Research On Finite Field Arithmetic And Scalar Multiplication Of Elliptic Curve Cryptographys

The finite field arithmetic, elliptic curve scalar multiplication and the related algorithms are investigated in this dissertation. With the emphasis on the results obtained by the author, the basic scheme of the finite field theory and elliptic curve cryptosystems theory is sketched. It is consists of eight chapters, which are enumerated as follows:In chapter one, some introductive materials are presented, including the motivations and developments of the finite field arithmetic, Elliptic Curve Crypto-systems(ECC) and elliptic curve scalar multiplication, the intentions of the research work, the main contents of the paper and the list of the results obtained by the author.In chapter two, some necessary and basic materials of the finite field arithmetic theory and elliptic curve cryptosystems theory are introduced.In chapter three, the modular arithmetic algorithms over prime fields are studied. To the moduli with special forms (such as Mersenne numbers, pseudo-Mersenne numbers, generalized Mersenne numbers), we analysis and study modular arithmetic laws, and obtain the conclusions as follows: for Mersenne numbers and pseudo-Mersenne numbers, we get modular arithmetic formulas, and determine exact expressions of the number of modular addition to generalized Mersenne numbers generated by irreducible monic trinomials and pentanomials. Depending on the coefficient of the polynomial, we can easily compute the number of modular addition of A mod p by the formulas for any given irreducible monic trinomial and monic pentanomial.In chapter four, the multiplication algorithms over finite extension fields are investigated. By replacing pseudo-Mersenne numbers with generalized Mersenne numbers, we propose a new notion-Generalized Optimal Extension Fields(GOEFs), and study the fast arithmetic about multiplication and modular arithmetic in GOEFs. At last, we deduce common formulas for multiplication and some more general formulas for modular arithmetic in GOEFs. The results in this paper extend the corresponding work on arithmetic in Optimal Extension Fields(OEFs) made by Bailey, Mihailescu and...