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cryptography and useless mathematics. part 1

Knowledge of logarithms and integrals helped few people in everyday life. We perfectly calculate the interest on a mortgage or the size of a discount in a store without it.

Perhaps you came across teachers at school who were more consciously able to explain the importance of this science: gymnastics for the brain, broadening your horizons, understanding the mechanics of the world around you, and so on. If you didn’t come across such teachers or you didn’t believe them, we hope we can convince you today. But be careful: there will be mathematics, probably even too much mathematics. But not terrible.

Useless math

The practical usefulness of mathematics is often not obvious in itself. Sometimes it is not obvious even to mathematicians themselves. Take at least number theory: this is a branch of mathematics in which they study the magical interaction of numbers and notice interesting patterns. However, it is absolutely unsuitable for practical use.

Understanding the uselessness of mathematics is impossible without mathematics itself. Therefore, you can either take our word for it, or attempt to figure out the mechanism of action yourself. Let's conditionally call such mathematical or not so mathematical mechanisms black boxes: we will try to explain each of such simple black boxes that fit into the article in an accessible language.

Pierre Fermat was a cunning guy and left behind theorems that other mathematicians had to prove. For example, in the black box of Fermat's Little Theorem is the statement:

There is a number a, it is an integer, i.e. not fractional. And there is a number p, it is simple, i.e. is only divisible by 1 and itself. If a is not divisible by p, then ap-1-1 is divisible by p. Let's take the numbers 7 and 3. When dividing 7 by 3, you won't get an integer. But if you raise 7 to the power of 3-1 and subtract 1, you get 48. And 48 is perfectly divisible by 3 without a remainder.

An interesting theorem and, at first glance, absolutely useless. And especially in the opinion of mathematicians, who at one time simply did not understand how to use it. Minor spoiler: this theorem protects your information every day, you just don't know it. Don't suspect yet, but we'll get there.

Actually, this article is about cryptography

Man has always had his secrets. And if it is not difficult to retell the secret in the ear of a friend, then it is already much more difficult to convey it at a distance. The statesmen had especially many secrets: this was influenced by the conditions of the war, conspiracies and constant court intrigues. Thus, the first ciphers were invented precisely for political purposes.

The Jews invented the Atbash cipher. The encryption rule was as follows: the n-th letter in the word was replaced by the letter i-n+1. Those. the first letter of the alphabet was replaced by the last, the second by the penultimate, and so on.

The name of this cipher comes from the letters "alef", "tav", "bet" and "shin", that is, the first, last, second and penultimate letters of the Hebrew alphabet.

A little more complicated was the cipher of the Spartans - Skital. To encrypt and decrypt messages, they used special tools - cylinders of different diameters. By spiraling a narrow strip of parchment around a stick, they wrote down the message horizontally. When the parchment was unwound, the phrase was encrypted.

At any lecture on cryptography, you will definitely be told about the Caesar cipher. Sending secret messages to his associates, Caesar shifted the letters in the alphabet to an arbitrary value. For example, in Cyrillic, when shifted three positions forward, the letter "A" will become encrypted "G". Well, if the approximate knew the meaning of this very shift. Otherwise, the Roman emperor doomed his subjects to sleepless nights of unraveling, and himself to the prospect of becoming the salad of the same name.

The Caesar cipher with a shift of thirteen is currently used in the ROT13 algorithm. The number thirteen has no underlying reason: there are simply 26 letters in the Latin alphabet, and when shifted by 13, the encryption algorithm coincides with the decryption algorithm. And Sir Arthur Conan Doyle dabbled in it in his story "The Dancing Men".

All ciphers that use letter shifting are called monoalphabetic ciphers. They were good on the battlefields of Caesar against Obelix and Asterix. Today, any smartphone can decipher them by brute force, but even in those days they were solved without any computers.

Grammar betrayed us all

The secret sooner or later becomes clear. And if your secret that you are a fan of "My Little Pony" does not cause much unrest even among relatives, then revealing the army's offensive plans can be fatal. This information is very important. And, of course, there is always someone who wants to intercept them.

For example, Aristotle was brought in to decipher the Spartan Scytals. And quite successfully. He used a cone, winding an encrypted note around it and changing the diameter. Sooner or later, the words acquired meaning, and the cipher itself lost this very meaning. Other ciphers cannot be cracked this way.

