Putting down the paper in his hand, Xu Chuan quietly looked at the title on the homepage, recalling the entire reading process.
For people like him, seeing a good paper in a new field is no less than eating a delicacy that has never been enjoyed by ordinary people, which is enough to last a lifetime.
The polynomial decomposition problem of large positive integer factors undoubtedly meets this standard.
In fact, the factorization problem of large numbers is one of the most basic and oldest problems in mathematics, and it is still one of the problems that people pay attention to but cannot be completely solved.
Its importance and difficulty in the field of number theory are not weaker than the existence of the Yang-Mills equation in the field of partial differential equations.
Because large integers may be prime numbers or composite numbers, the prerequisite for solving this problem is to first judge the given large number, determine whether the given number is a prime number (i.e., the primality determination problem) and decompose the large composite number
There are two aspects to decomposing large numbers into prime factors.
In mathematics, it is very similar to the qualitative detection problem, but qualitative detection has been completely proved to be solvable in polynomial time, while the problem of factoring large numbers is still unsolved.
Even, for hundreds of years, the problem of factoring large numbers has not been proven to be a P problem solvable in polynomial time, nor has it been proven to be an NP-complete problem.
However, in the paper in front of him, Xu Chuan saw a detailed answer, or in other words, a path leading to one of the ultimate questions in number theory.
.......
After carefully reviewing the paper in his hand, Xu Chuan opened his eyes, dragged the computer from the corner of the desk, and clicked on the prestige chat box.
"I have read the paper once, and it is very good!"
His fingers tapped lightly on the keyboard, and a compliment was transmitted across the computer screen thousands of kilometers away.
This was not against his will, but a sentiment from the bottom of his heart.
Although he had known for a long time that she was very talented in mathematics and computers, he never thought that one day she would be able to enter this field.
In academia, or on the Internet, when people discuss a subject, if it has high research value and practicality in some aspects, is difficult enough to learn, and has certain difficulties in the job market,
It will be called "Tiankeng Professional".
These majors are usually considered to be basic subjects, which are difficult to learn, and their employment prospects and salary packages are often not as good as other majors.
For example, the four most common "biochemical environmental materials" sinkholes.
However, many times, the most basic mathematics major in the natural sciences is rarely recorded, or few people call it a sinkhole major.
It's not that it's not difficult enough, it's that it's too difficult.
If other majors are a sinkhole, you can see that there are many people (scholars) at the bottom of the sinkhole struggling to climb up.
The mathematics major is like a cliff. There is no bottom below, and the clouds and mist are so deep that there is no echo even if you throw something. You can’t see how deep it is, and you can’t see how many people are inside. You can only see a few.
Countless bulls were flying around above the clouds and mist close to the top of the cliff...
In the words of the mathematical world, these giants flying above the clouds and mist are all gods in the mathematical world.
Xu Chuan himself is the one who flies the highest.
Now, after solving the polynomial algorithm problem of positive integer factorization, Liu Jiaxin has also leapt from the abyss of mathematics to the top of the clouds.
Although this is not a complete solution to the millennium problem of P=NP?, it is only one of the phased results, but its difficulty and influence on the world are extremely great.
Because, in addition to being an important issue in mathematics and computing theory, any kind of proof will be of great significance to mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, and even philosophy, economics and many other fields.
have a profound impact on the field.
To switch to another area that arguably touches everyone: “Passwords!”
Nowadays, whether it is a mobile phone, a computer, or an email that requires information exchange, or anything that involves account security, it all involves the existence of passwords.
In computer cryptography, currently, the most important public key algorithm is RSA.
It is the cornerstone of computer communication security, ensuring that encrypted data cannot be decrypted. RSA encryption is asymmetric encryption and can complete decryption without directly transmitting the key.
Simply put, it is a process of encryption and decryption using a pair of keys, called public key and private key respectively.
Assumption: Party A and Party B communicate with each other. Party B generates a public key and a private key. Party A obtains the public key and encrypts the information (the public key is public and can be obtained by anyone). Party A uses the public key to encrypt the information
.
