Efficient algorithms for the basis of finite Abelian groups

Efficien t algorithms for t he basi s of fin ite Ab elian gr o ups Gregor y Karag iorg o s a and Dimitr ios Poulakis b , 1 a Dep a rtment of Inform atics and T ele c ommunic a tions University of A thens Panepistimioup olis 157 84, A thens, Gr e e c e b Dep a rtment of Mathematics, Aristo tle University of Th essaloniki, Thessaloniki 541 24, Gr e e c e Abstract Let G b e a finite abelian group G with N elemen ts. In this pap er w e give a O ( N ) time algorithm for computing a basis of G . F urthermore, we obtain an algorithm for computing a basis from a generating system of G with M elemen ts ha ving time complexit y O ( M P p | N e ( p ) ⌈ p 1 / 2 ⌉ µ ( p ) ), where p run s o v er all the prime divisors of N , and p e ( p ) , µ ( p ) are the exponent and the num b er of cyclic groups whic h are direct factors of the p -primary comp onent of G , resp ectiv ely . In case where G is a cyclic group ha ving a generat ing sys tem with M elemen ts, a O ( M N ǫ ) time alg orithm f or the computation of a basis of G is obtained. Key wo r d s: Algorithmic group theory , ab elian group, generating system, basis of ab elian group. 1 In tro duction In recen t y ears, interes t in studying finite ab elian groups has raised due to t he increasing significan t of its relationship with public k ey cryptography , quan- tum computing and error-correcting co des. Abelian groups as the groups Z ∗ n of in v ertible elemen ts of Z n , the m ultiplicativ e groups of finite fields, the groups of elemen ts of elliptic curv es ov e r finite fields, the class groups of quadratic fields and others ha v e b een used for the designation of public k ey cryptosystems [17]. On the other hand, in quantum computing, the famous hidden subgroup 1 Email: greg@di.uoa.gr, p oulakis@math.auth.gr Preprint su bmitted to Elsevier 23 No v em b er 201 8 problem in case of a finite ab elian gr o up has b een solv ed b y a p olynomial time quan tum algorithm [6,16,19]. The Shor’s algorithm to fa ctorize in tegers is one v ery imp ortant sp ecial case [20]. R ecen tly , an in teresting application o f finite ab elian groups has b een given in the construction of efficien t error correcting co des [7]. The problem of determining if t w o groups are isomorphic is one o f the funda- men tal problems in group theory and computation. The group isomorphism problem is related to the graph isomorphism problem. In teresting results ha v e b een obtained for the case of abelian gro ups [2 4,14,8]. Recen tly , Ka vitha [15] pro v ed tha t group isomorphism for ab elian groups with N elemen ts can b e determined in O ( N ) time. Let ( G, +) b e a finite ab elian gro up. Let H 1 , . . . , H r b e subgroups of G . The set of elemen ts x 1 + · · · + x r , where x i ∈ H i ( i = 1 , . . . , r ) is denoted b y H 1 + · · · + H r and is a subgroup of G called the sum of H 1 , . . . , H r . It is called dir e ct sum if for ev ery i = 1 , . . . , r w e hav e H i ∩ ( H 1 + · · · + H i − 1 + H i +1 + · · · + H r ) = { 0 } . In this case w e write H 1 ⊕ · · · ⊕ H r . If x ∈ G , then the set < x > of ele men ts ax where a ∈ Z is a subgroup of G called the cyclic gr oup generated by x . Let S ⊆ G . The group < S > = P x ∈ S < x > is called the gr oup gener ate d b y S . In case where G = < S > , the set S is called a gener ating system for G . Suppose no w tha t G has N elemen ts and N = p a 1 1 · · · p a k k is the prime factorization of N . It is w ell kn ow n that G = G ( p 1 ) ⊕ · · · ⊕ G ( p k ), where G ( p i ) is a subgroup of G of order p a i i ( i = 1 , . . . , k ) [18, Theorem 16, page 96] and is called the p i - primary c omp onent of G . F urthermore, for ev e ry i = 1 , . . . , k , G ( p i ) can b e decomp osed to a direct sum of cyclic groups < x i,j > ( j = 1 , . . . , µ ( p i )) with prime-p o w er order. The set of elemen ts x i,j ( i = 1 , . . . , k , j = 1 , . . . , µ ( p i )) is called a b asis of G . The smallest prime p o w er p e ( p i ) i suc h that p e ( p i ) i x = 0, for ev ery x ∈ G ( p i ) is called the exp on ent of G ( p i ). The elemen ts of a basis with its orders fully determine the structure of a finite ab elian gro up. Th us, an algorithm f or finding a basis of a finite a b elian group can b e easily conv e rted to