In the summer of 2020, traveling to Port Ravit, the capital of the country.
"The Jade Khan people are particularly cunning, and their intelligence bureau is full of talents. It is not easy to deceive them." Loew looked at Kezi who nodded frequently, turned around and continued to report to Shamron II:
"After discussing with Kezi, combined with his characteristics as a standard science and engineering man, we formulated a plan to lure the Yuhan people to the bait."
"If we send Kezi abroad, it is obviously inconsistent with his expertise and the nature of his work, and it is difficult not to be suspected. But in Port Ravi, if he is incited to rebel face to face, it is simply a fantasy. Therefore, we hope to lure Yuhan People are instigating rebellion against Ketz through the Internet." Loew continued his analysis:
"Katz is one of the core designers of the 'Copper Wall' missile defense system. We believe he must be on the Yuhan people's attention list. Katz was recently punished, and the Yuhan people must also have noticed."
"Do you want the Yuhan people to take the initiative to contact Kezi on the dark web?" Shamron II asked.
"Yes, I am a programmer and a science enthusiast. I operate anonymously on the dark web, but I intentionally allow interested people to crack and trace my home IP address." Katz said confidently.
"We plan to have Kezi go to a forum where programmers do mental gymnastics. Kezi will take the initiative to post brain-burning posts. I believe there will be Jade Khanate agents in the comments," Loew said.
"Well, then you select the person to chat with in the thread, and then find the target through chat?"
The three of them looked at each other and smiled.
The so-called dark web does not refer to a specific website, but a general term for networks that are anonymous and difficult to trace.
Usually senior players are network masters, even hardcore players at the hacker level.
The dark web is of course the first choice for those who want to gamble, gamble, drug, or buy murderers and guns.
But not all people on the dark web are bad people. Many programmers who are under high work pressure and fearful of society like to post some brain-burning questions there, waiting for experts to crack them.
Loew watched Katz skillfully type on the keyboard, defining his residential IP address as the bottom layer, and then performing layer upon layer encryption.
After Katz wandered around, he finally opened the dialog box and entered his screen name:
"A quartet of log(n)-Fermat tests".
Katz stood up from the chair leisurely and walked to the window to open it for some fresh air.
Loew looked puzzled and asked:
"How come you just enter the name and don't get a puzzle? How can you get people hooked?"
Katz smiled mysteriously and said:
“My screen name is Puzzle, just wait.”
Jade Khanate Plateau City
Hamid called Bashir and Roxana and assigned a task:
"Ketz, one of the core designers of the 'Copper Wall' defense system in Chicago, was punished for a mistake and is probably dissatisfied. The intelligence center found that he logged into a darknet today. You can chat with him anonymously. , test it out.”
"The core designer of the 'Copper Wall' system? No matter how wronged the people of Luzhi are, they won't be able to surrender to us, right?" Bashir shook his head in doubt.
"I don't think it's possible, but it's abnormal for technical agents from the country of Chicago to access the darknet. We're not that easy to deceive. Let's analyze it while chatting." Rosanna nodded in agreement with Bashir's opinion, and then asked Hami De said:
"Dad, give us the URL link, his online name, and chat history."
"There is no chat record, only one online name, log(n)-Fermat's test quartet." Hamid couldn't help laughing.
"Interesting, Bashir, this is your strong point. It should be a puzzle about number theory, right?" Rosanna blinked at Bashir and looked at him expectantly.
Bashir was thinking and explaining to Roxana.
Fermat is a famous amateur mathematician. He is remembered and familiar to the world, mainly because of his seemingly simple Fermat's Last Theorem, which has puzzled the mathematical community for nearly 300 years and was not proven until 1995.
Although Fermat's Little Theorem is not as well-known, its contribution to number theory and cryptography is no less important. It can be said to be the basis for the study of prime numbers.
