Cryptographic algorithms used to ensure information security when interacting on the INTERNET.

Cryptographic algorithms used to ensure information security when interacting on the INTERNET.

Alexey Alekseevich TERENIN, PhD in Engineering Sciences

CRYPTOGRAPHIC ALGORITHMS USED TO ENSURE INFORMATION SECURITY WHEN INTERACTING ON THE INTERNET

A brief overview of the most common encryption algorithms is presented on today, their description, as well as problems that arise during their implementation and significant aspects in practical use.

Protection of information by cryptographic transformation methods consists of changing its constituent parts (words, letters, syllables, numbers) using special algorithms or hardware solutions and key codes, i.e., bringing it to an implicit form. To get acquainted with encrypted information, the reverse process is used: decoding (decryption). The use of cryptography is one of the common methods that significantly increases the security of data transmission in computer networks, data stored in remote memory devices, as well as when exchanging information between remote objects.

For transformation (encryption), some algorithm or device implementing a given algorithm, which may be known to a wide range of people, is usually used. The encryption process is controlled by a periodically changing key code, which ensures an original presentation of information each time when using the same algorithm or device. Knowing the key allows you to decrypt the text simply and reliably. However, without knowing the key, this procedure may be practically impossible even with a known encryption algorithm.

Even a simple transformation of information is a very effective means of hiding its meaning from most unskilled intruders.

A Brief Historical Review of the Development of Encryption

The origins of cryptography go back to Egyptian hieroglyphs. Since ancient times, when Egypt and Persia flourished, messengers were used for the most important state and military missions, who carried the text of the message either on parchment or in their heads to convey it verbally, the latter method was more preferable. Even then, more or less successful methods of protecting transmitted information from the encroachments of interceptors appeared. Let us cite a well-known legend from the Ancient World. A certain king, having been captured, made a tattoo on the head of a slave — a message to the allies. When the hair grew back, the slave moved to the recipients of the message and the king was freed. The prototype of today's steganography.

The ancient Greeks used round sticks of the same diameter, on which strips of parchment were wound. The inscription was made longitudinally along the length of the stick. It was possible to fold the text into a readable one only if you had a stick of the same diameter.

In Ancient Rome, the science of cryptography, translated from Latin as secret writing, was already clearly beginning to form. The Caesar cipher appeared, when each letter was replaced by a letter three letters away in the alphabet.

In medieval intriguing Europe and Central Asia, there was a rapid development of cryptography and cryptanalysis – methods for breaking encrypted texts. The first systematic work on cryptography is considered to be the book of the architect Leon Battisti Alberti (1404 — 1472). One of the first cryptanalysts was François Viète (1540 — 1603), at the court of the King of France Henry IV. At the same time, advisers from the Adgenti family, who can also be called cryptanalysts, served at the court of the Pope. The entire period up to the middle of the 17th century is full of works on cryptography and cryptanalysis.

In the 19th and first half of the 20th centuries, for secret diplomatic correspondence, many countries, including Russia, use encryption methods, the keys for which were made up of excerpts from certain texts of ordinary books (cipher books).

Since the beginning of the 20th century — with the First World War — special encryption machines have begun to be used.

The German Enigma machine, the code of which was broken by the British, is widely known. In order not to reveal the fact that the German cipher had been broken, the British government made great sacrifices among the civilian population by not warning the residents of two large cities about the impending bombings. But this later helped to gain a significant advantage in the northern naval battles with Germany, when the invincible German submarines and cruisers were destroyed.

After World War II, computers took up cryptography. For a long time, it was the domain of the most powerful supercomputers of their time.

Publications on this topic were strictly classified and the use of scientific research in this area was a national prerogative. Only the textbook work of Von Neumann from the 1940s was publicly available, describing, in addition to the principles of building computing systems, some other possible malicious methods of influence to disrupt the “legal” computing process, as well as the classic work of Shannon, which laid the foundations of computer cryptography.