In ancient times, people did not know what cryptographic frequency analysis was, but, without suspecting it, they actively used it - grammar helped them in this. Every language has its weaknesses: the most obvious example is articles in English. They are often repeated in the text and thus make it more vulnerable. Highlighting the same fragments of the cipher, we can assume that this is an article, find out the shift and use the brute force and guesswork method to get the answer.

For the secret to remain secret, you need to better encrypt, Caesar.

Encryption Strikes Back

And then the polyalphabetic cipher was invented. It uses a set of monociphers: not one shift, but several. A good example is the Vigenère cipher. It uses a special table and a code word to encrypt the text. This is how the cipher table for English looks like.

Let's encrypt the phrase "encrypted" in it. Let's think of a key. It can be any word or phrase no longer than the cipher itself: the number of letters in the key must equal the number of letters in the cipher. Let the word "geese" be the key. We repeat the letters in the key as many times as there are letters in the encrypted message. It turns out "gusigusigu". Now we encrypt. At the intersection of the first letters of our phrase and the key there will be the letter "s", at the intersection of the second letters there will also be "s", then "e" and so on. The finished cipher will look like this: “yёshchyudefvo”.

The Vigenère cipher can also be subjected to frequency analysis, but it is much more difficult to trace the patterns in it, because each letter is shifted in a random order. This suited everyone who conveys messages. The enumeration took a lot of time, and it was extremely difficult to decipher the message in time - the information simply became outdated.

Nevertheless, the endless war of encryption and decryption continued. At the beginning of the twentieth century, cryptographic fashion trends changed. Electromechanical machines began to encrypt for us. And decrypt for us too.

The most difficult and interesting thing in cryptography is just beginning.

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Policy on the Processing of Personal Data

 

1. General Provisions

This Personal Data Processing Policy is drafted in accordance with the requirements of Federal Law No. 152-FZ dated July 27, 2006, “On Personal Data” (hereinafter referred to as the Personal Data Law) and defines the procedure for processing personal data and the measures taken to ensure the security of personal data by LLC “AISIU+” (hereinafter referred to as the Operator).

1.1. The Operator considers the observance of human and civil rights and freedoms during the processing of personal data, including the protection of the right to privacy, personal and family confidentiality, to be its primary objective and essential condition for carrying out its activities.

1.2. This Policy of the Operator regarding the processing of personal data (hereinafter – the Policy) applies to all information that the Operator may receive about visitors to the website .

2.9. Personal data made publicly available by the subject of personal data – personal data to which the subject has granted access to an unlimited number of persons by giving consent to the processing of personal data made publicly available, in accordance with the procedure established by the Personal Data Law (hereinafter – personal data made publicly available).

2.10. User – any visitor of the website.

7.2. The Operator is also entitled to send the User notifications about new products and services, special offers, and various events. The User may at any time opt out of receiving such informational messages by sending an email to the Operator at hello@icu.agency with the subject “Unsubscribe from notifications about new products, services, and special offers.”

7.3. Anonymized data of Users collected through web analytics services is used to gather information about User behavior on the website, improve the quality of the website, and enhance its content.

8. Legal Grounds for Personal Data Processing

8.1. The legal grounds for the processing of personal data by the Operator are:
– Federal Law “On Information, Information Technologies and Information Protection” No. 149-FZ dated July 27, 2006;
– federal laws and other regulatory legal acts in the field of personal data protection;
– the Users’ consent to the processing of their personal data, including the processing of personal data permitted for distribution.

8.2. The Operator processes the User’s personal data only if it is provided and/or submitted by the User voluntarily through special forms available on the website . By filling out the relevant forms and/or sending their personal data to the Operator, the User consents to this Policy.

8.3. The Operator processes anonymized data about the User if this is permitted in the User’s browser settings (such as enabling the storage of cookies and the use of JavaScript technology).

8.4. The personal data subject independently decides to provide their personal data and gives consent freely, by their own will, and in their own interest.

9. Conditions for Personal Data Processing

9.1. Personal data is processed with the consent of the personal data subject to the processing of their personal data.

9.2. The processing of personal data is necessary to achieve the purposes provided for by an international treaty of the Russian Federation or by law, as well as to fulfill the functions, powers, and duties imposed on the Operator by the legislation of the Russian Federation.

