In this study, we developed a residue number system (RNS), a numeral system consisting of an arbitrary number of pairwise coprime integers representing a specific integer by its value. We discovered that this system reduced the runtime or training speed of image processing. In addition, we observed that the characteristic operations of the core functions of the decentralized congruential system derived from the proposed system reduced the overall execution time or training speed of image processing. Both the speed of calculation and the properties of the congruential system ensured the accuracy and security of information after distributed processing. Overall, this approach enabled the use of different algorithms on microdevices while ensuring confidentiality. In summary, we developed a system capable of increasing the operation speed of image processing by 50% through core function precomputation, with a modified image data input, a modified core function, and a reverse core function.