| Summary: | A cryptography system was developed previously based on Cipher Polygraphic Polyfunction transformations, <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>C</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>j</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>≡</mo> <msubsup> <mi>A</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>i</mi> </mrow> <mi>t</mi> </msubsup> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>j</mi> </mrow> </msub> <mspace width="0.277778em"></mspace> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mspace width="0.277778em"></mspace> <mi>N</mi> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>j</mi> </mrow> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>j</mi> </mrow> </msub> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>i</mi> </mrow> </msub> </semantics> </math> </inline-formula> are cipher text, plain text, and encryption key, respectively. Whereas, <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is the number of transformations of plain text to cipher text. In this system, the parameters (<inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mrow> <mi>i</mi> <mo>×</mo> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>) are kept in secret by a sender of messages. The security of this system, including its combination with the second order linear recurrence Lucas sequence (LUC) and the Ron Rivest, Adi Shamir and Leonard Adleman (RSA) method, until now is being upgraded by some researchers. The studies found that there is some type of self-invertible <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mrow> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mrow> </msub> </semantics> </math> </inline-formula> should be not chosen before transforming a plain text to cipher text in order to enhance the security of Cipher Tetragraphic Trifunction. This paper also seeks to obtain some patterns of self-invertible keys <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mrow> <mn>6</mn> <mo>×</mo> <mn>6</mn> </mrow> </msub> </semantics> </math> </inline-formula> and subsequently examine their effect on the system of Cipher Hexagraphic Polyfunction transformation. For that purpose, we need to find some solutions <inline-formula> <math display="inline"> <semantics> <msub> <mi>L</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>L</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> <mn>2</mn> </msubsup> <mo>≡</mo> <msub> <mi>A</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> <mspace width="3.33333pt"></mspace> <mo form="prefix">mod</mo> <mspace width="0.277778em"></mspace> <mspace width="0.277778em"></mspace> <mi>N</mi> </mrow> </semantics> </math> </inline-formula> when <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> </semantics> </math> </inline-formula> are diagonal and symmetric matrices and subsequently implement the key <inline-formula> <math display="inline"> <semantics> <msub> <mi>L</mi> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </msub> </semantics> </math> </inline-formula> to get the pattern of <inline-formula> <math display="inline"> <semantics> <msub> <mi>A</mi> <mrow> <mn>6</mn> <mo>×</mo> <mn>6</mn> </mrow> </msub> </semantics> </math> </inline-formula>.
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