8/8/2019 Decomposing Finite Abelian Groups

1/6

arXiv:cs/0

101004v1

[cs.DS]

5Jan2001

Decomposing Finite Abelian Groups

Kevin K. H. Cheung Michele Mosca

February 1, 2008

Abstract

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups.Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm.Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficientalgorithm for computing class numbers (known to be at least as difficult as factoring).

1 Introduction

The work by Shor [9] on factoring and finding discrete logarithms over Zn can be generalized tosolve the Abelian Hidden Subgroup Problem (see for example [10], [4], [7] ). These algorithmsfind the hidden subgroup of a function f : G S, where G = ZN1 ZNl , for some integersN1, N2, . . . , N l. Any Abelian group G is isomorphic to such a product of cyclic groups, however itis not always known how to find such an isomorphism efficiently. Consider for example the groupZN, the multiplicative group of integers modulo N. This is an Abelian group we can compute in

efficiently, yet no known classical algorithm can efficiently find its decomposition into a productof cyclic groups. Consider also the class group of a quadratic number field. This group is alsoAbelian, and finding its decomposition into a product of finite cyclic groups will give us the size of

the group and therefore the class number of the quadratic number field. As Watrous [11] pointsout, assuming the generalized Riemann Hypothesis we can apply the algorithm in this paper andefficiently find class numbers (a problem known to be at least as hard as factoring).

In this paper, we show how we can make use of the solution to the Abelian Hidden SubgroupProblem to decompose a finite Abelian group. Such decompositions makes it possible to apply theAbelian Hidden Subgroup algorithm to a larger class of Abelian groups.

2 Integer Arithmetic Basics

A nonsingular integral matrix U is called unimodular if U is has determinant 1. It is easy tocheck that U is unimodular if and only if U1 is unimodular.

The following operations on a matrix are called elementary (unimodular) column operations:1. exchanging two columns; 2. multiplying a column by -1; 3. adding an integral multiple of onecolumn to another column.

Ph.D. student, Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo,

Waterloo, Ontario, N2L 3G1, Canada. E-mail: kkhcheun@math.uwaterloo.ca. Research of this author was supported

by NSERC PGSBDepartment of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo,

Ontario, N2L 3G1, Canada. E-mail: mmosca@cacr.math.uwaterloo.ca. Research of this author was supported by

NSERC

1

http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1http://arxiv.org/abs/cs/0101004v1

8/8/2019 Decomposing Finite Abelian Groups

2/6

We define elementary row operations similarly.

Theorem 1 For any integral matrix A, one can find in polynomial time using elementary row

and column operations unimodular matrices U and V such that U AV =

D 00 0

where D =

Diag(d1,...,dk) with positive integers such that d1|d2

|...

|dk and for each i, the product d1...di is

equal to the g.c.d. of the subdeterminants of A of order i. We call the matrix

D 00 0

the Smith

normal form (abbreviated as SNF) of A.

Proof : See Kannan and Bachem [5].

3 Group Theory Basics

Recall that a group G is said to be Abelian if for all a, b G, a b = b a. In this paper, all groupsare finite Abelian unless otherwise stated. G is said to be cyclic if there exists a G, such thatG =

{an

|n

Z

}. Here, we call a a generator of G. H

G is called a subgroup of G if H is a

group under the operation induced by G. In this case, we write H G. Let a G. The setaH = {g G|g = ah for some h H} is called a coset of H in G determined by a.

Let a G. If an = e for some n N, then a is said to have finite order. The smallest such nis called the order of a, denoted by ord(a). It is easy to see that the elements in {e,a,a2,...,an1}form a subgroup. We call this subgroup the the cyclic subgroup generated by a and we denote it bya.

Let G1, G2 be groups such that G1 G2 = {e}. The set {a1a2 | a1 G1, a2 G2}, denotedby G1 G2, is called the direct sum of G1 and G2. Note that G1 G2 is a group under binaryoperation such that (a1b1)(a2b2) = (a1b1)(a2b2).

Let G be a group and p be a prime number. Let P G. Then P is called a Sylow p-subgroupof G if |P| = p for some N such that p divides |G| but p+1 does not.

We first quote a few classical results without proof. The interested reader can refer to a standardtext on group theory.

Theorem 2 If N is a subgroup of an Abelian group G, then the set of cosets of N forms a groupunder the coset multiplication given by

aNbN = abN

for alla, b G. The group is denoted by G/N.

Theorem 3 Let N be a subgroup of a finite Abelian group G. If a1,...,ak generates G, then

a1N,...,akN generates G/N.

Theorem 4 A finite Abelian group can be expressed as a direct sum of its Sylow p-subgroups.

Theorem 5 Let K be a subgroup of G = Gp1 Gpl where Gpi is a Sylow pi-subgroup fori = 1,...,l and p1,...,pl are distinct primes. Then there exists Kpi Gpi, i = 1,...,l, such thatK = Kp1 Kpl.

2

8/8/2019 Decomposing Finite Abelian Groups

3/6

The next theorem is an important result on finite Abelian groups.

Theorem 6 (Fundamental Theorem of Finite Abelian Groups). Any finite Abelian group can bedecomposed as a direct sum of cyclic subgroups of prime power order.

In this paper, we shall give an algorithm to find the decomposition.

The next theorem is an integral part of the algorithm.Theorem 7 Given a generating set{a1,...,ak} of a finite Abelian group G and a matrix M suchthat ax1

1 axkk = e if and only if x = (x1,...,xk)T intcol(M) where intcol(M) denotes the set of

vectors obtainable by taking integer linear combinations of columns of M, we can find in polynomialtime (in the size of M) g1,...,gl with l k such that G = g1 gl.Proof : (Adapted from Algorithm 4.1.3 in [3].) By Theorem 1, we can find in polynomial time

unimodular matrices U and V such that U1MV =

D 00 0

where D is a diagonal matrix with

diagonal entries d1,...,dm. Since V is unimodular, intcol(M V)=intcol(M). Thus, ax11

axkk = e ifand only if x = (x1,...,xk)

T intcol(M V). For each i = 1,...,k, set ai = aU1i1 aUkik . Then

a1

x1

ak

xk

= e (x1,...,xk)T intcol(U1M V) di|xi for i = 1,...,m, and xi = 0 for i = m + 1,...,k

Since G is finite, we must have m = k. Otherwise, G will have an element of infinite order. Let jbe the smallest index such that dj > 1. Set gi = a

i+j1 for i = 1,..,l where l = m j + 1. It is

clear that g1,...,gl still generate G and gi has order di+j1 for i = 1,...,l. Therefore, if 0 xi