Researches On Existence Of (m,n)-generalized Bent Functions

Generalized bent functions are applied in many fields such as communica-tion theory,coding theory,cryptography theory and combinatorial design theory because of its good properties,so that it has always attracted the attention of scholars.To study the existence and construction of generalized bent function with good properties are the two most important problems.Domestic and for-eign scholars have made many deep and extensive researches on it and a wealth of plentiful research achievement have been obtained over the years.The nonex-istence of the?m,n?-GBF studied in this paper is a generalization of the liter-ature[4],by using some elementary methods in elementary number theory and algebraic number theory and the methods mentioned in relevant literature,some new results are obtained.Then,this paper constructs some concrete?m,n?-GBFs by useing and generalizing the previous ideas and methods.The main results are as follows:Let m=p₁^a1p₂^a2p₃^a3,a_i?1,p₁,p₂,p₃are distinct odd prime numbers,i=1,2,3,n is odd number.1.Let p₁?3?mod 8?,p₂?7?mod 8?,p₃?5?mod 8?,?p₂/p₃?=-1.The conditions of the nonexistence of?m,n?-GBF are calculated and obtained.2.Let p₁?3,5?mod 8?,p₂,p₃?7?mod 8?,?p₃/p₁?=?p₂/p₁?=-1.The conditions of the nonexistence of?m,n?-GBF are calculated and obtained.3.?m,n?-GBF is constructed by concatenating two generalized semi-bent functions.4.By using Boolean permutations,presenting a construction of generalized bent function defined on with valued in Z₈.