Efficient algorithms for the basis of finite Abelian groups

arXiv:0808.3331v1 [cs.DS] 25 Aug 2008 Efficient algorithms for the basis of finite Abelian groups Gregory Karagiorgos a and Dimitrios Poulakis b,1 aDepartment of Informatics and Telecommunications University of Athens Panepistimioupolis 157 84, Athens, Greece bDepartment of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Abstract Let G be a finite abelian group G with N elements. In this paper we give a O(N) time algorithm for computing a basis of G. Furthermore, we obtain an algorithm for computing a basis from a generating system of G with M elements having time complexity O(M P p|N e(p)⌈p1/2⌉µ(p)), where p runs over all the prime divisors of N, and pe(p), µ(p) are the exponent and the number of cyclic groups which are direct factors of the p-primary component of G, respectively. In case where G is a cyclic group having a generating system with M elements, a O(MN ǫ) time algorithm for the computation of a basis of G is obtained. Key words: Algorithmic group theory, abelian group, generating system, basis of abelian group. 1 Introduction In recent years, interest in studying finite abelian groups has raised due to the increasing significant of its relationship with public key cryptography, quan- tum computing and error-correcting codes. Abelian groups as the groups Z∗ n of invertible elements of Zn, the multiplicative groups of finite fields, the groups of elements of elliptic curves over finite fields, the class groups of quadratic fields and others have been used for the designation of public key cryptosystems [17]. On the other hand, in quantum computing, the famous hidden subgroup 1 Email: greg@di.uoa.gr, poulakis@math.auth.gr Preprint submitted to Elsevier 23 November 2018 problem in case of a finite abelian group has been solved by a polynomial time quantum algorithm [6,16,19]. The Shor’s algorithm to factorize integers is one very important special case [20]. Recently, an interesting application of finite abelian groups has been given in the construction of efficient error correcting codes [7]. The problem of determining if two groups are isomorphic is one of the funda- mental problems in group theory and computation. The group isomorphism problem is related to the graph isomorphism problem. Interesting results have been obtained for the case of abelian groups [24,14,8]. Recently, Kavitha [15] proved that group isomorphism for abelian groups with N elements can be determined in O(N) time. Let (G, +) be a finite abelian group. Let H1, . . . , Hr be subgroups of G. The set of elements x1+· · ·+xr, where xi ∈Hi (i = 1, . . . , r) is denoted by H1+· · ·+Hr and is a subgroup of G called the sum of H1, . . . , Hr. It is called direct sum if for every i = 1, . . . , r we have Hi ∩(H1 + · · · + Hi−1 + Hi+1 + · · · + Hr) = {0}. In this case we write H1 ⊕· · · ⊕Hr. If x ∈G, then the set < x > of elements ax where a ∈Z is a subgroup of G called the cyclic group generated by x. Let S ⊆G. The group < S >= P x∈S < x > is called the group generated by S. In case where G =< S >, the set S is called a generating system for G. Suppose now that G has N elements and N = pa1 1 · · · pak k is the prime factorization of N. It is well known that G = G(p1) ⊕· · · ⊕G(pk), where G(pi) is a subgroup of G of order pai i (i = 1, . . . , k) [18, Theorem 16, page 96] and is called the pi-primary component of G. Furthermore, for every i = 1, . . . , k, G(pi) can be decomposed to a direct sum of cyclic groups < xi,j > (j = 1, . . . , µ(pi)) with prime-power order. The set of elements xi,j (i = 1, . . . , k, j = 1, . . . , µ(pi)) is called a basis of G. The smallest prime power pe(pi) i such that pe(pi) i x = 0, for every x ∈G(pi) is called the exponent of G(pi). The elements of a basis with its orders fully determine the structure of a finite abelian group. Thus, an algorithm for finding a basis of a finite abelian group can be easily converted to an algorithm of checking the isomorphism of two such groups. Therefore, the development of efficient algorithms for the deter- mination of the basis of a finite abelian group has fundamental significance in all the above applications. In [4], Chen gave an O(N2) time algorithm for finding a basis of a finite abelian group G. Recently, in [5], Chen and Fu showed an O(N) time algorithm for this task. In case where G is represented by an explicit set of M generators, a O(MN1/2+ǫ) time algorithm is given by Iliopoulos [11] and O(MN1/2) time algorithms are obtained by Teske [23] and by Buchmann and Schmidt [3]. When G is represented by a set of defining relations that is associated with an integer matrix M(G), the computation of the structure of G can be reduced to computing the Smith Normal Form of M(G). A such approach can be found 2 in [13]. Finally, in [2], an algorithm is given for computing the structure of G based on Gr¨obner bases techniques. In this paper we give a simple deterministic algorithm for the computation of a basis of a finite abelian group. More precisely, we prove the

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