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• DEVELOPMENT OF ALPHABETICAL DISSYMMETRIC TRIGRAM CRYPTOSYSTEM BASED ON SOLVING A NORMAL SYSTEM OF DIOPHANTINE EQUATIONS OF THE 5TH DEGREE OF DIMENSION SIX OVER THE RING OF GAUSSIAN INTEGERS

  V.О. Оsipyan , Е.S. Fursina , E.Т. Alghareeb

  91-99

  2025-07-24

  Abstract ▼

  The aim of the work is to develop a mathematical model of an alphabetic cryptosystem based on a general two-parameter solution of a normal system of Diophantine equations of the fifth degree of dimension six over the ring of Gaussian integers and to write a program demonstrating the capabilities of such a cryptosystem. The paper implements the idea of K. Shannon to develop a mathematical model of a cryptosystem containing Diophantine difficulties encountered in solving normal and other multistep systems of Diophantine equations (MSDE) of the Tarry-Escott type. K. Shannon noted that cryptosystems containing Diophantine difficulties have the greatest uncertainty in selecting keys. The peculiarity of such MSDEs is that general non-exhaustive methods for solving them based on a negative solution to Hilbert's 10th problem on the algorithmic undecidability of an arbitrary Diophantine equation in integers are unknown.
  It should also be noted that Diophantine equations are a powerful tool in cryptography due to their complexity, but their use requires a deep understanding of the mathematical apparatus of Diophantine analysis with possible methods of solutions to prevent vulnerabilities in such cryptosystems. Solutions are key factors for ensuring the security and reliability of cryptographic systems based on these equations. We provide for the use of strategies and approaches depending on the values of the dimension and degree of such MSDE to increase the share of resistance of alphabetic information security systems (ISS), including the number of parameters included in its general parametric solution, taking into account either the complexity of the algorithm for solving the system of equations, or the solution itself, or both at the same time. The paper presents a mathematical model of an alphabetic dissymmetric trigram cryptosystem based on a general two-parameter solution of a normal system of Diophantine equations of the fifth degree of dimension six over a ring of integer Gaussian numbers, among the numerical values of the parameters of which are both numerical equivalents of elementary messages and keys, for finding which an illegal user will need to look for a general two-parameter solution of a normal system of Diophantine equations.
  The mathematical model of the alphabetic dissymmetric trigram cryptosystem presented in the paper contains Diophantine difficulties, so it has good cryptographic resistance: an illegal user will not be able to reduce the set of keys being tried, he needs to solve a system of Diophantine equations in Gaussian numbers, which is a difficult-to-calculate problem without having the corresponding secret keys. Also, the use of three-symbol (trigram) encryption of plaintext instead of symbolic encryption of plaintext further increases the cryptographic resistance of the system. A software implementation of the specified cryptosystem using the Python language is provided.