The Structure,Algorithms Of Finite Dimension Polynomial Algebra And Its Application In Coding And Cryptology

The idea of this disseration roots in both solving algebraic equations and algebraic immunity research.To solve algebraic equations is related to the properties of multinomial function. This relationship is represented by truth table in Boolean ring. Algebraic immunity is related to both ideal property and multiplication property. Coding and cryptology concerns with the Higher Boolean ring . The properties of element and ideal depict the algebraic structure and algorithm in the Higher Boolean ring.There exist three approachs for researching the Higher Boolean ring. Firstly, we can obtain the structure constant, subideal, subalgebra, direct sum ,because the Higher Boolean ring is an Artinian algebra.We can grasp the algebraic characteristic from this approach,but there is no good expression and algorithm from this approach.Secondly,we can obtain the generators of an ideal by computer algebra,such as Gr?bner bases,Wu method,and resultant.This approach can give the exact expression,but the algorithm complexity is double exponent.Thirdly,we can obtain the expressions of elements and the rules of multiplication operation from multivariate interpolation polynomials,but the poisedness of interpolation problems and the algorithm of multivariate interpolation polynomials are complicated .In this dissertation,we synthesize three methods for researching the Higher Boolean ring and give some applications.The main work and achievements are given below.1. The necessary and sufficient condition that generators of a monomial ideal are the regular sequence of the ideal is obtained. The regular sequence of zero dimension ideals is gotten from its Gr?bner bases. From these results, we pointed that the security of a secret sharing scheme based on computer algebra is imperfect.2. We introduce the definition of higher Boolean ring, describe the relationship among power product bases, multiplication factoring bases and interpolation bases ,and demonstrate that higher Boolean ring is principle ideal ring.3. The linearization expression of element in higher Boolean ring is given. The linearization expresses that polynomial function equations can be solved by linearization. Furthermore, the algorithm complexity of Boolean function equation systems is given.4. The relationship between weight and degree of element in Boolean ring is obtained.5. The definition of higher Boolean ring is extended, the substitution of Boolean ring is put forward.6. The weight distributions of two classes of linear codes based on all known perfect nonlinear functions from a finite field to itself are completely solved.