Since the 70s, open publications have appeared: Heffey-Dillman in 1976. In 1970, there was a secret invention of James Ellis (Great Britain) in the field of cryptography. The most famous algorithm of asymmetric cryptography is RSA, developed by Ronald Rivest, Eddie Shamir and Len Edleman in 1977. The RSA algorithm is of great importance, since it can be used both for public key encryption and for creating an electronic digital signature.

This was a revolutionary period in the development of cryptographic science. The latest methods of secret distribution of key information in open computing systems appeared, and asymmetric cryptography was born.

But even after that, for a long time, the prerogative of using cryptography in data protection was with government services and large corporations. The computing equipment of that time, which had the power necessary for cryptographic transformations, was very expensive.

At that time, the main state standards of cryptographic algorithms appeared (USA and some European countries), the use of which was prescribed when working with information classified as a state secret.

The veil of secrecy surrounding these technologies even led to the fact that in the United States cryptographic algorithms were equated with weapons, a ban was introduced on the export of encryption hardware and software. Then export restrictions were introduced on the length of the key used in encryption algorithms outside the United States, which allowed American intelligence agencies to decrypt messages on the available computing power without knowing the shortened key. Since March 1, 2001, export restrictions were lifted. Due to the events of September 11 of the same year, there is a tightening of government control. The US government is considering options for the return of export controls over encryption tools.

Let's go back to the 70s. Since then, neither scientific research nor the development of computing tools have stopped. The computing power of supercomputers increases several times every few years. The personal computer appears. The power of a personal computer is approximately equal to the power of a supercomputer ten years ago. Nowadays, personal computers have become even more powerful.

Since the 1980s, ordinary users have had the opportunity to use cryptographic tools on their computers, which government agencies have been fiercely preventing, and it has become more difficult to monitor the activities of citizens of the country, including criminals.

The release of PGP (Pretty Good Privacy) by Phil Zimmermann (version 1.0 was released in 1991) and its provision for open and free use gave great opportunities to ordinary computer users. Phil Zimmermann was even declared an enemy of the state, he was sentenced to imprisonment.

Ever-increasing computing power forced the use of increasingly complex cryptographic algorithms or increasing the length of the keys used in encryption.

Standards for cryptographic algorithms became obsolete and unreliable. Information protected by a certain key could no longer be kept confidential for a long enough time — as long as required by government regulations. For example, keeping information completely secret in encrypted form for 5 years meant that an adversary with the most powerful computing power, constantly trying out possible keys, would not be able to find the right key to decrypt the stored information within that period with a fairly high probability.

Competitions began to be held to crack some information encrypted using the algorithm of one of the standards. The winner was awarded a solid cash prize, as well as worldwide fame in the information community. By combining ordinary computers in a computer network to work in parallel to solve a given problem, users gathered in groups and selected the key together.

A key length of 48 bits means that 248 attempts must be made. Increasing the key length, for example, by only 16 bits, means that 216 times more must be tried.

But even such a key size allowed the problem of cracking the cipher to be solved by the combined groups in days and even hours of parallel work. Later, it was necessary to switch to keys that were several times longer than those mentioned. But this was only a temporary measure, and new standards for cryptographic transformation algorithms (AES in the USA) were recently adopted.

Currently, the press has published many publications devoted to this problem. Numerous books are published, both translated and by Russian authors. Cryptography allows solving the problem of protecting information from disclosure and modification. The mathematical apparatus of modern cryptography surpasses in complexity that used to develop nuclear weapons and space systems.

Modern cryptography is divided into symmetric and asymmetric. Symmetric — into stream cipher, block and composite. Asymmetric cryptography is more resource-intensive, and in symmetric there is a problem of effective key distribution. Modern secure exchange systems are based on the use of mixed cryptography. At the beginning of the exchange session, the parties send each other secret session keys using asymmetric cryptography, which are then used for symmetric encryption of the transmitted data. The asymmetric cryptography system allows distributing keys in symmetric encryption systems.

