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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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1. An overview of data protection


The following gives a simple overview of what happens to your personal information when you visit our website. Personal information is any data with which you could be personally identified. Detailed information on the subject of data protection can be found in our privacy policy found below.

Data collection on our website

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How do we collect your data?

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What do we use your data for?

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2. General information and mandatory information

Data protection

The operators of this website take the protection of your personal data very seriously. We treat your personal data as confidential and in accordance with the statutory data protection regulations and this privacy policy.

If you use this website, various pieces of personal data will be collected. Personal information is any data with which you could be personally identified. This privacy policy explains what information we collect and what we use it for. It also explains how and for what purpose this happens.

Please note that data transmitted via the internet (e.g. via email communication) may be subject to security breaches. Complete protection of your data from third-party access is not possible.

Notice concerning the party responsible for this website
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Telephone: +66815434565
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3. Data protection officer

Statutory data protection officer

We have appointed a data protection officer for our company.

Alexander Rusin
Telephone: +66815434565
E-Mail: hello@icu.agency
Company registration number: 0845566007085

4. Data collection on our website


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Google Analytics cookies are stored based on Art. 6 (1) (f) DSGVO. The website operator has a legitimate interest in analyzing user behavior to optimize both its website and its advertising.

IP anonymization

We have activated the IP anonymization feature on this website. Your IP address will be shortened by Google within the European Union or other parties to the Agreement on the European Economic Area prior to transmission to the United States. Only in exceptional cases is the full IP address sent to a Google server in the US and shortened there. Google will use this information on behalf of the operator of this website to evaluate your use of the website, to compile reports on website activity, and to provide other services regarding website activity and Internet usage for the website operator. The IP address transmitted by your browser as part of Google Analytics will not be merged with any other data held by Google.

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For more information about how Google Analytics handles user data, see Google’s privacy policy: support.google.com

Outsourced data processing

We have entered into an agreement with Google for the outsourcing of our data processing and fully implement the strict requirements of the German data protection authorities when using Google Analytics.

Demographic data collection by Google Analytics

This website uses Google Analytics’ demographic features. This allows reports to be generated containing statements about the age, gender, and interests of site visitors. This data comes from interest-based advertising from Google and third-party visitor data. This collected data cannot be attributed to any specific individual person. You can disable this feature at any time by adjusting the ads settings in your Google account or you can forbid the collection of your data by Google Analytics as described in the section “Refusal of data collection”.

Google Analytics Remarketing

Our websites use the features of Google Analytics Remarketing combined with the cross-device capabilities of Google AdWords and DoubleClick. This service is provided by Google Inc., 1600 Amphitheater Parkway, Mountain View, CA 94043, USA.

This feature makes it possible to link target audiences for promotional marketing created with Google Analytics Remarketing to the cross-device capabilities of Google AdWords and Google DoubleClick. This allows advertising to be displayed based on your personal interests, identified based on your previous usage and surfing behavior on one device (e.g. your mobile phone), on other devices (such as a tablet or computer).

Once you have given your consent, Google will associate your web and app browsing history with your Google Account for this purpose. That way, any device that signs in to your Google Account can use the same personalized promotional messaging.

To support this feature, Google Analytics collects Google-authenticated IDs of users that are temporarily linked to our Google Analytics data to define and create audiences for cross-device ad promotion.

You can permanently opt out of cross-device remarketing/targeting by turning off personalized advertising in your Google Account; follow this link: https://www.google.com/settings/ads/onweb/?hl=en.

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We will, therefore, process any data you enter onto the contact form only with your consent per Art. 6 (1) (a) DSGVO. You can revoke consent to the storage of your data and email address as well as their use for sending the newsletter at any time, e.g. through the “unsubscribe” link in the newsletter. The data processed before we receive your request may still be legally processed.

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Data processing is based on Art. 6 (1) (a) DSGVO. You may revoke your consent at any time by unsubscribing to the newsletter. The data processed before we receive your request may still be legally processed.

The data provided when registering for the newsletter will be used to distribute the newsletter until you cancel your subscription when said data will be deleted from our servers. Data we have stored for other purposes remains unaffected.

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