Only the private key can be cracked, so as long as the private key is not leaked, the security of the information can be guaranteed.
Therefore, it is widely used in various fields, and its security depends on the difficulty of decomposing large integers.
When all the factors of the composite number are very large, it is very difficult to obtain the specific factors using brute force, and this is the core of the RSA system theory.
However, after solving the polynomial algorithm problem of large positive integer factorization, the algorithm of the RSA encryption system can quickly collapse into a 'solution' after finding a method.
What this means is naturally self-evident.
Of course, this is only theoretical. In fact, it is impossible to treat encryption algorithms such as RSA as nothing, even with this paper.
Perhaps when quantum computers mature in the future, and then cooperate with this paper, it will probably truly dominate the field of traditional computers.
As for now, I can only say that it still needs time to ferment.
But one can imagine how much impact this paper will have on the entire world. Computer communication codes alone will usher in a complete change.
Those encryption methods based on traditional positive integer factorization may be abandoned and replaced by various countries.
After all, it's no longer theoretically safe.
This chapter is not over, please click on the next page to continue reading!......
Late at night, in the study, the click of authority sounded softly. After sending a message, Xu Chuan made a video call.
After waiting for a while, the video was connected. On the opposite side, Liu Jiaxin, who was also in the study, appeared on the phone, revealing a slender swan neck and pale white pajamas.
Looking at the senior sister on the opposite side of the video, Xu Chuan's eyes naturally fell on the exposed skin that was whiter than her pajamas. He was stunned for a moment and forgot to speak.
Although the two of them often interacted with each other because of work and mathematical matters, the two of them basically met during the day, and there was no such time as seeing each other in pajamas.
Opposite me, Liu Jiaxin noticed Xu Chuan's gaze, and then realized that she was wearing pajamas at home. She pursed her lips and adjusted the buttons of her coat in embarrassment.
"Cough~"
Xu Chuan came back to his senses, coughed slightly and said: "I have read the paper in detail, and so far, it is very excellent! Although I cannot say for sure that you have completely solved this problem, after all, it has not yet been solved
It's been peer-reviewed, but if you asked me to give my opinion, there's no doubt that you did it."
"Thank you." On the other side of the video call, Liu Jiaxin said with a smile: "Sorry to trouble you, I'm still asking you to help me so late."
"No, no, no, don't say that!"
Hearing this, Xu Chuan quickly shook his head and said: "This is not a trouble. If it is, then I hope there will be more trouble like this!"
For a mathematician, if he can see such a paper, let alone not sleeping, even if he is woken up by someone while sleeping, he will not have any opinions. He failed to read it in the first place.
That would feel like a pity.
Of course, for a girl, this may not be a standard answer.
But it was obvious that neither of them were paying attention to anything other than academic matters at this moment. Both of their thoughts were focused on the paper in their hands.
"... make in-depth changes to the quadratic sieve factorization method and introduce the Hamiltonian graph determination method and polynomial function algorithm, so that the problem of the existence of complex zero points can be converted into a linear equation system solution problem,
Then the complexity of the algorithm for determining the existence of complex solutions to the system of equations f1 = 0,..., fk=0 is given."
"...According to Fermat's little theorem, if p is a prime number, then a^(p-1)≡1(mod p) holds for all a∈[1,n-1]. So if in [1
.n-1], randomly pick one out, and find that it does not satisfy Fermat’s little theorem, then prove that n must be a composite number.”
"..."
During the video call, Liu Jiaxin explained that the positive integer factorization has the core and ideas for solving polynomial algorithm problems, while Xu Chuan asked some questions of his own from time to time across the screen.
Although the paper has completely described the proof process of the polynomial algorithm problem of the factorization of large positive integers, reading the paper alone and listening to the creator's explanation based on the paper are two completely different concepts.
If all the problems could be understood by reading the papers, then the mathematical community would not require provers to give lectures after solving these world-class conjectures.
Time ticked by in the middle of the night, and it wasn't until after midnight that the two of them stopped.
