Disymmetric Bigram Cryptosystem Mathematical Model & Solutions
Explore the development of a mathematical model for a disymmetric bigram cryptosystem based on parametric solutions of Diophantine equations. Delve into properties and parameterization methods for multi-degree systems, as well as the cryptosystem's structure and encryption algorithms.
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Development of the mathematic model of disymmetric bigram cryptosystem based on a parametric solution family of multi-degree system of Diophantine equations Valeriy O. Osipyan Kuban State University Russian Federation v.osippyan@gmail.com Kirill I. Litvinov Kuban State University Russian Federation lyrik-1994@yandex.ru
DIOPHANTINEEQUATION ? ?1,?2, ,?? = 0 In the paper we will consider the next type of Diophantine Equaltion system: ?1+ ?2+ + ??= ?1+ ?2+ + ?? ?,? < ? ?+ ?2 ?+ + ?? ?= ?1?+ ?2?+ + ?? ?1 or in short notation ??1,?2, ,??; ??= ???. ?1,?2, ,??= Solution of the equation system can be written in the form ??1,?2, ,??; ??= ???. ?1,?2, ,??=
MSDE SOLUTIONS PROPERTIES Now we proceed to consider a more general MSDE that allow establishing the equivalence of numeric sets or sets of ordered parameters. Due to the exceptional importance of these statements, we give them below regarding MSDE with given allowable m and n values. n??, then ??, ?? 1= n??, then ?,? ,???+ ? = n??, then ?,? ,???+ ???= n??, then ,??,??+ = n?? n?? ?, or in particular ??, ?? 1= n???+ ?, n???+ ??? If ??= If ??= If ??= n+1??,??+ . If ??=
PARAMETERIZATION METHOD FOR MULTI-DEGREE SYSTEMS Let us demonstrate the parameterization method for MSDE of ? dimension and ? order ??= n??,? < ? n?6,? = 1,2,4,6 can be found as: A family of two-parameter solutions of MSDE ?6= ?1= 5? + ?,?2= ? 5?,?3= 7? + 5?, ?4= 5? 7?,?5= 2? + 8?,?6= 8? 2?, n?4,? = 2,4,6 A family of three-parameter solution of MSDE: ?4= ?1= ? 7?,?2= 3? + ?,?3= ? 2? + ?,?4= 3? + 2? + ?, ?1= ? + 7?,?2= 3? ?,?3= ? 2? ?,?4= 3? + 2? ?,
MATHEMATICAL MODEL OF CRYPTOSYSTEM ON THE BASIS OF DIOPHANTINE EQUATION Proposed cryptosystem is based on the Diophantine equation system of the form: ??1,?2, ,??, ?1,?2, ,??= Let s the system has the following parametric solution ??= ??,? = 1..?, ??= ??,? = ? + 1..2?, ???+1, ,?2?; ?1,?2, ,??, ??+1, , ?2? 1= ??2? ?1,?2, ,??= ? ??+1 ?= ?? ? ? ?+ + ?? ? ????+1 = ??= ?1 ? ?? solution of equation ?2? ?2?
MATHEMATICAL MODEL OF ALPHABETIC DBC BASED ON 2-PARAMETER SOLUTION Let s the system has the following parametric solution ??= ???,? = ??,? = 1..?, ??= ???,? = ??,? = ? + 1..2?, M is a numeric equivalent of message (or message s block) ? = 27??+ ??+1,??,??+1 numeric equivalents of ??,??+1 d is a secret key Encryption algorithm: Decryption algorithm: ? ??+1 ? ? ?+ ?2 ?+ + ?? ? = ? ????+1 = ???,? = ?1 ? solution of equation ?2? ?2? 1 ?= ?
MATHEMATICAL MODEL OF ALPHABETIC DBC BASED ON 3-PARAMETER SOLUTION Let s the system has the following parametric solution ??= ???,?,? = ??,? = 1..?, ??= ???,?,? = ??,? = ? + 1..2?, ??,??+1 numeric equivalents of ??,??+1 d is a secret key Encryption algorithm: Decryption algorithm: ? ?2? 1 ? ?+ ?2 ?+ + ?? ? = ? ????+1 = ????,??+1,? = ?1 ? solution of equation ?2? ?= ?
CONCLUSION As the work progressed, the following results were obtained: We developed a mathematical model of DBC containing Diophantine difficulties in solving MSDE of given dimension and order. As noted above, to determine the numeric equivalents of elementary messages, a legal user solves a simple equation of given degree, but an illegal user solves a multivariable MSDE of given dimension and order A new approach to the development of DBC based on the parametric solution of MSDE is proposed that generalizes the principle of building the public key cryptosystems: for direct and inverse transformations of the processed information, the parametric solution of MSDE is preliminarily divided into two parts: one part is used according to the given algorithm for direct transformation of the plain text, and the other part - for reverse transformation of the closed text using blocks of given length, for example, bigrams.