All prime numbers satisfy Fermat's Little Theorem, but conversely, integers that satisfy Fermat's Little Theorem are not necessarily prime numbers. These integers that are not prime numbers are called pseudoprimes.
Modern cryptography is inseparable from prime numbers. Cryptographers can use two large known prime numbers A and B at will, and the product C can be easily obtained.
The person sending the password only needs to issue C, which is the so-called "public key" we are familiar with.
Anyone who intercepts C wants to know A or B, unless he has a codebook, otherwise he will need to use a very large amount of calculations to perform difficult integer decomposition.
When C is large enough (such as 2^1024), integer decomposition requires months or even years of calculation time, thus achieving the purpose of confidentiality.
In order to ensure that A and B are prime numbers (otherwise, the difficulty of decomposition will decrease exponentially), the problem of prime number determination has become an urgent topic in number theory and cryptography research.
There are many ways to use a computer to check whether a large integer n is prime. The goal of either method is to shorten the inspection time as much as possible.
The integer n used in cryptography is very large. Even if a computer is used, the number of calculations cannot be related to n (the number of digits will crowd out the memory), and can only be related to log(n) at most.
In 2002, three mathematicians proved that within polynomial time log^12(n), later optimized to log^7.5(n), a deterministic primality test can be performed on any integer n.
This test method is named AKS test method after the first letters of the last names of three mathematicians.
Unfortunately, this inspection method consumes too much computer memory and cannot be used on a computer. It can only stay at the paper level.
Currently, for cryptography used in military, communications, and finance, the underlying primality testing program uses the probability testing method.
The more popular algorithm is the composite algorithm based on the Miller-Rabin test.
Because there are too many Fermat pseudoprimes, Fermat's little theorem cannot be used for primality testing only for encryption.
Bashir's introduction made Hamid drowsy. He quickly stopped talking, pointed at the strange online name and said:
"As part of number theory research, some mathematics enthusiasts still use the Fermat test to explore the extremely interesting properties of integers. For example, I once saw an interesting conjecture." Bashir continued:
"Round down any integer n from binary to log(n) and perform Fermat's test. In addition to the Carmichael number, the pseudo-prime numbers that can pass the test must have n=(a+1)(2a+ 1) form.”
"Some enthusiasts posted on the Internet and announced 47 pseudo-primes within 2^64, all of which satisfy the above conjecture."
"The smallest n=242017633321201=11000401×22000801."
"Are the two factors of these 47 numbers prime?" Rosanna asked curiously.
"You have reached the key point. According to conjecture, a+1 can be a prime number or a composite number. If I remember correctly, 46 of the numbers have only two prime factors, and only one n of a+1 is a three-factor composite. Number, 2a+1 is a prime number, and this n is a composite number composed of four prime factors."
Rosanna finally understood and asked:
"Quartet refers to four prime factors? Perform Fermat's test for all integers less than 2^64, and the carry system is from 2 to log (n). The only pseudo-prime number that can pass the test is a four-factor composite number that is not a Carmichael number. What is the smallest four-factor composite number that satisfies the conditions?"
Bashir opened his computer, found a table containing 47 numbers from his favorites, and copied the only four-factor pseudo-prime number on the blackboard:
n=168562580058457201=103×307×9181×580624801
Among them, a+1=103×307×9181=290312401.
"This is the quartet of log(n)-Fermat tests!" Bashir said proudly.
Hamid looked at Bashir with approval and asked:
"The content of your reply to that Kezi was these four numbers, right?"
Bashir nodded in approval, and Roxana said thoughtfully:
"Replying these four numbers only solved the riddle he created. In order to keep the chat going, we also need to come up with an online name with its own riddle to test him."
"This is interesting." Bashir left the screen name input field blank and entered the chat content below:
"103, 307, 9181, 580624801"
Bashir pushed the keyboard to Roxana and made a playful "please" gesture. Rosanna thought for a while and entered in the screen name field:
“O(√n ln(n))-the triad of Riemann’s hypothesis.”
