No one is interested in my tax return, there is no big deal. It is a good thing that an attacker was quite easily perceived when I printed encrypted communication between my laptop and printer in recent years.
Introduction to Information Technology Security Hanno Böck researchers in the process of cracked some of these encryptions in the process 2023 Prepress paperPublished a CryptoGic Research International Association‘s Cryptology EPRINT Archive. Century Pierre de Fermat developed by French wisers.
Fermat: The most famous for her mysterious “Last theorem“Experts in decades allowed the world of science, helped them in the world of science in his life. For example, he put the foundations for the theory of probability, as well as those values that only distribute themselves.
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Mathematicians could use Fermat’s work to break down encryption, and Böck showed that case.
Complex security issues
Modern encryption systems are based on difficult math problems. They work like a padlock: Problem (Lock) Unable to fix (key). The ordinary procedure is called RSA cryptographythat is associated with the prime numbers. It is difficult to decompose large numbers removed by large numbers, making it useful keys.
The main numbers are mentioned in the atoms of splitters of construction blocks that are built in natural numbers. Any other primer number can be written as a unique product, such as 15 = 3 × 5 or 20 = 2 × 2 × 5. For small values, it is easy to determine the first divisors. But what about, say, 7.327.328.314? So far, no computer programs quickly calculate the main divisors of large numbers of arbitrary numbers.
This limit is that RSA exploits cryptography. To understand how this type of protocol works, consider simplified examples, where the RSA is used with the help of large amounts of data encrypted data. Suppose that a person wants to send the word science, which is made up of seven letters, an encrypted form to a recipient. To do this, they use a large number of seven digits, such as 6,743,214 and each letter of science according to the corresponding digit. The end result is an encrypted word CJMHPDI. One sender can send another person to another person capable of decoding the message.
The recipient, however, should be able to determine the science of the original words, with a key calculation (6,743,214) or key. As the first is always endangered, an attacker maintained communication between the two parties, thus providing key-rsa cryptographic mode to reconstruct safe mode. The basic idea is a secret message, creating the key to the information available before sending the sender and receiver. Safety ensures that the sender and recipients use each secret to large numbers, which are multiplied together and send the results of this calculation to each other. EavesDropper needs the first number to create the key. But because of this person can only detect products and cannot be factored, because it is not helpless. (The real RSA protocol of the key creation is more difficult, but that is the general idea behind it).
Fermat Factorization
Nearly four centuries ago, he was working on problems related to Fermath. He wanted to know how to factorizing numbers in its first ingredients. He did this from mathematical curiosity, at the time, no cryptographic method of exchange key exchange.
In fact, Fermat found a way to factorize large numbers that are the product of two first products. Its method is not complex; You can do it with a calculator (even Fermat, unexpectedly, did not one). To surprise his contemporaries, Fermat showed the method using the example number n = 2.027.651.281.
Fermat Factorization works as follows: You take the number n, In this case 2,027,651.281, and take its root. In general, this will lead to a strange value, as is here: √2.027.651.281 ≈ 45.029.45. You go round to reach 45,030. This number is box, and the original value n It is removed from the result: 45,0302 – 2.027.651.281 = 49.619. Now you need to check whether the result is square square. As it happens, 49,619 is not square.
So you continue. Start again with rounded root 45,030, add 1 and then square result to remove the original value n From there, this is 45,0312 – 2.027.651.281 = 139,680-and check if the result is square square. Again, this is not like that.
So you repeat the whole thing. This time 2 to 45,030 and square result, because you remove the original value n: 45,0322 – 2.027.651.281 = 229.743. Again, this is not the number of squares.
Fermat had to be a great patience. In his example, you must perform the procedure 12 times until the square number is found: 45.0412 – 2.027.651.281 = 1.040.400 = 1.0202.
And how does this help help? In the previous equation, a number of squares y2 (In this case 45.0412) minus n It has another number x2 (In this case, 10.202). Equation y2 – n = x2 can reorganize y2 – x2 = n. The left side corresponds to an equation known as the third formula binomial (y – x) · (y + x) = n. This automatically factorizes the number n In two numbers y – x and y + x. With example n = 2,027.651.281, therefore, the two factors 45,041 – 1.020 = 44,021 and 45,041 + 1.020 = 46,061. Both are the main numbers.
Attacking the printer
In fact, this method of factoring is always odd n. Computers can do enough quick enough if two factors n They are not far away. And that, in fact, found Böck in a library that used several companies at the time. The first numbers created for encryption were not enough random, and the program often selected two default numbers next to each other. This means that Fermat’s factoring method can be used to prevent encryption.
Böck realized that some companies used inappropriate encryption. They used RSA cryptography, such as a network of confidential documents sent to the printer. After finding in 2022, these companies were issued Alerts and Repairs to deal with the problem. We can expect other companies close to safety gaps.
In any case, many companies will need to rethink the regulations of encryption in the coming years. Even if the ordinary computers are not large numbers, Will be different with strong quantum computers. Fermato had never dreamed of more than 380 years after the discovery, for their calculations can be used by computers based on the complicated principle of quantum mechanics.
This article originally appeared Science spectrum and reproduced with permission.
