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APPLICATION OF TRIGRAMS IN ENCRYPTION OF OBJECTS USING THE POLYBIUS SQUARE AND A DOUBLE KEY

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APPLICATION OF TRIGRAMS IN ENCRYPTION OF OBJECTS USING THE POLYBIUS SQUARE AND A DOUBLE KEY

Gafurov M.Kh., Rajabova A.S., Ghiyosov R.B.

Tajik Technical University named after academician M.S. Osimi

The works [1-2] consider the method of composing trigrams, its identification frequency (repetition) in Tajik literature and its use in identifying the author of a text in the Tajik language. The work [3] considers the methods of developing a set of encryption alphabets, the elements of which consist of symbols, the methods of creating an arbitrary encryption key and the method of encryption using symbols (unigrams) of the language. In [4] a method for creating an arbitrary operator-key is considered, which uses language symbols and the Polybius square. The work [5] considers the method of encrypting an object using a double key, which uses language symbols, and the work [6] presents a method of encrypting language elements.

The works [7] consider the method of creating left-hand trigrams, creating an arbitrary version of the encryption key from these elements (trigrams) and the method of encrypting a text object using a double key.

We will consider encryption of a given public object using trigrams, development of a variant of an arbitrary operator key based on the Polybius square and a second encryption key.

  1. Let the plaintext be given as an object Ω.
  2. Using the auxiliary key Кσ, the elements of which in this object represent special symbols, punctuation marks, spaces and paragraphs that are not literal symbols of the text (it is possible to use ASCII or Unicode symbols) are replaced and the given plaintext is presented in the canonical form of a sequence of symbols, i.e. in the form of object Ω1.
  3. Let us break the object Ω1, starting with the first symbol, into left-hand trigrams (as shown in [7]), the set of which constitutes the encryption sets (elements of the encryption alphabet) of this object and have the following form:

M = {уi, i = ;  уi }                                                     (1)

  1. According to the instructions given in [4], an arbitrary version of the key operator is developed based on the use of the Polybius square. Note that an arbitrary number of Polybius squares of arbitrary size is selected taking into account the number of elements of the set of encryption of the object (1), which has the following form:

К = К1 ⋃  К2 ⋃  ...                                                                   (2)

  1. The elements of the object encryption set – trigrams (1), are sequentially or arbitrarily placed in the cells of an arbitrarily developed version of the operator key (2).
  2. Each element (trigram) of the object Ω1, which is placed in the cells of the developed version of the operator key, is replaced by the symbols standing in the section of the row and column (or column and row), as a result of which the encrypted object Ω2 is obtained.
  3. To create an arbitrary version of the second encryption key R, the encrypted object Ω2, starting from the initial symbol, is divided into trigrams. Then, the second set of object encryption M1 is created, the elements of which are trigrams of the encrypted object Ω2. After that, symbols are arbitrarily selected from the symbols of the ASCII or Unicode code and the set M2 is created, the number of symbols of which is equal to the number of elements of the encryption set M1. Then, as indicated in [3], one of the arbitrary version of the second encryption key R is created, which has the following form:

                                      (3)

  1. Using the second created version of the arbitrary encryption key R, the encrypted object Ω2 is encrypted, resulting in a doubly encrypted object Ω3.
  2. To decrypt the doubly encrypted object, that is, to obtain the original object Ω from the encrypted object Ω3, it is sufficient to sequentially perform the above steps, from end to beginning. That is, having access to the encrypted object Ω3 and the second encryption key R, it (object Ω3) is translated into the encrypted object Ω2. After that, the obtained object (encrypted once) Ω2 is divided from beginning to end into pairs of symbols, where each pair of symbols are elements representing symbols standing at the intersection of the row and column (column and row) of the first key operator K. Replacing the intersection of the row and columns of the key K in the encrypted object Ω with elements (trigrams), we obtain object Ω1. Finally, using the auxiliary key Kσ, the object Ω1 is easily reduced to the original object Ω.

We will consider the encryption of an object using this method in the following example.

  1. Encryption of an object using trigrams.

A1). Let the open object Ω be given in the following form (rubai from Omar Khayyam):

                            (А)

A2). To bring the object Ω into a canonical form, we develop an auxiliary key Кσ using existing symbols and signs that are not included in the alphabetic characters of the object's text, i.e. the signs comma, period, dash, space (˽), paragraph (¿) and empty symbol (⌀), which are found in the given object Ω and accordingly replacing them with the following symbols of the Unicode code, i.e. the symbols ©, @, ®,%, $, £, we have:

Кσ = {, , . ,  ˽ , ¿  $, ⌀   }                               (А*)

Now, using the created version of the auxiliary key Кσ, we reduce this object Ω to the canonical object Ω1, which has the following form:

          (А**)

A4). From the created set M it follows that it consists of 49 elements. Now, according to the instructions given in the work [4], based on the use of the Polybius square, we will create one of the variants of an arbitrary operator-key for encrypting an object, where the number of cells in them is not less than the number of elements in the set M.

