license (http://creativecommons.org/licenses/by/4.0) ENCODING SYRIAC LETTERS IN PARTITION THEORY USING EXTENDED

Since the topic of e-Abacus diagram appeared in 1978 through its application within the partition theory, where this representation was one of the ideas of cryptography, many researchers study this subject from a purely theoretical perspective without practical application to it. Many researchers have shown interest in it by starting to apply purely mathematically, but starting in the past few years they have been searching for ideas that serve the topic. This research deals with the Vigenere Cipher, which is one of the multi­alphabet ciphers and in the past it was one of the most popular ciphers. For its simplicity and resistance to frequency analysis tests for messages encoded with simple ciphers such as Caesar's Cipher through. Its application to the Syriac letters, which is one of the oldest ancient languages, used exclusively when reading religious hymns, for Christians in some regions of the world and in other regions as a trading language. In 2017 it was carefully studied through the above diagram and made the application more confidential among its users in terms of trying to find a suitable agreement between the English language letters and Syriac. Special signs and symbols were used in the Syriac language to be able to address the problem of the difference in the number of letters between the two languages. As well as the possibility of using the Cipher between the letters of the Syriac language


Introduction
Let r be a nonnegative integer, the sequence ( ) 1 2 , ,..., n µ = µ µ µ is said to be the composition of r if 1 n j j r = µ = µ = ∑ where 1 2 , , n µ µ µ  are non-negative integers, the composition is said to be a partition of r if it is 1 j j + µ ≥ µ for every 1 j ≥ [1]. β-number for any partition μ to r is i i b i β = µ + − for all 1 i b ≤ ≤ where b is a positive integer greater or equal to the number of parts of μ, and the set { } is called a set β-number of the partition μ. In 1978, [2] provided a diagram called the e-Abacus, this is to represent any partition μ where each partition has its own diagram. The number of runner depends on the integer e≥2 as shown below: Each number of β-numbers will be represented by a bead (•) in diagram (A), but if not, the value in the diagram is represented by a blank (-). For Fig. 1. e-Abacus diagram

The aim and objectives of the study
The aim of the study is to complement the ideas put forward in the Sami and Mahmood research, including which is there an additionl way to make coding Syriac letters more secure.
To achieve this aim, the following objectives are accomplished: -are there previously approved methods that can be adopted (as encryption) and can be applied to Syriac letters by using the coding techniques; -is it possible to combine more than one coding at the same time.

Vigenere Cipher
It is one of the best known manual methods for multi-letter cipher, Vigenere uses the English letters arranged by a table named Vigenere table (Table 2). Consisting of 26 rows and 26 columns, this table is used for encryption and decryption. The first row consists of the English alphabet (26 letters) arranged in the order from letter (A) to (Z), which represents the letters that make up plain text. The first column from the left also consists of alphabetical letters arranged in the column; the letters represent an encryption key, each containing the row specified by the letter, and are the repetition of the alphabet that appeared from that letter. You can learn more about the history of the cipher and how to use it by [7][8][9][10]. Table 2 Vigenere Table   A   It can also be represented mathematically by using the following two equations: where C represents the encrypted text, P represents the plain text and K represents the key. Equation (1) for encrypted text and equation (2) for decryption. Used by sequencing each letter from 0 to 25.

