LUT | Academic Journals and Conferences

Analysis of multiplication algorithms in Galuis fields for the cryptographic protection of information

The mathematical basis for processing a digital signature is elliptic curves. The processing of the points of an elliptic curve is based on the operations performed in the Galois fields GF(p^m). Fields with a simple foundation are not well-studied and very interesting for research. In this paper, a comparison of the complexity of algorithms for the realization of the multiplication operation in Galois fields GF(p^m) with different bases is carried out. Conducts a comparison of the 3 most common multiplication algorithms.

Multiplier realization in FPGA of the high level Galois fields.

In this paper, the implementation of matrix multipliers of the Galois fields with basics 2, 3, 5, 7, 13 and the analysis of the implementation of multipliers with a higher basis on the FPGA Xilinx Spartan-6 and Altera – Cyclone-5 is considered. It is shown that the smallest hardware costs will be in multiples of Galois fields with a base 2. For the implementation of the Guild cells with a large foundation, the core generator of the modified Guild cells was implemented.

Апаратні витрати помножувачів полів Галуа GF (dm) з великою основою

The paper compares realised on modern FPGA Galois fields GF (dm) elements
multipliers hardware costs for great basis d to determine the field in which the multiplier has
the lowest hardware complexity. Guild cell internal structure consisting of modul n multiplier
and adder. It is shown that hardware costs will have a constant value 4 which tends to increase
when the foundations of the field.

Definition of the extended Galois field GF(dm) with multiplier minimal hardware complexity

The paper compares realised on modern FPGA Galois fields multipliers hardware costs to select Galois field GF(dm) with approximately the same number of elements and the lowest multiplier hardware complexity. The total increase in hardware costs depending on the increase of the basics of the field has been demonstrated. Local minimums for odd d correspond to d = 2i-1 and the global minimum for analysis based on Guild cell with realization like single unit corresponds to the value d = 3 and based on Guild cell with its multiplier and adder separate realization – the value d=7.