A kind of signature scheme based on class groups of quadratic fields

Quadratic-field cryptosystem is a cryptosystem built from discrete logarithm problem in ideal class groups of quadratic fields(CL-DLP). The problem on digital signature scheme based on ideal class groups of quadratic fields remained open, because of the difficulty of computing class numbers of quadratic fields. In this paper, according to our researches on quadratic fields, we construct the first digital signature scheme in ideal class groups of quadratic fields, using q as modulus, which denotes the prime divisors of ideal class numbers of quadratic fields. Security of the new signature scheme is based fully on CL-DLP. This paper also investigates realization of the scheme, and proposes the concrete technique. In addition, the technique introduced in the paper can be utilized to realize signature schemes of other kinds.

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H( m } of a real quadratic field Q (m)to contain a cyclic subgroup of order n.Some criteria satisfying the condition are also obtained.And eight types of such fields are proved to have this property,e.g.fields with m=(zn+t+12)+4t(with t|zn-1),which contains the well-known fields with m=4zn+1 and m=z2n+4 as special cases.

Subgroups of ideal class groups of real quadratic algebraic function fields

Necessary and sufficient condition on real quadratic algebraic function fields K is given for theirideal class groups H(K) to contain cyclic subgroups of order n. And eight series of such real quadratic functionfields K are obtained whose ideal class groups contain cyclic subgroups of order n. In particular, the ideal classnumbers of these function fields are divisible by n.

THE DETERMINATION OF SUBGROUPS IN IDEAL CLASS GROUPS OF REAL QUADRATIC FIELDS

Series of results about ideal class groups H （m） and class numbers h （m） of real quadratic fields Q (m1/2) can be obtained from [6]. Some of them will be shown in this note. We denote by Cn =Z/nZ the cyclic group of order n. Let m ∈ Z denote a square free positive integer, and let z1, z, t ∈Z be arbitrary integers with z1 odd and t】0.

Ideal Class Groups and Subgroups of Real Quadratic Function Fields

In this paper, we study the real quadratic function fields K=k(D), given a necessary and sufficient condition for the ideal class group H(K) of any real quadratic function field K to have a cyclic subgroup of order n, and obtained eight series of such fields. The ideal class numbers h(O K) of K in the series all have a factor n.[

On Tame Kernels and Ideal Class Groups

It is well known that there is a close connection between tame kernels and ideal class groups of number fields. However, the latter is a very difficult subject in number theory. In this paper, we prove some results connecting the p^n-rank of the tame kernel of a cyclic cubic field F with the p^n-rank of the coinvariants of μp^n×CI（δE,T） under the action of the Galois group, where E = F（ζp^n ） and T is the finite set of primes of E consisting of the infinite primes and the finite primes dividing p. In particular, if F is a cyclic cubic field with only one ramified prime and p = 3, n = 2, we apply the results of the tame kernels to prove some results of the ideal class groups of E, the maximal real subfield of E and F（ζ3）.

Infinite Family of Imaginary Cyclic Fields of Degree p-1 with the p-Rank of the Ideal Class Groups of at Least Two

Imaginary cyclic fields of degree p-1 which have two distinct unramified cyclic extensions of degree p are produced using elementary properties of the Lucas sequences. An infinite family of imaginary cyclic fields of degree p-1 are then given with the p-rank of the ideal class groups of at least two.

On imaginary quadratic function fields with the ideal class group to be exponent ≤2

A necessary condition is presented for the ideal class group of an imaginary quadratic function field K=k(D) (k=F-q(x), 2q) to be of exponent ≤2. The condition is proved to be sufficient in some cases. An analogue of Louboutin’s result in function field case is particularly presented.

关于丢番图方程D_1x^2+2~mD_2=y^n的解数

设Ｄ１、Ｄ２、ｍ、ｘ、ｙ是适合Ｄ１＞１，Ｄ２＞１，２Ｄ１Ｄ２，ｇｃｄ（Ｄ１，Ｄ２）＝ｇｃｄ（ｘ，ｙ）＝１的正整数，ｎ是适合ｎｈ的奇素数，其中ｈ是虚二次域Ｑ（－２ｍＤ１Ｄ２）的类数．本文主要证明了：方程Ｄ１ｘ２＋２ｍＤ２＝ｙｎ至多有５·１０１６组例外解（Ｄ１，Ｄ２，ｘ，ｙ，ｍ，ｎ），而且这些解都满足７ｎ＜８．５·１０６以及ｙｎ＜ｅｘｐ（ｅｘｐ（ｅｘｐ４６））．

关于椭圆曲线上的点作成群的另一个证明

给出了特征不为2的有限域上的椭圆曲线上的点按通常的方式(见文[1])作成一个群的又一个证明.证明仅用到了Dedekind整环的理想类群的知识.

