| Permutation polynomial is a very important and popular research object in algebra,and it is also widely used in cryptography,which provides theoretical support for the encryption and decryption process of cryptography.In the study of the permutation polynomials EA-equivanlent to monomial polynomials,Li and Wang obtained the condition that a polynomial x + L(x)becoming a permutation polynomial on F when n is an odd number.In this paper,we will adopt a new method,that is,using Hermite criterion combined with Hamming weight method,to prove some of their conclusions again.On the one hand,it makes the operation method more direct,on the other hand,it is also a useful exploration for the study of permutation polynomials.The specific research contents are as follows:The chapter 1 is composed of the introduction and the research status of domestic and foreign.We introduce the brief history of algebra and basic theories of cryptography.And we also list some relevant literature and some methods of research;In chapter 2,we introduce the notations and prelimlinaries.We introduce the the basic theories of algebra and properties of permutation polynomials on finite field.And some relevant lemmas are given;In chapter 3,we study the permutation polynomials over F when n is equal to 3.By using the condition of Hermite criterion and Hamming weight in the process of polynomial expansion,the concrete form of linear polynomial L(x)in this case is obtained.In this part,we mainly use the power expansion of 2 to find the terms that meet the requirements in the possible situations of the expansion,and the final results are consistent with the known results;In chapter 4,we use the method similar to that in the third chapter to get the specific form of linear polynomial L(x)when n is 5.Moreover,by using induction and summary,we further get the conditions that x + L(x)becoming a permutation polynomial on F when n is odd.Although the methods are similar,there are many differences in the implementation process. |
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