文章指出了论文“A wide family of nonlinear filter functions with a large linear span”存在的问题,给出一个反例,同时得到如下结论:设a是F2上以f(x)为极小多项式的n级m-序列,α为f(x)的一个根,αδ为F2n在F2上的正规元,1≤δ<2...文章指出了论文“A wide family of nonlinear filter functions with a large linear span”存在的问题,给出一个反例,同时得到如下结论:设a是F2上以f(x)为极小多项式的n级m-序列,α为f(x)的一个根,αδ为F2n在F2上的正规元,1≤δ<2n-1。设2≤k≤n-2,令Γ,δk={xδxδ2dxδ22d…xδ2(k-1)d+G(x0,x1,…,x2n-2)|deg(G(x0,x1,…,x2n-2))<k,gcd(d,n)=1},则以Γ,δk中的任意布尔函数作为非线性过滤函数作用在a上得到的输出序列线性复杂度下界为nk。记F2n在F2上的正规元的数量为v(n),则集合{xδxδ2dxδ22d…xδ2(k-1)d|1≤δ<2n-1,αδ为F2n在F2上的正规元,d<n,gcd(d,n)=1}中互异函数的个数为φ(n).v(n)/2。展开更多
The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP ...The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.展开更多
文摘文章指出了论文“A wide family of nonlinear filter functions with a large linear span”存在的问题,给出一个反例,同时得到如下结论:设a是F2上以f(x)为极小多项式的n级m-序列,α为f(x)的一个根,αδ为F2n在F2上的正规元,1≤δ<2n-1。设2≤k≤n-2,令Γ,δk={xδxδ2dxδ22d…xδ2(k-1)d+G(x0,x1,…,x2n-2)|deg(G(x0,x1,…,x2n-2))<k,gcd(d,n)=1},则以Γ,δk中的任意布尔函数作为非线性过滤函数作用在a上得到的输出序列线性复杂度下界为nk。记F2n在F2上的正规元的数量为v(n),则集合{xδxδ2dxδ22d…xδ2(k-1)d|1≤δ<2n-1,αδ为F2n在F2上的正规元,d<n,gcd(d,n)=1}中互异函数的个数为φ(n).v(n)/2。
基金The National Natural Science Foundation of China(No.11371089)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20120092110020)the Natural Science Foundation of Jiangsu Province(No.BK20141327)
文摘The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.
