Same Bit-Size Moduli Formation of Residue Number System for Application in Asymmetric Cryptography
Mykhailo M. Kasianchuk, Ihor Yakymenko, Vasyl Yatskiv, Stepan Ivasiev, А. С. Сверстюк · 2021
This paper presents three methods (factorization, exhaustive search and replacement) of four equal bit size moduli sets formation in a residue number system modified perfect form. This allows using the bit grid registers more efficiently. Such problem is relevant for asymmetric cryptography and noise-protected coding algorithms. The theoretical bases of residue number system, its perfect and modified perfect forms are considered, their advantages and disadvantages are defined. It is shown that the most commonly used moduli in the form of power of two, Mersenne numbers and Fermat numbers require searching for the inverse element and multiplying by it, which makes it difficult to recover a decimal number from its residues using the Chinese remainder theorem. A modified perfect form of the residue number system simplifies this procedure. The graphical dependency of the fourth modulo on two prior ones with one known modulo is presented. Different bit sizes of moduli sets are considered. It is shown that in sets of four modulo with the same bit size in a modified perfect form of residue number system, the first and fourth moduli are negative, second and third are positive.