Let p be an odd prime,and n,k be nonnegative integers.Let Dn,k(1,x)be the reversed Dickson polynomial of the(k+1)-th kind.In this paper,by using Hermite's criterion,we study the permutational properties of the rev...Let p be an odd prime,and n,k be nonnegative integers.Let Dn,k(1,x)be the reversed Dickson polynomial of the(k+1)-th kind.In this paper,by using Hermite's criterion,we study the permutational properties of the reversed Dickson polynomials Dn,k(1,x)over finite fields in the case of n=mp* with 0<m<p-1.In particular,we provide some precise characterizations for Dn,k(1,x)being permutation polynomials over finite fields with characteristic p when n=2p^(s),or n=3p^(s),or n=4p^(s).展开更多
This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almos...This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form x^(2+2^α) is almost perfect nonlinear.It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.展开更多
Functions with difference uniformity have important applications in cryptography. Some planar functions and almost perfect nonlinear（APN） functions are presented in the note. In addition, an upper bound of the unifo...Functions with difference uniformity have important applications in cryptography. Some planar functions and almost perfect nonlinear（APN） functions are presented in the note. In addition, an upper bound of the uniformity of some power mappings is provided by using an interesting identity on Dickson polynomials. When the character of the finite field is less than 11, the upper bound is proved to be the best possibility.展开更多
By using a powerful criterion for permutation polynomials, we give several classes of complete permutation polynomials over finite fields. First, two classes of complete permutation monomials whose exponents are of Ni...By using a powerful criterion for permutation polynomials, we give several classes of complete permutation polynomials over finite fields. First, two classes of complete permutation monomials whose exponents are of Niho type are presented. Second, for any odd prime p, we give a sufficient and necessary condition for a-1xdto be a complete permutation polynomial over Fp4 k, where d =(p4k-1)/(pk-1)+ 1 and a ∈ F*p4k. Finally, we present a class of complete permutation multinomials, which is a generalization of recent work.展开更多
Let F_(q) be a finite field and F_(q)^(s) be an extension of F_(q).Let f(x)∈F_(q)[x]be a polynomial of degree n with g c d(n,q)=1.We present a recursive formula for evaluating the exponential sum∑c∈F_(q)^(s)χ^((s)...Let F_(q) be a finite field and F_(q)^(s) be an extension of F_(q).Let f(x)∈F_(q)[x]be a polynomial of degree n with g c d(n,q)=1.We present a recursive formula for evaluating the exponential sum∑c∈F_(q)^(s)χ^((s))(f(x)).Let a and b be two elements in F_(q) with a a≠0,u be a positive integer.We obtain an estimate for the exponential sum∑c∈F^(∗)_(q)^(s)χ^((s))(ac^(u)+bc^(−1)),whereχ^((s))is the lifting of an additive characterχof F_(q).Some properties of the sequences constructed from these exponential sums are provided too.展开更多
This paper is devoted to the study of semi-bent functions with several parameters flexible on the finite field F2n.Boolean functions defined on F2n of the form f（r）ab（x） =Trn1（axr（2m-1））＋Tr41（bx（2n-1）/5） ...This paper is devoted to the study of semi-bent functions with several parameters flexible on the finite field F2n.Boolean functions defined on F2n of the form f（r）ab（x） =Trn1（axr（2m-1））＋Tr41（bx（2n-1）/5） and the form g（rs）abcd（x）=Trn1（axr（2m-1））＋Tr41（bx（2n-1）/5）＋Trn1（cx（2m-1）1/2＋1）＋Trn1（dx（2m-1）s＋1） where n = 2m,m = 2（mod 4）,a,c ∈ F2n,and b ∈ F（16）,d ∈ F2,are investigated in constructing new classes of semi-bent functions.Some characteristic sums such as Kloosterman sums and Weil sums are employed to determine whether the above functions are semi-bent or not.展开更多
基金supported by National Natural Science Foundation of China(No.12226335)by China's Central Government Funds for Guiding Local Scientific and Technological Development(No.2021ZYD0013).
文摘Let p be an odd prime,and n,k be nonnegative integers.Let Dn,k(1,x)be the reversed Dickson polynomial of the(k+1)-th kind.In this paper,by using Hermite's criterion,we study the permutational properties of the reversed Dickson polynomials Dn,k(1,x)over finite fields in the case of n=mp* with 0<m<p-1.In particular,we provide some precise characterizations for Dn,k(1,x)being permutation polynomials over finite fields with characteristic p when n=2p^(s),or n=3p^(s),or n=4p^(s).
基金supported by the National Basic Research Program of China under Grant No.2011CB302400
文摘This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form x^(2+2^α) is almost perfect nonlinear.It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.
文摘Functions with difference uniformity have important applications in cryptography. Some planar functions and almost perfect nonlinear（APN） functions are presented in the note. In addition, an upper bound of the uniformity of some power mappings is provided by using an interesting identity on Dickson polynomials. When the character of the finite field is less than 11, the upper bound is proved to be the best possibility.
基金supported by National Natural Science Foundation of China(Grant Nos.61272481 and 61402352)the China Scholarship Council,Beijing Natural Science Foundation(Grant No.4122089)+1 种基金National Development and Reform Commission(Grant No.20121424)the Norwegian Research Council
文摘By using a powerful criterion for permutation polynomials, we give several classes of complete permutation polynomials over finite fields. First, two classes of complete permutation monomials whose exponents are of Niho type are presented. Second, for any odd prime p, we give a sufficient and necessary condition for a-1xdto be a complete permutation polynomial over Fp4 k, where d =(p4k-1)/(pk-1)+ 1 and a ∈ F*p4k. Finally, we present a class of complete permutation multinomials, which is a generalization of recent work.
文摘Let F_(q) be a finite field and F_(q)^(s) be an extension of F_(q).Let f(x)∈F_(q)[x]be a polynomial of degree n with g c d(n,q)=1.We present a recursive formula for evaluating the exponential sum∑c∈F_(q)^(s)χ^((s))(f(x)).Let a and b be two elements in F_(q) with a a≠0,u be a positive integer.We obtain an estimate for the exponential sum∑c∈F^(∗)_(q)^(s)χ^((s))(ac^(u)+bc^(−1)),whereχ^((s))is the lifting of an additive characterχof F_(q).Some properties of the sequences constructed from these exponential sums are provided too.
基金supported by the National Natural Science Foundation of China under Grant No.11371011
文摘This paper is devoted to the study of semi-bent functions with several parameters flexible on the finite field F2n.Boolean functions defined on F2n of the form f（r）ab（x） =Trn1（axr（2m-1））＋Tr41（bx（2n-1）/5） and the form g（rs）abcd（x）=Trn1（axr（2m-1））＋Tr41（bx（2n-1）/5）＋Trn1（cx（2m-1）1/2＋1）＋Trn1（dx（2m-1）s＋1） where n = 2m,m = 2（mod 4）,a,c ∈ F2n,and b ∈ F（16）,d ∈ F2,are investigated in constructing new classes of semi-bent functions.Some characteristic sums such as Kloosterman sums and Weil sums are employed to determine whether the above functions are semi-bent or not.