9.3. The processing of personal data is necessary for the administration of justice, the execution of a court decision, or the decision of another authority or official subject to execution in accordance with the legislation of the Russian Federation on enforcement proceedings.

9.4. The processing of personal data is necessary for the performance of a contract to which the personal data subject is a party, a beneficiary, or a guarantor, or for the conclusion of a contract at the initiative of the personal data subject or a contract under which the personal data subject will be a beneficiary or a guarantor.

9.5. The processing of personal data is necessary for the exercise of the rights and legitimate interests of the Operator or third parties, or for the achievement of socially significant goals, provided that this does not violate the rights and freedoms of the personal data subject.

9.6. Personal data is processed that has been made publicly available by the personal data subject or at their request (hereinafter – publicly available personal data).

9.7. Personal data is processed that is subject to publication or mandatory disclosure in accordance with federal law.
10. Procedure for the Collection, Storage, Transfer, and Other Types of Personal Data Processing

The security of personal data processed by the Operator is ensured through the implementation of legal, organizational, and technical measures necessary to fully comply with the requirements of current legislation in the field of personal data protection.

10.1. The Operator ensures the safekeeping of personal data and takes all possible measures to prevent unauthorized access to personal data.

10.2. The User’s personal data will never, under any circumstances, be transferred to third parties, except in cases related to the fulfillment of applicable legislation or if the personal data subject has given the Operator consent to transfer the data to a third party for the purpose of fulfilling obligations under a civil law contract.

10.3. If any inaccuracies in the personal data are identified, the User may update them independently by sending a notification to the Operator at the email address hello@icu.agency with the subject line “Personal Data Update.”

10.4. The period of personal data processing is determined by the achievement of the purposes for which the personal data was collected, unless a different period is specified by contract or applicable law. The User may withdraw their consent to the processing of personal data at any time by sending a notification to the Operator via email at hello@icu.agency with the subject line “Withdrawal of Consent to Personal Data Processing.”

10.5. All information collected by third-party services, including payment systems, communication tools, and other service providers, is stored and processed by those parties (Operators) in accordance with their own User Agreements and Privacy Policies. The personal data subject and/or User is solely responsible for reviewing such documents in a timely manner. The Operator is not responsible for the actions of third parties, including the service providers mentioned in this clause.

10.6. Any restrictions established by the personal data subject on the transfer (except access), processing, or conditions of processing (except access) of personal data permitted for distribution do not apply in cases where personal data is processed for state, public, or other socially significant interests as defined by the legislation of the Russian Federation.
10.7. The Operator ensures the confidentiality of personal data during processing.

10.8. The Operator stores personal data in a form that allows identification of the personal data subject for no longer than is necessary to achieve the purposes of personal data processing, unless a longer storage period is established by federal law or a contract to which the personal data subject is a party, beneficiary, or guarantor.

10.9. The grounds for termination of personal data processing may include the achievement of the processing purposes, the expiration of the data subject’s consent, the withdrawal of consent by the personal data subject, or the identification of unlawful processing of personal data.

11. List of Actions Performed by the Operator with Collected Personal Data

11.1. The Operator carries out the collection, recording, systematization, accumulation, storage, clarification (updating, modification), retrieval, use, transfer (distribution, provision, access), depersonalization, blocking, deletion, and destruction of personal data.

11.2. The Operator performs automated processing of personal data with the receipt and/or transfer of the obtained information via information and telecommunication networks or without such transfer.

12. Cross-Border Transfer of Personal Data

12.1. Before initiating the cross-border transfer of personal data, the Operator must ensure that the foreign country to which the data is to be transferred provides reliable protection of the rights of personal data subjects.

12.2. Cross-border transfer of personal data to foreign countries that do not meet the above requirements may only occur if there is written consent from the personal data subject for the cross-border transfer of their personal data and/or for the execution of a contract to which the personal data subject is a party.

13. Confidentiality of Personal Data

The Operator and other individuals who have access to personal data are obligated not to disclose or distribute personal data to third parties without the consent of the personal data subject, unless otherwise required by federal law.

14. Final Provisions

14.1. The User can obtain any clarifications regarding the processing of their personal data by contacting the Operator via email at hello@icu.agency.

14.2. Any changes to the Operator’s personal data processing policy will be reflected in this document. The Policy is valid indefinitely until replaced by a new version.

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