Kummer函数域的理想类群(英文)

得到了二次函数域的理想类群的２－秩公式和一般ｌ次Ｋｕｍｍｅｒ函数域理想类群ｌ－秩的下界．由此完全确定出类数不被ｌ整除的所有ｌ次Ｋｕｍｍｅｒ函数域．其次，还证明了不存在这样的ｌ（ｌ＞２）次Ｋｕｍｍｅｒ函数域。

关于Euler生成多项多2x^2+pq的一个注记(英文)

假设Ω(n)为正整数n的素因子个数,K=Q(-2pq),p,q互不相同的奇素数,为虚二次域,记EK=max{Ω(2x2+pq);x∈Z∩[0,pq-1]}.本文证明了Cl(K)Z/2Z Z/2Z当且仅当EK=2.

一类有关虚二次域理想类群的指数丢番图方程

设Ｄ１、Ｄ２是适合Ｄ１≥１，Ｄ２＞１，ｇｃｄ（Ｄ１，Ｄ２）＝１的无平方因子正整数，ＩＫ是虚二次域Ｋ＝Ｑ（－Ｄ１Ｄ２）的理想类群。本文证明了：如果存在正整数ａ、ｂ、ｋ以及奇素数ｐ，可使Ｄ１ａ２＋Ｄ２ｂ２＝ｋｐ，ｇｃｄ（ａ，ｂ）＝１，ｂ∈Ｎｐ，２ｋ，其中Ｎｐ是所有不含２ｔｐ±１之形素因数的正整数的集合，则当ｐ＞５且ｍａｘ（Ｄ１，Ｄ２）≥１０１０１９７时。

理想类群指数≤2的虚二次函数域

本文给出虚二次函数域K=k^（1/2）D（k=Fq（x）,2+q）的理想类群指数≤2的一个必要条件.在某些条件下,这个条件也是充分的.特别地,给出了Louboutin[1]一个结果在函数域情形的模拟.

分裂模的Euler类的对偶定理(英文)

利用相交理论的观点研究秩为1的投射模的弱Whitney类映射.特别地,对于可裂模Q≌l1l2…ld,证明了E0(Q)=(-1)-dE0(Q*),其中E0(Q)和E0(Q*)分别指投射模Q及其对偶模Q*的弱Euler类这一结果是关于适量丛Chern类的对偶定理的代数对应.

二次域理想类数 生成元的求解及计算机实现

二次域上的理想类数是有限的,文章利用理想类、类群的相关性质,通过计算机编程,求解出二次域Z(D)的类数及理想的代表。

道德自律与诗意生存——宋代雅集图背后的文人群体人格分析

宋代是一个具有“独立之精神,自由之思想”的时代。宋代的士大夫阶层,一方面具有中国历史上最为强烈的淑世精神和入世理想,另一方面也具有最高雅的精神追求与审美品位,这一切都直观地反映在了宋代的一系列“文人雅集图”作品里。文章将宋代文人雅集图作品从历史事件、事实层面上升到政治文化层面进行探讨,分析文人雅集图程式化画面背后的政治、审美理想,以及宋代文人精英阶层的群体人格。

理想剩余类群上Gauss定理的推广

利用代数数论的方法,把整数环上的Gauss定理推广到理想m的剩余类群R(m)上,得到同余式,∏bij∈R( m)1≤i1 <i2 <…<ik≤hbi1 bi2 …bik ≡1(m od m) , 若e(2 )≠1,(- 1) Ck- 1 φ( m) - 1 (mod m) , 若e(2 ) =1,其中e(2 )是R(m)中阶为2的幂次的基的个数,h =φ(m)是R(m)中与m互素的理想个数.

费尔玛-欧拉定理的几种证明

本文应用初等数论知识及近代数学工具,对整数论中著名的费尔玛-欧拉定理,给出较完整的异于传统的几种证明。

The Selmer Groups of Elliptic Curves

Let E / K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map uK and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.