In the study, Xu Chuan's eyes were bright with some thoughts. After pondering for a moment, he came back from his distraction, looked at Liu Jiaxin on the other side of the video call, and said with a smile:
"It is an excellent proof that sublimating the quadratic sieve factorization method, introducing the Hamiltonian graph determination method and the polynomial function algorithm while twisting and collapsing large integers, this can be said to be a new mathematical tool. Based on the previous work
Yes, you did an even better job than I imagined."
On the opposite side, Liu Jiaxin pursed her lips and shook her head slightly, saying: "But I can't find a method that can transform NP problems into P problems, nor can I solve NP problems and NPC problems."
Looking at the senior student opposite, Xu Chuan smiled and said jokingly: "Thinking of solving P=NP at once? Guess? You are too greedy."
After a slight pause, he continued: "Among the P=NP? problems, the problem of polynomial decomposition of large positive integer factors is itself one of the two most difficult problems. If you can solve this, the remaining problems may not be far away from you.
It’s not very far away.”
Opposite me, Liu Jiaxin thought for a while, hesitated and then said: "But I think this problem is still far away, maybe it will never be solved."
Hearing this, Xu Chuan paused, raised his eyebrows in surprise, and asked, "Do you think P≠NP?"
Although he has not studied this problem for a long time and with full concentration, he has naturally explored the few remaining conjectures among the seven millennium problems.
Although it is not very in-depth, to be honest, his view on this issue is not that P=NP, but P≠NP.
That is, there is no simple key that can solve all the problems in this world.
This can be regarded as his implicit mathematical intuition.
Even after reading the proof of the polynomial decomposition problem of large positive integer factors tonight, which showed that P=NP was a big step forward, he still retained his own opinion and felt that P≠NP.
Of course, Xu Chuan never believed that his opinion on an unresolved issue must be right.
After all, he is just a person who has learned a little more knowledge than ordinary people. He is not an omniscient and omnipotent god.
But on the P=NP? problem, or on P-type problems and polynomial decomposition problems of large positive integer factors, the senior student in front of me should be one of the people who have gone the furthest so far, or in other words, the person who has gone the furthest.
.
What if she thinks P=NP? The conjecture may be incorrect. Combined with the opinions of most people in the mathematical community and his own intuition, maybe P=NP does not exist.
That is to say, NP-type problems can never collapse into P-type problems.
Some people may wonder that since the polynomial decomposition problem of large positive integer factors has been confirmed, why is P not equal to NP? Shouldn't it be a step closer to P=NP?
For this problem, can we only say that P=NP? The conjecture itself is not a completely defined mathematical problem.
Among the seven millennium problems of the Clay Mathematics Institute, it is called the 'Non-deterministic Polynomial problem, that is, the non-deterministic problem of polynomial complexity.'
P=NP? In the conjecture, P and NP on both sides are not fixed. It targets endless polynomial and non-deterministic problems. In this case, it is not easy to prove that P≠NP.
If P=NP, you need to ensure that every NP-type problem can collapse to a P-type problem. If P≠NP, then you need to prove that every potential algorithm will fail.
The algorithms and problems here do not only refer to the present, but also include everything in the past and future.
So rather than saying that the P=NP? problem is a mathematical conjecture, it is better to say that it is a way of thinking, a method of classifying and understanding problems based on their inherent difficulty.
.......
Opposite me, Liu Jiaxin nodded and said softly: "Well, maybe this problem has no solution. We can neither prove P=NP nor P≠NP."
"I tried to solve an NP-complete problem in the past, but found that it was impossible to find an algorithm that could solve the problem in all situations. I could only try my best to achieve the best results."
Xu Chuan nodded and said with a smile: "It seems that we have reached a consensus."
Smiling, he leaned back in his chair and continued: "If we talk about the problem alone, it is not just the P=NP? problem, there are many problems that are the same, and often we cannot solve it directly. But
Many times, the process of studying them is the most essential thing."
"For example, now, the problem of polynomial decomposition of large positive integer factors has given us a general framework and tools, which helps us think about how to deal with difficult problems arising from actual needs, and can also help us better improve mathematics.