Government and military telecommunications systems use exclusively symmetric encryption (usually using one-time keys). This is due to the fact that the strength of public key systems has not been proven mathematically, but the opposite has not been proven either.

Encryption of information should not be taken as a panacea for all information threats. It should be perceived as one of the mandatory measures to protect information as part of a comprehensive information security system. The use of encryption should be combined with legislative, organizational and other security measures.

Symmetric encryption algorithms

Encryption algorithms are designed to solve the problem of ensuring the confidentiality of information. Currently, cryptographic methods are intensively used to close information. Since ancient times, encryption has been and remains the most effective form of protection.

Encryption is defined as the mutual conversion of unprotected (open) information into encrypted (closed) form – ciphertext [1], in which it is not presented fully accessible to an attacker. Keys are used during encryption, the presence of which means the possibility of encryption and/or decryption of information. It is important to note that the encryption method itself does not need to be kept secret, since knowledge of it alone will not allow decryption of the ciphertext.

Modern cryptosystems can be clearly divided by the method of using keys into cryptosystems with a secret key (symmetric) and with an open key (asymmetric). If the same key is used for encryption and decryption, such a cryptosystem is called symmetric.

Symmetric cryptosystems include DES [2], AES, GOST 28147-89 [3], etc. A new direction in cryptography was the invention of asymmetric cryptosystems with an open key, such as RSA, DSA or El-Gamal [4, 5].

In asymmetric cryptosystems, different keys are used for encryption and decryption, which are practically not derivable from each other, one of which (the decryption key) is made secret, and the other (the encryption key) is made public. This makes it possible to transmit secret messages over an unprotected channel without first transmitting the secret key. It was public key cryptography that broke the vicious circle of symmetric ciphers, when in order to organize the exchange of secret information it was first necessary to distribute secret keys.

Public key cryptosystems will be discussed in detail later, but now let us return to symmetric cryptosystems (CS).

The most important component of CS are ciphers [1, 4] or procedures for the mutually inverse transformation of plaintext M into ciphertext M’:

M’ = E(M),
M = D(M’),

where E is the encryption function and D is the decryption function.

The generally accepted approach in cryptography is to construct a cipher in which its secrecy is determined only by the secrecy of the key KS (Kerkoff's rule). Thus, the cipher must be resistant to hacking, even if a potential cryptanalyst knows the entire encryption algorithm, except for the value of the key used, and has the full text of the intercepted ciphergram.

Practice has shown that the more the algorithm is known, the more people have worked with it, the more tested, and therefore reliable it becomes. Thus, publicly known algorithms now withstand the struggle with time, but classified state ciphers reveal many errors and shortcomings, since it is impossible to take everything into account.

The generally accepted scheme for constructing symmetric cryptosystems is cyclic permutations and substitutions of bits in a block of fixed length, the algorithm of which is determined by the secret key.

Fig. 1. Scheme of construction of symmetric cryptosystems

An encryption algorithm is considered secure if, given the private data and knowing the secret key, it is impossible to obtain information about the public data. It has been rigorously proven that it is impossible to construct an absolutely secure cipher, except for the case when the size of the secret key is equal to (or greater than) the size of the encrypted data [1]. This case is difficult to implement in practice, since the cryptographic protection tools actually used and available on the market use ciphers for which the task of restoring the open text from the closed text is difficult to calculate, that is, it requires such large resources that the attack becomes economically impractical.