a n alg o rithm of chec king the isomorphism of t w o suc h groups. Therefore, the dev elopmen t of efficien t algorithms for the deter- mination of the basis of a finite ab elian group has fundamen tal significance in all the a b o v e applications. In [4], Chen gav e an O ( N 2 ) time alg orithm f or finding a basis of a finite ab elian group G . R ecen tly , in [5], Chen and F u sho w ed an O ( N ) time algorithm fo r this ta sk. In case where G is represen ted b y an explicit set of M generators, a O ( M N 1 / 2+ ǫ ) time algo rithm is give n b y Iliop oulos [11 ] and O ( M N 1 / 2 ) time algorithms are o bt a ined b y T esk e [23 ] and by Buc hmann and Sc hmidt [3]. When G is represen ted b y a se t of defining relations that is asso ciat ed with an in teger matrix M ( G ), the computation of the structure of G can b e reduced to computing the Smith Normal F orm o f M ( G ). A suc h approach can b e found 2 in [13]. Finally , in [2], an algorithm is giv en for computing the structure of G based on Gr¨ obner bases tec hniques. In this pa p er we giv e a simple deterministic algorithm for the computatio n of a basis of a finite ab elian group. More precisely , we pro v e the f o llo wing theorem. Theorem 1 L et G b e an ab elian gr oup G with N elements. Ther e is an O ( N ) time algorithm for c omputing a b asis of G . An immediate consequence of the abov e theorem is the following corolla ry . Corollary 1 Gr oup isomo rphism for ab elian gr oups with N elements c an b e determine d in O ( N ) time. Adapting our metho d in case where a generating system of G is kno wn, w e obtain the follo wing theorem. Theorem 2 L et G b e an a b elian gr oup with N elements having a g ener ating system with M elements. L et N = p a 1 1 · · · p a k k b e the prime factorization of N and µ ( p i ) , e ( p i ) as ab ove . The n, ther e is an al gorithm for c omp uting a b asis of G with time c omplexity O ( M P k i =1 e ( p i ) ⌈ p 1 / 2 i ⌉ µ ( p i ) ) and Ω( P k i =1 p ( µ ( p i ) − 1) / 2 i ) . In the sp ecial case where the g roup G is cyclic, we ha v e the follo wing b etter result. Theorem 3 L et G b e a cyclic gr oup with N elements having a gener ating system with M elemen ts. Then, ther e is an algorithm for c omputing a b asis of G with time c omplexity O ( M N ǫ ) , wher e ǫ is arbitr ary smal l. The a lgorithm of Chen and F u [5], whic h computes a basis of G , is relied on an algorithm of Kavitha [15] for computing the orders of all elemen ts in G . Our approac h is completely differen t. The basic idea of our paper is the construction of bases of successiv ely larger subgroups of the p -comp onen ts G ( p ) of G , using an algorithm of Beynon and Iliop oulos [1] and an algorithm of T esk e [22], un til a basis of G ( p ) is obtained. Note that [12] yields that the time complex it y in Theorem 1 and 2 is reasonable. W e assume that G is giv en b y its m ultiplication t a ble. This is equiv alen t to the mu ltiplication ora cle where eac h group op eration can b e p erformed in constan t time. Since the exp ected running t ime t ha t an inte ger N can b e factorized into pro duct o f prime num bers is sub exp onen tial (see [9, Chapter 19] and [2 5 , Section 2.3 ]), w e assume that the prime factorization of N , whic h is the size of ab elian g roup, is kno wn. T o measure the running time of our algorithm w e coun t the n um ber of nume rical op erations, comparisons and the n um b er of group operations. 3 The pap er is organized as follows. In section 2 w e giv e some auxiliary results on finite ab elian groups whic h a re necessary for the presen tation o f our al- gorithms. In section 3, 4 and 5 w e g iv e the algorithms whic h corresp ond to Theorems 1, 2 and 3. 