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.

ON IDEAL CLASS GROUPS AND UNITS IN TERMSOF THE QUADRATIC FORM x^2 + 32y^2

For quadratic number ?elds F = Q(√2p1 ···pt?1 ) with primes pj ≡ 1 mod 8, the authors study the class number and the norm of the fundamental unit of F. The resultsgeneralize nicely what has been familiar for the ?elds Q(√2p) with a prime p ≡ 1 mod 8, including density statements. And the results are stated in terms of the quadratic form x2 + 32y2 and illustrated in terms of graphs.

8-Ranks of Class Groups of Quadratic Number Fields and Their Densities

For F = Q（√εpq）, ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2^n, the quartic residue symbol （1/q）4 and binary quadratic forms such as q^h（-2p）/^4 = x^2 ＋ 2py^2 where h（-2p） is the class number of Q（√-2p）. The results are very useful for numerical computations.

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.[

8-rank of the class group and isotropy index

Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class group of F in terms of a quadratic form Q over F2 which is defined by quartic symbols. In particular,we show that r8 is bounded by the isotropy index of Q.

8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let F = Q（√-p1p2） be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol （p1/p2） = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold： （1） The quartic residue symbols （p1/p2）4 = （p2/p1）4 = 1; （2） Either both p1 and p2 are represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=ε（2x^2-p1y^2）,x,y∈Z,ε∈{±1},where h＋（2p1） is the narrow class number of Q（√2p1）,Moreover, we also generalize these results.

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.

SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.

Equivariant Mapping Class Group and Orbit Braid Group

Motivated by the work of Birman about the relationship between mapping class groups and braid groups,the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper.Their construction is based on the exact sequence given by the fibration F_(0)^(G) M→F(M/G,n).The conclusion is closely connected with the braid group of the quotient space.Comparing with the situation without the group action,there is a big difference when the quotient space is T^(2).

A New Boundary of the Mapping Class Group

Based on the action of the mapping class group on the space of measured foliations,we construct a new boundary of the mapping class group and study the structure of this boundary.As an application,for any point in Teichmüller space,we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmüller space.We also construct some new points in the Gardiner-Masur boundary of Teichmüller space.

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.

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.

Modification of Cohen-Lenstra heuristics for ideal class groups and numbers of certain real quadratic fields

THE famous Cohen-Lenstra heuristics aroused wide insterest and research. Here for a certaintype of real quadratic fields with elements P of potential order p in their ideal classes, modifi-cations of the Cohen-Lenstra heuristics for the probability that the class number h is a multipleof p, and the probability that P is of order p, are presented. Via a quite large amount ofcomputations, it was found that both of these probability predictions agree fairly well with thenumerical data.

A relationship between the nonexistence of generalized bent functions and class groups

A new result on the nonexistence of generalized bent functions is presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations. At the same time,a method is given which can be used to simplify the known results. Then we give the bounds and the meaning in algebraic number theory of the parameters in our results.

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.

PROOF OF A CONJECTURE RELATED TO THE PARABOLIC CLASS NUMBERS OF SOME FUCHSIAN GROUPS

This paper proves a conjecture given in [6], which is concerning with the parabolic class numbers of some Fuchsian groups.

The Class Number of Derived Subgroups and the Structure of Camina Groups

A finite group G is called a Camina group if G has a proper normal subgroup N such that gN is precisely a conjugacy class of G for any g ∈E G - N. In this paper, the structure of a Camina group G is determined when N is a union of 2, 3 or 4 conjugacy classes of G.

Index-<i>p</i>Abelianization Data of <i>p</i>-Class Tower Groups

Given a fixed prime number p, the multiplet of abelian type invariants of the p-class groups of all unramified cyclic degree p extensions of a number field K is called its IPAD (index-p abeliani- zation data). These invariants have proved to be a valuable information for determining the Galois group of the second Hilbert p-class field and the p-capitulation type of K. For p=3 and a number field K with elementary p-class group of rank two, all possible IPADs are given in the complete form of several infinite sequences. Iterated IPADs of second order are used to identify the group of the maximal unramified pro-p extension of K.

Students' In-group Interaction in Chinese ESL Class

The traditional teacher-centered ESL class has been prevalent in China for years. Under this circumstance, teachers tend to adopt a spoon-fed teaching method, and become the dominator in ESL class by setting the same instructional pace, explaining a grammar point, leading drill work; while the students' in-class participation is relatively low. As a result, students with different learning styles often feel suffocated in this teacher-dominated class; they desire to have more opportunities to participate in the free interactions or group discussions.Based on the theoretical framework of interactive teaching approach suggested by Lightbown and Spada(1993),the research aims to analyze the merits of students' in-group interaction and to examine the feasibility of implementing it in Chinese ESL class. The research reveals that in-group interaction can cultivate students' independent learning habits by granting them autonomy to bring about the problem and tackle them. Nonetheless, different perceptions from several students and teachers upon group work may influence the usefulness. The research strives to be instructive for the more efficient application of in-group interaction in Chinese ESL class.