Information security is a very important science, especially after the continuous increase in cybercrime of all kinds, which caused very large financial, economic and political losses, so it was necessary to find a solution to protect the data and keep it secret. Data encryption is one of the most important solutions to preserve data and reduce cybercrime. The encryption process means converting the plain text using a private key into symbols and numbers that make this text incomprehensible to others. There are many algorithms that are used for encryption, and the public key algorithms are the strongest in the encryption process because they depend on two keys, the public key and the private key. One of the most important problems facing this type of algorithm is the lack of a clear mathematical model that facilitates understanding and how to deal with these keys. In this thesis, a mathematical model has been proposed in which two functions are defined, the public key and the function of the private key, based on the concepts and specific characteristics of the function. A general mathematical model was constructed by defining two functions, one for the public key and the other for the private key, using the properties of a function in mathematics, to produce strong keys used in the public key algorithm, to protect data and keep it secret, and this model was applied to the RSA and ELGamal algorithms. The most important result was that in the case of applying the mathematical model, it is easy to understand and deal with the public key and the private key mathematically.
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