Research Of Complete Permutation Monomials And Optimal Cyclic Codes Over Finite Fields

Complete permutation polynomials over finite fields are widely used in cryptography,coding and combinatorial design.Most block ciphers show that the relationship between plaintext and ciphertext is essentially permutation.In addition,permutations with good cryptographic properties are often used to design s-boxes,which are the only nonlinear components in block cryptography.Complete permutation polynomial is a special kind of permutation which has excellent cryptographic properties.The block cipher algorithm SMS4 published in China in 2006 is based on complete permutation.SMS4 is used for WAPI of wireless LAN.It is the national cryptography industry standard and plays an extremely important role in the cryptography industry.In recent years,complete permutation polynomials have been used to construct cryptographic functions with good cryptographic properties,and gradually become a research hotspot in cryptography.Sparse complete permutation polynomials over finite fields are favored by scholars for their simple algebraic form and easy realization.Substitution polynomials over finite fields can also construct cyclic codes.Cyclic code is an important subclass of linear block codes.Although its errorcorrecting ability is not as good as other linear codes,it is widely used in consumer electronic products,data storage systems and communication systems because of its clear algebraic structure,simple coding and decoding,and easy to implement.Based on the intensive study of permutation polynomials and cyclic codes in finite fields,this paper summarizes the existing research results of complete permutation monomials and optimal cyclic codes in finite fields in detail,and constructs a new class of complete permutation monomials.In addition,several conjectures of optimal cyclic codes are proposed.The main innovations of this paper are as follows:(1)A new class of complete permutation monomials with exponents d1=(pn-1)/4+1 and d2=(3·pn+1)/4 over a finite field with odd characteristics is constructed by using an effective criterion of permutation polynomials.Therefore,this paper proves that when p=7,n=4,d(d-1)=601(1 801),and p=11,n=4,d(d-1)=3661(10 981)are the complete permutation monomials exponents.(2)By analyzing the parameters of cyclic codes C(1,e,s)constructed by monomial functions,and presents an effective and fast method to judge whether quinary cyclic codes C(1,e,s)is optimal.In addition,this paper constructs e=(pm-1)/2+2,e=3+/k·(5t/2-1)and e=2+3·(7t/2-1).cyclic codes C(1,e,s)is optimal.This paper provides data support for the study of complete permutation monomials and optimal cyclic codes,and enriches the existing constructions of complete permutation monomials and optimal cyclic codes.