Savel’ev Boris Aleksandrovich, Doctor of engineering sciences, professor, sub-department of data-computing systems, Penza State University (40 Krasnaya street, Penza, Russia), firstname.lastname@example.org
Bobrysheva Galina Vladimirovna, Candidate of engineering sciences, associate professor, sub-department of data-computing systems, Penza State University (40 Krasnaya street, Penza, Russia), email@example.com
Ubiennykh Anatoliy Gennad'evich, Senior lecturer, sub-department of data-computing systems, Penza State University (40 Krasnaya street, Penza, Russia), utolg@.ru
Background. Constructed on the basis of logic circuits or storage devices, the tools of multiplication of the finite field elements are widely used in com-munication systems and cryptographic protection of information. When implement-ing multiplication devices the multiplication of elements in a normal basis is of par-ticular interest, which is subject to analysis in this paper. The article aims at design-ing a multiplier in the field with a normal basis.
Materials and methods. The theoretical substantiation of statements on genera-tion of normal bases and the multiplier’s design is shown using a special mathemati-cal apparatus.
Results. The authors investigated a process of multiplication of the elements of a normal basis implemented by hardware. The researchers obtained a mathematical expression for determining a number of normal bases. It is proved that any irreduci-ble polynomial generates a normal basis; a number of the multiplier structures of el-ements in a normal basis for any field is equal to a number of classes of the associated elements, each of which is determined by the leading elements of the cyclotomic classes and does not depend on the structure of a generating polynomial. The article shows two ways of finding the leading elements for the fields and , providing one construction of multipliers.
Conclusions. The results of the theoretical and practical research of tools of mul-tiplications of the finite field elements showed that the elements of the finite field , presented in a normal basis, can be generated by any irreducible polyno-mial, and the complexity of the construction of multipliers is determined by the lead-ing elements of the cyclotomic classes. The design of multipliers in the field with a normal basis ensures the greatest regularity of structure, which is especially impor-tant in implementation of the multiplication device in the BIS or a programmable logic matrix.
finite field, normal basis, leading element, associated element, irreducible polynomial, multiplier, construction of multiplier.
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