Among the symmetric ciphers, the following are the most well-known and frequently used (the block size in bits is designated as b, the number of cycles is r, and the key length is l):

DES— US government standard [2] (b = 64, r = 16, l = 56). Currently, DES has been proven to be insufficiently resistant to brute force attacks [6, 7].
Triple DES and DESX (b = 64, r = 16, l = 168;112) — sequential application of the DES algorithm with different keys, which provides significant resistance to hacking [4, 8, 9].
IDEA — (b = 64, r = 8, l = 128) [4]. Active research into its resistance has revealed a number of weak keys, but the probability of their use is negligible.
RC5— a parameterized cipher with variable block size (b I [32, 64, 128]), number of cycles (r Ј 255), and number of key bits (l Ј 2040) [10]. Studies of its security [11] have shown that for b = 64 it is inaccessible to differential cryptanalysis for r і 12 and to linear cryptanalysis for r і 7.
GOST 28147-89— Russian data encryption standard [3] (b = 64, r = 32, l = 256). Many weak keys have been found for GOST, significantly reducing its effective security in simple encryption modes [12, 13]. Evaluation of GOST's cryptographic security is also complicated by the fact that the most important part of the algorithm — substitution nodes or S-boxes in the terminology of the DES cipher — is not described in the standard and the laws of its generation remain unknown. At the same time, it has been proven that there is a high probability of obtaining weak substitution nodes, simplifying the cryptanalysis of this cipher.
Blowfish— is a 64-bit block cipher developed by Schneier in 1993, implemented using key-dependent permutations and substitutions. All operations are based on XORs and additions on 32-bit words. The key has a variable length (maximum 448 bits) and is used to generate several subkey arrays. The cipher was designed specifically for 32-bit machines and is significantly faster than DES [14].

Now the new encryption standard AES has been adopted in the USA. A competition was held among encryption algorithms, in which the winner and the basis for AES was Rijndael. Rijndael is an iterative block cipher with variable block length and different key lengths. A more detailed description of this algorithm and the results of the competition are given in [18].

A fairly large number of symmetric algorithms have been developed, published and researched in the world (Table 1), of which only DES and its modification Triple DES have been sufficiently time-tested. The table does not include little-known and poorly studied algorithms, such as Safer, etc.

Table 1. Overview of symmetric encryption methods

Currently, symmetric algorithms with a key length of more than 100 bits (Triple DES and IDEA, etc.) are not unbreakable. The domestic GOST algorithm, in comparison with them, is distinguished by increased complexity both in generating replacement nodes and in generating keys. Also, for the GOST algorithm, there is a high probability of generating an unstable key, which in some encryption modes reduces its effective key length from 2256 to 262 [12].

Triple DES is a more proven algorithm than IDEA and provides acceptable speed of operation. The Triple DES algorithm is a threefold application of the DES algorithm to the same data, but with different keys.

DES has penetrated into Russia and is widely used in practice as an integral part of various software and hardware, the most widely known of which are the S.W.I.F.T. system, secret VISA and EUROPAY modules, secret ATM and POS modules, and, finally, smart cards. Particularly intense discussions around data encryption algorithms are caused by smart cards. At the same time, there are serious grounds to believe that the reliability of domestic cryptosystems of conversion origin will exceed foreign analogues [16].

However, Russian legislation, as well as the legislation of many other countries, only permits the use of national encryption standards.

The GOST 28147-89 algorithm is built on the same principle as DES, it is a classic block cipher with a secret key, but differs from DES in a longer key length, a larger number of rounds and a simpler scheme for constructing the rounds themselves. Table 2 shows its main parameters, for convenience — in comparison with DES parameters [17].

Table 2. Comparison of parameters of DES and GOST ciphers

If secret information needs to be exchanged between people who trust each other, i.e. who are part of the same organization, symmetric cryptography can be used. Of course, both (or more) parties must already have encryption keys for interaction.

If we briefly describe the information exchange scenario, it is as follows:

• an existing file containing secret information is created or used;
• the file is encrypted using a key known to both parties, determined by the encryption algorithm;
• the encrypted file is transferred to the subscriber, the storage medium is not so important, it can be a floppy disk, e-mail, a message on the network or a modem connection, it is very convenient, to reduce the risk, to also store all files containing secret information in encrypted form. Then, if the computer, laptop of an employee on a business trip, or a hard drive falls into the hands of an intruder, the files closed with a key will not be available for direct reading. Now in the world there are systems that automatically encrypt all information stored in a laptop, they also provide a forced entry mode, if an employee is forced to boot the laptop, then by entering a special password, instead of the usual one, you can destroy all the information, naturally, a recovery mode is provided after this action. The hard drive can simply be dismantled from the computer, it is not so difficult to take it out of the protected area (compared to an entire computer);
• On the receiving side, the legitimate recipient, having the key, opens the encrypted files for further use.

