RSA encryption algorithm, the main principle it utilizes 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 necessarily whether we can immediately see that it is equal to 17×13.
And the larger these two numbers are, the harder it is to crack.
Of course, as a sieve method specially used to factorize large numbers, it has become an important cracking method for this kind of encryption.
After all, the principle of the sieving method itself is to eliminate various factors by continuously multiplying them upward.
Therefore, in the RSA encryption system, there is an attack method called Number Field Sieve, which is recognized as the most effective encryption method to crack RSA.
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. For the modern RSA encryption algorithm, those very large numbers are used, so when using the sieve method, it is inevitable that there will be great deviations in the cracking process.
But now...
"Yes... it used to be very difficult to crack RSA passwords using the sieve method. After all, the parity issue is a big problem."
Maynard laughed and patted Xiao Yi on the shoulder and said, "But now, the impact of the parity check problem has been directly suppressed by your classification filter. Those who work on passwords will have a headache."
Terence Tao also smiled and said, "Don't those people outside always think that our mathematics has no practical application? Now, let's directly apply a password attack to them."
Seeing the two mathematicians gloating about their misfortune, the computer scientist Professor Kleinrock next to them said without curiosity, "Do you think so? 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. If we want to actually crack it, we are still far away. After all, even using the sieving method to crack it will take a lot of time." Terence Tao was not frightened, and waved his hands and said, "But we are quite happy to give them a headache."
People like them who study pure mathematics are often asked what applications their research has, which often causes friction between the pure mathematics community and other fields.
For example, after Zhang Yitang achieved a breakthrough in the twin quality conjecture, Google once invited him to give a speech, but he refused because he expressed fear that after giving a speech in the past, others would ask him what the use of his results was.
Similarly, after Perelman proved Poincaré Conjecture, he was invited by major schools to give lectures to explain his proof.
As a result, someone asked him what applications he had after solving this method. As soon as he heard this question, Perelman immediately became furious and said why someone asked such a stupid question. In the end he couldn't stand it anymore, so he went back to Russia directly.
All in all, those who engage in pure mathematics and other applications can basically not urinate in a pot.
Of course, this also forms the temperament of some pure mathematics people. If the things they create are less applicable to applications, they will be more complacent, thinking that this is the real "pure mathematics".
Hearing the discussion of the professors in front of him, Xiao Yi shook his head helplessly.
This was not his intention. How could he have thought that if he built a classification screen, it would bring the risk of cryptography being cracked.
"Okay, don't gloat about your misfortune. Now I want to ask you how to solve this problem. This kind of thing involving pure mathematics will eventually fall on you pure mathematicians."
Professor Kleinrock said.
Then the three mathematicians calmed down and waited for Kleinrock's explanation.
"Hmm... If I want to explain what Multi-Party Security Algorithm is, let's first start with a classic question."
"That is Millionaires' Problem, do you know this question?"
Terence Tao nodded. He also had some research on computers. He had developed something called Information Acquisition Guidance Theory more than ten years ago. Simply put, this is a digital compression imaging technology. In the end, this technology was widely used in the information field and other major aspects, which fully demonstrated his strong ability in applied mathematics.
However, Xiao Yi and James·Maynard seem a bit difficult.
The latter was okay, saying that he had heard of this, "I remember that the person who asked this question was a Turing Award winner."
"Haha, yes." Kleinrock nodded and said, "Speaking of which, the winner of Turing Award is also Chinese like Xiao Yi. His English name is Andrew Yao, and his Chinese name seems to be Yao Qizhi."
"To put it simply, Millionaires' Problem means, suppose there are two millionaires named Alice and Bob. Now they want to compare who is richer, but they don't want to reveal how much money they have to each other. So in this case, how should they compare their wealth?"
With that said, Kleinrock also wrote a description on the blackboard.
[Assume that the properties of two people, Alice and Bob, are i and j respectively, and the sizes of i and j are both between 1 million and 10 million. Then how can we compare the sizes of i and j without the other party knowing the specific number of i or j?]
Looking at this problem, Terence Tao knew how to solve it, but Maynard and Xiao Yi started to think about it.
This question seems quite interesting.
After a while, the two of them were given some time to think, but Kleinrock did not think about asking them to answer it on the spot, and then said, "Okay, we can discuss this question later..."
"Wait."
However, at this moment, Xiao Yi spoke, "I think this problem may be solved this way."
Then he walked to the blackboard, picked up the pen and started writing.
[Suppose M is a set whose elements are all N-bit non-negative integers, and QN is the permutation group from M to M.]
"If there is nothing wrong with my design, as long as they follow the following protocol for comparison, 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 retaining the inverse Da of Ea as the private key for itself."
"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 became slightly silent.
They looked at each other, somewhat caught off guard by the scene in front of them.
Especially James·Maynard.
Of course, if he really wanted to study this problem carefully, it wouldn't be a problem for him. As a similar mathematical modeling problem, its difficulty is just that.
After all, this problem has been completely concrete. From an abstract level, it is too childish compared with the problems above Analytic Number Theory studied by Maynard.
However, it would be impossible for him to react so quickly and start giving plans directly.
Seeing that Xiao Yi had already written half of the blackboard, Maynard experienced for the first time what is called [dimensionality reduction strike].
Maybe when he showed what [top mathematical logical thinking] was in front of his students before, his students felt the same way, right?
It's just a mess!
Kleinrock's reaction was similar to that of Maynard. Originally, when he raised this issue, he didn't expect them to solve it on the spot. He only left a few minutes to let them become a little familiar with the issue.
As a result, who would have thought, after only a few minutes passed, the young man had already started writing.
Forgive him as a computer scientist, but he is really not very good at mathematics. At least compared to these mathematicians, there is a saying that only those who are truly good at mathematics can stay rooted in pure mathematics. People who are not good at mathematics will change careers early.
But, Xiao Yi's "mathematics is very good" is a bit better than he imagined.
"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 guarantee that at least two of all z are not equal, otherwise x will be re-selected and p will be calculated..."
"Alice relabels the prime number p and [z1, z2,..., zi, zi+1]..."
"Bob checks zj`. If zj`=x mod p, it means i is greater than or equal to j, otherwise i is less than j."
He wrote the last stroke on the blackboard, then Xiao Yi went back and read it again from beginning to end. He nodded with satisfaction, then put down the pen, turned his head and said to the three people next to him.
"In this way, the comparison of the wealth of Alice and Bob is completed. Except when the wealth of both parties is equal, they did not reveal their true wealth to the other party."
"Professor Kleinrock, my method should be possible, but I don't know how to write the specific program." Xiao Yi said modestly, "I don't understand programming at all."
However, what puzzled him was that the three professors in front of him were all slightly silent at this time.
After a while, Kleinrock asked Terence Tao in a low voice, "Are all of you people who are good at mathematics like this? Have you solved a difficult problem in our computer field as... addition and subtraction within 100?"
Terence Tao waved his hands repeatedly. "Well... that's not true, but you know, there will always be some special situations in this world, and some small probability events will happen. Xiao Yi is such a small probability event, you know."