In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that comp...In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.展开更多
为了促进交换性的发展,根据半质环及半单环的相关资料,扩展了文献[1-2]的结论,得出了环的两个交换性定理:定理1:设R为一个半质环,若对▽x1,x2,…,xn∈R,有依赖于x1,x2的整系数多项式p(t)使得[…[[x1-x12p(x1),x2],x3],…,xn]∈Z(R),则R...为了促进交换性的发展,根据半质环及半单环的相关资料,扩展了文献[1-2]的结论,得出了环的两个交换性定理:定理1:设R为一个半质环,若对▽x1,x2,…,xn∈R,有依赖于x1,x2的整系数多项式p(t)使得[…[[x1-x12p(x1),x2],x3],…,xn]∈Z(R),则R为交换环。定理2:设R为一个kothe半单纯环,若对▽a,b,x2,…,xn∈R都有一正整数K=K(a,b),一含有x2和n=n(a,b)(≥K)个y的字fx(x,y)及一整系数多项式φx(x,y)使得[…[[∑ki=0αi bi abk-i-fx(a,b)φx(a,b),x2],x3],…,xn]∈Z(R)其中|∑ki=0αi|=1,则R为交换环.展开更多
文摘In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.
文摘为了促进交换性的发展,根据半质环及半单环的相关资料,扩展了文献[1-2]的结论,得出了环的两个交换性定理:定理1:设R为一个半质环,若对▽x1,x2,…,xn∈R,有依赖于x1,x2的整系数多项式p(t)使得[…[[x1-x12p(x1),x2],x3],…,xn]∈Z(R),则R为交换环。定理2:设R为一个kothe半单纯环,若对▽a,b,x2,…,xn∈R都有一正整数K=K(a,b),一含有x2和n=n(a,b)(≥K)个y的字fx(x,y)及一整系数多项式φx(x,y)使得[…[[∑ki=0αi bi abk-i-fx(a,b)φx(a,b),x2],x3],…,xn]∈Z(R)其中|∑ki=0αi|=1,则R为交换环.