Many modern methods of protective transformations can be classified into four large groups: permutations, replacements (substitutions), additive and combined methods. Permutation and substitution methods are usually characterized by a short key length, and the reliability of their protection is determined by the complexity of the transformation algorithms. Additive methods are characterized by simple transformation algorithms, and their cryptographic resistance is based on increasing the key length.

Cipher cracking

There is a method for cracking a cipher based on trying all the key variants. The criterion for the correctness of the variant is the presence of a “probable word” in the text.

The set of all possible keys is searched, the encrypted text is decrypted on each key. A probable word is searched for in the resulting “pseudo-open” text. If there is no such word, the current text is rejected, and the next key is moved on. If such a word is found, a key variant is displayed on the screen. Then the keys are searched until the entire set of variants is exhausted. It is possible to find several keys for which the “pseudo-open” texts contain a probable word.

After the search is complete, it is necessary to decrypt the text on the keys found. The “pseudo-open text” is displayed on the screen for visual control. If the operator recognizes the text as open, then the work on opening is finished. Otherwise, this key variant is rejected and the transition to the next key is carried out.

Increasing the length of the encryption key used allows you to combat the full search method. Moreover, increasing its length by only 8 bits increases the number of search options by 28 times, respectively by 64 bits — by 264 times.

Among the problems inherent in the use of cryptographic encryption algorithms, it is necessary to highlight the problem of key distribution. Before the interacting parties can send each other encrypted messages, they must exchange encryption keys via some secret channel. In addition, the information exchange system must maintain a huge number of keys in an up-to-date state.

Cryptographic encryption algorithms do not allow to establish the integrity of the received message (i.e. to make sure that the message was not modified during transmission). Authorship can be confirmed only by possession of a certain key, so anyone who becomes the owner of someone else's key will be able to pass off their messages as messages sent by another user.

The problem of distributing secret keys over a public communication channel can be solved by the Diffie-Hellman algorithm. But this algorithm is an asymmetric cryptographic algorithm. They use two keys: public and private.

Asymmetric cryptographic algorithms were rapidly developed in the 70s of the last century. Such algorithms can also solve the problems of authorship and authenticity verification, allowing the exchange of encrypted information between parties that do not trust each other. In addition, the use of asymmetric algorithms reduces by an order of magnitude the number of keys that must be distributed between the interacting parties. Asymmetric encryption systems include a public database of public keys that can be distributed over open communication channels and their disclosure will in no way lead to a compromise of the system, which is why they are called open.

Asymmetric encryption algorithms

Public key cryptosystems are usually built on the basis of a complex mathematical problem of calculating a function inverse to a given one. Such functions are called one-way, i.e. their inversion is a practically unsolvable problem. The essence of the encryption method is that the calculation of the function of the encrypted message in the forward direction is carried out using the public key of the receiving subscriber, and during decryption (calculation of the inverse function) his secret key is used. As expected, there are few known mathematical problems that satisfy the listed requirements, and only some of them were used to build ciphers used in practice. Let us consider a number of the most well-known public key cryptosystems.

• RSA [1]. The factorization problem (calculation of prime factors) of a large integer is used. It is based on the multiplication of two prime numbers of large capacity [2]. It is widely used in cryptographic protocols for information closure and authentication.
• El-Gamal [3, 4]. It is based on the discrete logarithm problem in a finite field [2]. It is used in the standards of electronic digital signature (EDS) DSS [5], GOST R34.10-94 [6], etc.
• Elliptic curves [7]. Based on the discrete logarithm problem on elliptic curves in a finite field.

