Advanced method of factorization of multi-bit numbers based on Fermat's theorem in the system of residual classes

Factorization is one of the most important problems in the theory of numbers and a modern asymmetric cryptography. Known methods of factorization are quite cumbersome; therefore, require significant computational resources for processing multi-bit numbers. The relevance of the factorization problem is dictated also by uncertainty regarding the theoretical foundation of stability for the disclosure of asymmetric cryptosystems. The advanced method of factorization of multi-bit numbers on the basis of Fermat's theorem with the help of the system of residual classes has been developed in this article. The operation of squaring is excluded from proposed method. Arithmetic operations are performed with numbers, which are smaller than the selected module. Last one allows to shifted zone of bit computing resources on several orders to deeper side and replaces the operation of finding the square root, which is caused of computational complexity of the Fermat's algorithm onto generating a binary key of factorization.