3.2. Some Solutions for a Symmetric Matrix
We investigate the key’s feature such that where is a symmetric matrix to secure our Cipher Hexagraphic Polyfunction system.
Proposition 2. Let and . Then with , and are solutions to a symmetric matrix .
Proof. Let
. Substitute
and
into Equations (
2)–(
10). Substituting Equation (
3) into Equation (
5), we get
.
Substituting Equation (
4) into Equation (
8), we have
Hence,
Substituting Equation (
7) into Equation (
9), we have
Hence,
From Equations (
28)–(
30), we get the following
and
Finally, we substitute Equations (
31)–(
33) into Equations (
2)–(
10) to get
in terms of
and
h. □
Next, we give an implementation of Proposition 2 as follows.
Example 2. We let . Then, by using Equations (31)–(33) we have Then, Now, from Proposition 2, we consider four cases when are symmetric as follows.
Case 1
From Equations (
31)–(
33), we let
,
and
. Thus, we get
,
and
, respectively. For this case, we get the following result.
Corollary 1. Let . If then is symmetric where and .
Proof. Let and
Therefore, is symmetric. □
Next, we give an implementation for Corollary 1.
Example 3. We let . Then, we have . Followed by
Case 2
From Equations (
31)–(
33), we let
,
and
. Thus, we get
,
and
, respectively. Followed by the following result.
Corollary 2. Let . If , then where and
Proof. The proving method is similar to Corollary 1. □
Example 4. We let . Then, we have Followed by
Case 3
From Equations (
31)–(
33),we let
,
and
. Thus, we get
,
and
, respectively. Followed by the following result.
Corollary 3. Let . If then where and
Proof. The proving method is similar to Corollary 1. □
Example 5. We let . Then, we have ≡ .
Followed by .
Case 4
From Equations (
31)–(
33), we let
,
and
. Thus, we get
,
and
, respectively. Followed by the following result.
Corollary 4. Let . If , then where and
Proof. The proving method is similar to Corollary 1. □
Example 6. We let . Then, we have Followed by
Now, we investigate the key’s feature
such that
where
is a symmetric matrix by subtituting
into Equations (
31)–(
33). We get the following result.
Corollary 5. Let . If then,
Proof. Let
Then,
where
We can clearly see that
□
Next, we give an implementation for Corollary 5.
Example 7. We let . Then, we have Followed by ≡ .
Futhermore, we investigate the key’s feature of
such that
where
is a symmetric matrix by subtituting
into Equations (
31)–(
33). We get the following result.
Corollary 6. If , where , then
Proof. The proving method is similar to Corollary 5. □
Lastly, we investigate the key’s feature
such that
where
is a symmetric matrix by substituting
into Equations (
31)–(
33). We get the following result.
Corollary 7. If , where , then
Proof. The proving method is similar to Corollary 5. □
3.3. Generation of Self-Invertible Matrix
In this section, we apply in the following example, the method of generating of self-invertible matrices that was mentioned earlier. In this paper, we choose .
Example 8. Consider and as two secret keys. Let with , where are relatively prime with N. This is the solution to a diagonal matrix using Proposition 1.
Now, let and . We choose , therefore and Since , then
Suppose , using similar procedure as in Example 8, we can get all the following self-invertible matrices produced by from Proposition 2 and Corrolaries 1–7.
From Propositions 2, we get where
and
From Corrolary 1, we get
From Corrolary 2, we get
From Corrolary 3, we get
From Corrolary 4, we get
From Corrolary 5, we get where .
From Corrolary 6, we get
From Corrolary 7, we get
3.4. Effect of Self-Invertible Key on Cipher Hexagraphic Polyfunction
Cipher Hexagraphic Polyfunction Transformation is constructed based on the following theorem.
Theorem 1. Let Cipher Hexagraphic Polyfunction Transformation be defined as Definition 1. Say that the determinant for is not a zero and , so have unique solutions and the decryption algorithms are as follows:where is the inverse matrix for which acts as the decryption key. Proof. Let Cipher Hexagraphic Polyfunction transformations be as follows.
There exist the inverse of
such that
when
. So
and
From Equations (
34)–(38), we get the decryption algorithm as follows:
From Equations (
39)–(43), if
so
have unique solutions. □
In Theorem 1, the repeated process occured (that is ) when . The sender can encryp the plain text until the transformation to make sure that the message is kept in secret. It is different with the effect of such a system when we consider . The following is an example of using this key. Of course the use of long transformation from plain text to cipher text is more suitable for cryptographic proposals. We begin with examining the patterns of cipher text when using the small number of transformations.
Suppose the plain text numbers are arranged into
. We choose any generated self-invertible matrix
. Before we proceed to do the encryption process, we need to make sure that the secret key that we have chosen fulfils the conditions as stated in Theorem 1; that is,
. Thus,
. Now, the encryption process is as follows:
The above process is continued such that and for .
This is because of
. Thus, the transforming process after
is not necessary. Now, we scrutinize the condition for
in Theorem 1. If we want to convert a plain text to its cipher text via the third transformation, it is necessary to consider condition
. Therefore, all nine patterns of self-invertible matrices (
) in
Section 3.3 should be avoided from the system of Cipher Hexagraphic Polyfunction before implementing
. This can enhance the security of the cipher message.
Next, we give an implementation for the self-invertible matrix from Example 8.
Example 9. We let . Since and , then we have We have to make sure that the secret key follows the conditions before proceeding to the encryption process. In this case, . Since satisfies , then has a unique solution and the decryption for uses . Let us say we use the phrase ‘IHaveOneSister, TwoBrothersAndANiece’ as the plain text and will be used. This message then be translated into the corresponding numbers based on ASCII (refer https://www.ascii-code.com) and [15] as follows: 73 72 97 118 101 79 110 101 83 105 115 116 101 114 44 84 119 111 66 114 111 116 104 101 114 115 65 110 100 65 78 105 101 99 101 46. The numbers are arranged into matrix of 6 rows and 6 columns as follows:Now, the encryption process of this massage is as follows: Therefore, the corresponding numbers of the cipher text from the first and third transformation is as follows:
À d - “ G ? … ë ¸ w Ê » Ï ó Ì Ê o Š 1/4 CR X Æ ETB / < 3 Ã < 0/00) Í t - EOT
Now, maybe the third parties can analyze this message using the nine patterns of self-invertible matrices mentioned in Section 3.3 even though they do not know the decryption keys. By using , they can expect that the entries’ element in the first row of arethe entries’ element in the second row of arethe entries’ element in the third row of arethe entries’ element in the fourth row of arethe entries’ element in the fifth row of areand the entries’ element in the sixth row of are Using the self-invertible such as in Example 8, there are combinations of and c from the first and fourth rows, combinations of and g from the second and fifth rows and combinations of and g from the last row that need to be tested before deriving the actual value of the plain text. The same method is repeated by using another eight types of self-invertible keys until the actual message is found. It is not impossible to get it so fast with the appropriate algorithm and high performance computer.
Previously, the study of self-invertible effects
on the system of Cipher Polygraphic Polyfunction was pioneered by [
7]. In this paper, we have the effect of using nine types of self-invertible keys
on the same system. Perhaps in the future, we can expect the self-invertible pattern for
for any even number
i. This scenario is aimed to strengthening the prerequisites for a secret key before sending the message.