Inverse problems of factorization and discrete logarithm are solved by methods close to exhaustive search, and are difficult to calculate for large numbers.
Public key cryptosystems are used mainly in three areas:

• information closure;
• authentication using digital signature;
• interception-protected distribution of public keys (Diffie-Hellman cryptosystem) [8].

The advantages and disadvantages of asymmetric cryptosystems are considered in more detail in [5, 6, 9, 16, 22, 30, 31].

Hash functions

Protocols for protecting integrity and authenticity when forming imitations and digital signatures use cryptographic «compressing» hash functions, which make it possible to obtain a value with a fixed number of bits from a data block of arbitrary length [10, 11, 12, 13, 14, 15].
In order to reduce the size of the digital signature and reduce the time for its formation and verification, it is applied to hash values, which are usually significantly shorter than the original messages. A number of requirements are imposed on cryptographic hash functions aimed at making it difficult to forge the digital signature by finding a modification of the data block in which the value of the hash function and, consequently, the digital signature remain unchanged.
The most widely used hash functions are those based on a system of cyclically repeated permutations and substitutions (the length of the generated hash value in bits is given in brackets):

• MD5 [16, 17, 18, 19] (128);
• SHA-1 [20] (160);
• GOST [21] (256).

Table 1. List and parameters of hash functions

Table 1 does not include rarely used and exotic hash functions, as well as hash functions built on symmetric block ciphers according to the Meyer – Matyas and Davies – Price schemes [9].
The mentioned hash functions are described in more detail in [11, 16 – 21].
Although public key cryptographic protection or asymmetric cryptosystems have been used especially widely since the late 1970s [22], they have a very serious drawback — extremely low performance. In this regard, in practice, a combined cryptographic protection scheme is usually used [23]. When establishing a connection and authenticating the parties, public key cryptography is used, then a session key is generated for symmetric encryption, which closes all traffic between subscribers. The session key is also distributed using the public key.

Fig. 2. Scheme of the algorithm of an asymmetric cryptosystem

Table 2. Asymmetric cryptosystems

The RSA method is currently the de facto standard in information security systems and is recommended by the CCITT (Consultative Committee in International Telegraphy and Telephony) in the X.509 standard [24]. RSA is used in many international standards (S-HTTP, PEM, S-MIME, S/WAN, STT, SSL, PCT, SWIFT, ANSI X.9.31, etc.) [24, 25, 26, 27], in credit card service systems, in operating systems for protecting network protocols.
A huge amount of scientific research has been conducted for the RSA and ElGamal methods, a large number of methods of their cryptanalysis, protection against attacks have been studied, cryptographic resistance has been calculated in detail depending on the key length and other parameters. Both methods have the same cryptographic resistance (with the same key length) and approximately the same operating speed. Considering that the elliptic curve method is undergoing testing and has not been subjected to as many hacking attempts as the RSA and ElGamal methods, the use of the latter two in encryption systems seems preferable.
A detailed description of these algorithms is given in [1, 7, 11, 16, 29, 31].

Electronic digital signature

If information is exchanged between parties that do not trust each other or are interested in carrying out actions directed against each other (bank and client, store and buyer), it is necessary to use asymmetric encryption methods, as well as the EDS method.
It is necessary to ensure not only confidentiality, but also the integrity of the message (the impossibility of replacing the message or changing anything in it), as well as authorship. In addition, it is necessary to prevent the possibility of the author of the message denying the fact of sending a signed message.
An electronic signature of a document allows its authenticity to be established. In addition, cryptographic means provide protection against the following malicious actions:

• denial (renegacy) — subscriber A claims that he did not send the message to B, although in fact he did;
• modification (reworking) — subscriber B changes the document and claims that he received this document (modified) from subscriber A;
• substitution – subscriber B creates a document (new) and states that he received it from subscriber A;
• active interception – an intruder (connected to the network) intercepts documents (files) and changes them;
• «masquerade» – subscriber B sends a document on behalf of subscriber A;
• repeat – subscriber B repeats a previously transmitted document that subscriber A sent to subscriber B.