Let an arbitrary version of the encryption key operator consist of two squares, and since the number of elements in the set M is 49, we therefore choose the first square K1(5,5) and the second square K2(5,6). Then an arbitrary version of the encryption key operator of the object has the following form:

К = К1(5,5) ⋃  К2(5,6)                                                             (1.2)

A5). In the cells of an arbitrary version of the encryption key operator (1.2), we insert trigrams from (1.1) by rows (or by columns) and designate the rows and columns by arbitrary symbols from the ASCII or Unicode code. Also, the extra cells in the squares (6 cells are extra) are arbitrarily accepted as closed cells. As a result, an arbitrary version of the encryption key operator of an object based on the Polybius square takes the following form:

 

A6). Now, to encrypt the object Ω1 and reduce it to a closed object Ω2, replacing the trigrams in the object Ω1 using an arbitrary version of the encryption key operator K, with the row and column (column and row) symbols in squares, which takes the following form:

                                      (1.3)

  1. Method of developing and using the second key in object encryption.

B1). To create an arbitrary version of the second encryption key R, we divide the encrypted object Ω2 into one-sided trigrams, starting from the beginning. Then we create a second set of object encryption M1, the elements of which are trigrams from the encrypted object Ω2, and it has the following form:

                      (2.1)

B2). Using the object encryption set (2.1), for each trigram (in our example the number of trigrams is 34) one arbitrary symbol is selected, for example, symbols from the ASCII or Unicode code. Let the selected set of arbitrary symbols M2, the number of symbols of which is equal to the number of elements of the object encryption set (2.1), has the following form:

                        (2.2)

B3). Now for each arbitrary element (trigram) of the set (2.1) we associate one arbitrary symbol from the set (2.2), according to formula (3), as indicated in [3], and create an arbitrary version of the second encryption key R, which has the following form:

  (2.3)

B4). Using an arbitrary version of the second encryption key (2.3), we encrypt the encrypted object Ω2, which is a doubly encrypted object Ω3 and has the following form:

                                        (2.4)

It is clear from the doubly encrypted object Ω3 that it consists of a sequence of symbols and does not represent any meaning. As noted above (item 9), to decrypt the doubly encrypted object Ω3, it is sufficient to perform the sequentially listed steps of the encryption process from the end to the beginning.

Conclusions

  1. Due to the fact that the length (size, volume) of an arbitrary version of the encryption operator key depends on the number of elements of the object encryption set (trigrams), this method is most likely suitable for encrypting small-sized objects, since for large-sized objects, the transfer of a large-sized encryption operator key version using special connections is expensive.
  2. In this method, the strength of the encrypted object Ω3 is high, since the frequency of repetition of trigrams in texts of an arbitrary language has not been studied, and the number of polybaic squares in the structure of an arbitrary version of the encryption operator key remains unknown to interested parties – hackers.

REVIEWER: Komilov O.O.,

Candidate of Mathematical Sciences,

 Associate Professor

REFERENCES

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APPLICATION OF BIGRAMS IN ENCRYPTION OF OBJECTS WITH USING POLYBEUS SQUARE AND DOUBLE KEY

With the advent of new computing power and the development of the relevant industry, a number of existing methods and methods of information protection need a qualitative update, which necessitates the conduct of comprehensive research that will contribute to the modernization of existing methods and methods and the development of new ones. These studies will primarily be applicable in law enforcement and security forces, the banking and financial sector and the implementation of non-cash payments for government services, where it is necessary to ensure the security and protection of information, in particular in the investigation and detection of crimes related to unauthorized access to computer information - cybercrimes. This article discusses a new method and method for encrypting given text objects and reducing them to a closed (confidential) object, which allows structures providing state secrets and information protection, based on the creation of a set of one-way trigrams of the encryption alphabet, development and use of a variant of the key operator on based on the Polybey square, the use of a second key to encrypt an object using the example of Tajik language text (applicable to the text of an arbitrary language), which has a high strength of the encrypted (closed) object.Key words: method, object, alphabet, encryption, decryption, trigram, sets, symbol, operator-key, variant, stability.

Information about the authors: Gafurov Mirshafi Khamitovich - Tajik technical university named after academician M.S. Osimi, сandidate of technical sciences, associate professor.  Address: 734042, Dushanbe, Tajikistan, Academician Rajabov Avenue, 10. Telephone:  (+992) 918631197. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Radjabova Asliya Saidovna –Tajik technical university named after academician M.S.Osimi, senior lecturer. Address: 734042, Dushanbe, Tajikistan, Academician Rajabov Avenue, 10. Telephone: (+992) 915536767.

Giyosov Rustam Bakhromovich - Tajik technical university named after academician M.S. Osimi, 1st year master's student. Address: 734042, Dushanbe, Tajikistan, Academician Rajabov Avenue, 10. Telephone: (+992) 111120047. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Article received 24.04.2024

Approved after review 03.06.2024

Accepted for publication 02.09.2024

   
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