Security of image transfer and innovative results for (<i>p,q</i>)-Bernstein-Schurer operators

Abstract

&lt;p&gt;With the advent of quantum computing, traditional cryptography algorithms are at risk of being broken. Post-quantum encryption algorithms, developed to include mathematical challenges to make it impossible for quantum computers to solve problems, are constantly being updated to ensure that sensitive information is protected from potential threats. In this study, a hybrid examination of a (&lt;italic&gt;p,q&lt;/italic&gt;)-Bernstein-type polynomial, which is an argument that can be used for encryption algorithms with a post-quantum approach, was made from a mathematical and cryptography perspective. In addition, we have aimed to present a new useful operator that approximates functions and can be used in cases where it is not possible to work with functions in the fields of technology, medicine, and engineering. Based on this idea, a new version of the (&lt;italic&gt;p,q&lt;/italic&gt;)-Bernstein-Schurer operator was introduced in our study on a variable interval and the convergence rate was calculated with two different methods. At the same time, the applications of the theoretical situation in the study were presented with the help of visual illustrations and tables related to the approach. Additionally, our operator satisfied the statistical-type Korovkin theorem and is suitable for variable interval approximation. This is the first paper to study the statistical convergence properties of (&lt;italic&gt;p,q&lt;/italic&gt;)-Bernstein-Schurer operators defined on a variable bounded interval, to obtain special matrices with the help of (&lt;italic&gt;p,q&lt;/italic&gt;)-basis functions, and to give an application of (&lt;italic&gt;p,q&lt;/italic&gt;)-type operators for encrypted image transmission.&lt;/p&gt;