All of the above types of malicious actions cause significant damage. In addition, the possibility of malicious actions undermines confidence in computer technology [28, 29]. The authentication problem can be solved based on a cryptographic approach by developing special algorithms and programs.
When choosing an authentication algorithm and technology, it is necessary to provide reliable protection against all the above types of malicious actions (threats). However, within the framework of classical (single-key) cryptography, it is difficult to protect against all the above types of threats, since there is a fundamental possibility of malicious actions by one of the parties that owns the secret key.
No one can prevent a subscriber, for example, from generating any document, encrypting it on the existing key, common to the client and the bank, and then claiming that he received this document from a legitimate transmitter.
An effective way is to use schemes based on two-key cryptography [29]. In this case, each transmitting subscriber has his own secret signature key, and all subscribers have non-secret public keys of transmitting subscribers.
These public keys can be treated as a set of verification relations that allow one to judge the authenticity of the transmitting subscriber's signature, but do not allow one to restore the secret signature key. The transmitting subscriber is solely responsible for his secret key. No one else can generate a correct signature. The transmitting subscriber's secret key can be considered a personal seal, and the owner must in every possible way restrict access to it by unauthorized persons. [28].
To practically implement the idea of ​​open encryption, it was necessary to find specific and constructive answers to the following questions:

• How to «mix» the user's individual key with the contents of the document so that they become inseparable?
• How to verify that the content of the document being signed and the user's individual key are authentic without knowing either one in advance?
• how to ensure the possibility of multiple use by the author of the same individual key for digitally signing a large number of electronic documents?
• how to guarantee the impossibility of restoring the user's individual key for any number of electronic documents signed with it?
• how to guarantee the authenticity of the verification of the digital signature and the contents of the electronic document?
• How to ensure the legal validity of an electronic document with digital signatures that exists without a paper duplicate or other substitute?

It took about 20 years to answer all these questions since this idea was first formulated in 1976 in an article by Whitfield Diffie and Martin Hellman. Now we can definitely say that all these questions have been resolved: there is a full arsenal of technical means for authorizing electronic documents, called a digital signature. The modern principles of building a digital signature system are simple and elegant:

• the methods for calculating and verifying digital signatures for all users of the system are the same and are based on well-known mathematical problems;
• the methods for calculating digital signature verification keys and individual digital signature generation keys are also the same for everyone and are well known;
• individual digital signature generation keys are selected by the users themselves at random from a large set of all possible keys;
• for a specific digital signature algorithm, its strength can be assessed without involving any “closed” information, based only on known mathematical results and reasonable assumptions about the computing power of a potential hacker.

Cryptographic protection tools ensure the authenticity and reliability of information, in addition to solving the problem of maintaining its confidentiality. These functions are performed by digital signature technology [6].
The operation diagram of a digital signature is shown in Fig. 3.

Fig. 3. Electronic digital signature algorithm

The algorithm receives a file, not necessarily a text one, as input. The main requirement for the input parameters of the digital signature is a fixed length, for which a hash function is used.
Theoretically, the use of various encryption tools promises bright prospects for all companies that use the Internet in their activities, but here a new problem arises – finding a compromise with the state and its laws, this problem is covered in detail in [30].
In accordance with the Federal Law «On Electronic Digital Signature» No. 1-F3 of January 10, 2002, an electronic digital signature in an electronic document is recognized as equivalent to a handwritten signature in a document on paper. It also provides legal regulation for organizing electronic document management, distributing public and private keys, building certification centers, and determines the responsibilities of the parties.
The adoption of this law, although it contains some uncertainties, made it possible to regulate the use of asymmetric encryption tools, in this case, EDS, to protect data on the Internet.

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