James Maynard.

Although, if one really studies this problem carefully, it will not be difficult for him. As a mathematical modeling problem, its difficulty is just that high.

After all, this problem has been completely concretized, and in terms of abstraction, it is too trivial compared to the problems in analytic number theory that Maynard studied.

However, it is impossible for him to react so quickly and start to come up with solutions right away.

Looking at what Xiao Yi had written on half of the blackboard, Maynard experienced for the first time what "dimensionality reduction attack" meant.

Perhaps his students felt the same way when he demonstrated what "top mathematical and logical thinking" was in front of them?
It's simply cheating!
Kleinlock's reaction was similar to Maynard's. Originally, when he raised this question, he had not intended for them to solve it on the spot. He had left them a few minutes just to let them become a little more familiar with the problem.

As a result, who would have thought that just a few minutes later, the young man had already started writing.

Forgive him for being a computer scientist, he is not really good at mathematics, at least compared to these mathematicians. After all, there is a saying that only people who are really good at mathematics can always be rooted in pure mathematics, while people who are not good at mathematics will change their careers early.

However, Xiao Yi's "goodness in math" was a little beyond his imagination.

“Step 4: Alice randomly generates an N/2-bit prime number p and calculates Zu = yu mod p, where u = 1, 2, …, 10.”

"Then we ensure that at least two of all zu are not equal, otherwise we re-select p to calculate..."

“Alice relabels the prime numbers p and [z1, z2, …, zi, zi+1] … as …”

"Bob checks the zj`th element. If zj`=x mod p, then i is greater than or equal to j, otherwise i is less than j."

After writing the last stroke on the blackboard, Xiao Yi turned around and read it again from beginning to end. He nodded with satisfaction, put down the pen, turned around and said to the three people next to him.

“In this way, the wealth of Alice and Bob is compared, without revealing their true wealth to each other except in the case where their wealth is equal.”

"Professor Kleinlock, my method should be OK, but I don't know how to write the specific program." Xiao Yi said modestly: "I don't know anything about programming."

However, what puzzled him was that the three professors in front of him were all a little silent at this time.

After a moment, Kleinrock asked Tao quietly, "Are all of you who are good at math like this? Do you treat the difficult problems in our computer field as if they were... addition and subtraction within 100?"

Tao Zhexuan waved his hands repeatedly: "Uh... That's not the case, but you know, there are always some special cases in this world, some low-probability events. Xiao Yi is such a low-probability event, you know."

(End of this chapter)