Vigenere cipher with Syriac letters
Syriac is a Semitic language derived from Aramaic, which in the 6th century BC was the only spoken language in the Fertile Crescent until after the birth of Christ. Its new name acquired Syriac in the fourth century, coinciding with the spread of Christianity in the Levant. Syriac is the mother tongue of the Assyrian, Syriac, Chaldean communities in Iraq and Syria in specific. However, it is used not only by them but also by a number of Christian clergies in their books such as Arabs, Persians, Turks, Armenians, etc. The researchers [5,6] indicate that there is a constant connection between the Syriac language and the Arabic language because of migration and trade and cultural exchange between the Levant and the Arabian Peninsula.
The Syriac language, like all other languages, has 22 letters and contains symbols as shown in Table 3.
Before we start applying Vigenere cipher to Syriac letters, we know that any person, who knows the Syriac language and the sequence of its letters, is able to break the cipher. So in this research, the researchers decided to rearrange the letters according to the partition and beads of each letter that were found by Sami and Mahmood [4]. It is arranged by knowing the number of beads formed for each letter, then we start from the letter that has the lowest number, gradually to the last letter that has the most number of beads. If there is a repetition in the number of beads, we will look for the smaller 1 м where it is placed in the first, and if it is found in some letters that the number of beads is equal and also the value of μ 1 , the researchers will resort to μ 2 and so… Table 4 below shows the Syriac letters, the number of beads and the partition of each letter arranged from smallest to largest (Table 4). Now we will create a Vigenere table on Syriac letters, but it should be noted here that the table will carry the same old characteristics of the Vigenere table but the difference will be here in the Syriac letters and also the arrangement of the Syriac letters as shown below (Table 5). It can also be calculated mathematically by the sequence of each letter in the new order, so the equation will be: where is j sequence of the letter from 1 to 22, C represents the encrypted text, P represents the plain text, K represents the key. For example, the message to be encrypted is ‫,ܒܝܬܐ‬ and the key word is ‫,ܩܢܝܐ‬ the encrypted text would be ( Table 6).
The Vigenere table of Syriac letters can be used by intersecting plain text letters with a key letter, as shown below (Table 7).
To decrypt, we use equation (2), where we will use the encrypted text and the key to get the plain text, as shown below (Table 8).
And this method is very important for solving our problem.

Syriac and English in the Vigenere Cipher
In this part, we have a process to strengthen the cipher and make it more difficult to break, which is the process of linking the letters of the Syriac language and English language, where the work is done by entering the plain text in Syriac and the key is in English and therefore the encrypted text will be in Syriac. The researchers found that the number of letters in the Syriac language is 22, which is less than the number of English letters. So we added letters that can change when they come in the middle and the end of the word and also added a number of symbols used in the Syriac language; the number became 52 letters and symbols. As for the letters in English, we will repeat them with an addition (') to denote a new letter. For example, the letter A, when repeated, becomes a new sequence A '. Syriac letters and symbols will be arranged in the same way we did in (3) depending on the number of beads and partition of each letter. In [4], only Syriac letters partition has been found, since we added new symbols, so we need to find partition for it, as shown below (Fig. 4).
Where (Sy) represents the symbols, (Be) the number of beads, μ the partition and (Le) represents the English letter (Table 9).
Vigenere table, because of the large number of letters, we have divided it, so became (Table 10).
It can be calculated using mathematical equations: where s represents the sequence of letters from 1 to 52.   For example, the message to be encrypted is ‫ܡܐ‬ ܶ ‫ܫܠ‬ ܰ ‫ܝܢܐܘ‬ ܰ ‫,ܫ‬ and the key word is GOOD LUCK, the encrypted text would be (Table 11). It is also possible to validate the solution through the Vigenere table by intersecting the plain text with the key.

Continuation of
To decrypt, we will use the second equation to find the plain text, as shown below (Table 12).

Discussion of experimental results
Among the most important discussions that we came out with are: The Syriac letters can play a pivotal role in the applications of types of encryption; especially as it is a language that is currently less used except in certain regions of the world and therefore it can be a lifeline for rescue.
It is quite useful that the choice of the Vigenere cipher was a good choice because it contains special conditions and flexibility, which made working on it completely useful.
It is quite natural that the experience of Syriac letters with other types with encoding operations will give us different results from those found here in this work, knowing that until the preparation of this research, we only tried two types of encoding and it was noticed that the results did not differ by a large percentage, so it was limited speak on Vigenere.

Conclusions
1. The ability to apply Vigenere Cipher not only to the English language letters (or similar ones) exclusively, but also possible with the letters of any language used, but it just needs to make a kind of common basis for the coding to be appropriate.
2. In the research, work was done (pairing) between the letters of the English language, which led to a significant development in the use of this technology significantly. The letters, signs and symbols used in the Syriac language were used as one unit, which made a great development, as it made difficult for non-speakers to reveal the mysteries of this topic.