2 Auxiliary Results In this section we g iv e some results very useful for the design of our algorithms. W e denote b y | A | the cardinalit y of a finite set A . If x is a real n um ber, then w e denote b y ⌈ x ⌉ , as usually , the smallest in teger z suc h that x ≤ z . Let ( G, +) b e a finite abelian gro up. F or x ∈ G , the or der of G , denoted b y ord( x ), is the smallest p ositive in teger µ suc h that µ x = 0 (where 0 is the identit y elemen t of G ). If S = { x 1 , . . . , x k } ⊆ G , then w e also write < S > = < x 1 , . . . , x n > . Lemma 1 If x ∈ G has or der m = p b 1 1 · · · p b k k , then for i = 1 , . . . , k the element x i = ( m/p b i i ) x has or der p b i i and we have < x > = < x 1 > ⊕ · · · ⊕ < x k > . PR OOF. See [18, page 96 ]. ✷ Let | G | = N . W e a ssume that the prime factorization of N = p a 1 1 · · · p a k k is giv en. The p o sitive divisors of N are the in tegers d giv en by d = p b 1 1 · · · p b k k , where 0 ≤ b i ≤ a i ( i = 1 , . . . , k ). W e denote b y τ ( N ) the n um b er of p ositiv e divisors of N . Put B = { ( b 1 , . . . , b k ) ∈ Z k / 0 ≤ b i ≤ a i , ∀ i = 1 , . . . , k } . The follo wing alg o rithm pro vide us with the p ositiv e divisors of N . DIVISORS Input: N a nd its prime factorization N = p a 1 1 · · · p a k k . Output: D ( N ) = { ( d i , ( b i, 1 , . . . , b i,k )) /i = 1 , . . . , d τ ( N ) } , where d 1 < · · · < d τ ( N ) are the p ositiv e divisors of N sorting in increasing order and ( b i, 1 , . . . , b i,k ) ∈ B with d i = p b i, 1 1 · · · p b i,k k . (1) F or ( b 1 , . . . , b k ) ∈ B (a) Compute d ( b 1 , . . . , b k ) = p b 1 1 · · · p b k k . (b) Store d ( b 1 , . . . , b k ) and the corresp onding k -tuple ( b 1 , . . . , b k ) in bi- nary searc h tree. 4 (2) Using t he bina r y tree sort, sort the divisors of N in increasing or der d 1 < · · · < d τ ( N ) . The computation of eac h divisor d requ ires O ((log d ) 2 ) arithmetic op erations. Th us the time of computation of all p o sitive divisors of N is O ( τ ( N )(lo g N ) 2 ). The time complexity of binary searc h tree and binary tr ee sort algor ithms is O ( τ ( N )). F urther, b y [10, Theorem 3 1 5, page 260 ], w e ha v e τ ( N ) = O ( N ǫ ), for ev ery p ositive ǫ . Hence, the time complexit y of DIVISORS is O ( N ǫ (log N ) 2 ). Let x ∈ G and m = ord( x ). By Lagrang e’s theorem [18, page 35], m divides N . F urthermore, by Lemma 1, if m = p b 1 1 · · · p b k k , then for i = 1 , . . . , k the elemen t x i = ( m/p b i i ) x has order p b i i . The follo wing algorithm computes corr ectly m and m/p b i i ( i = 1 , . . . , k ). ORDER Input: x ∈ G and D ( N ). Output: [ m, m 1 , . . . , m k ], where o r d( x ) = m = p b 1 1 · · · p b k k and m i = m/p b i i . (1) F or l = 1 , . . . , τ ( N ) test w ether or not d l x = 0. If d l x = 0, then stop. Put m = d l . (2) F or i = 1 , . . . , k compute m i = m/p b i i . F or the computation o f d l x , w e need O (lo g N ) operatio ns in G [9, page 69] and so, the computation of all d l x needs O ( τ ( N )(log N ) 2 ). The computation of ev ery m i needs O ((log m )(log p b i i )) bit op eratio ns and hence, the computation of all m i requires O (( lo g N ) 2 ) bit op erations. Therefore, the time complexit y of the ORD ER algorithm is O ( N ǫ (log N ) 2 ). Supp ose that B = { b 1 , . . . , b n } is a subset of G suc h that the group H = < B > is the direct sum of the cyclic groups < b i > ( i = 1 , . . . , n ). The extending discr ete lo garithm pr oblem (EDLP) is the following pro blem: Giv en a set B ⊆ G as ab o v e and w ∈ G , determine the smallest p ositiv e in t eger z with z w ∈ H and positive integers z 1 , . . . , z n satisfying z w = n X i =1 z i b i . Note that z ≤ o rd( w ). If z = ord( w ), then H ∩ < w > = { 0 } . In [2 2], an algorithm is pres en ted whic h solve s EDLP with running time O (max {⌈ p 1 / 2 ⌉ n e ( p ) } ) , where the ma ximum is t ak en ov er all prime divisors of N and p e ( p ) is the exp o nen t o f the p -comp onen t of G . It is called SOL VE-EDLP . Th us, w e ha v e SOL VE-EDLP( w , B ) = ( z , z 1 , . . . , z n ). On the o ther hand, [21] implies t hat in 5 case where the order of b i is a p o w er o f a prime p , the expression of a giv en elemen t on a giv en basis of H requires at least Ω( p n/ 2 ) op erations. Let p be a prime divisor of N and G ( p ) the p - comp onen t of G . Suppo se that B = { b 1 , . . . , b n } b e a subset of G ( p ) suc h that the group H = < B > is the direct sum of the cyclic g roups < b i > , ( i = 1 , . . . , n ). If x ∈ G ( p ), then w e denote b y H ⋆ the group generated by the set B ∪ { x } . Supp ose that the orders of ele men ts of B are kno wn and w e hav e a relation of the form p k x = n X i =1 δ i b i , where δ i ∈ Z and 0 ≤ δ i < ord( b i ) ( i = 1 , . . . , n ). In [1], an algorithm is given called BASIS whic h computes a basis f o r H ⋆ with running time O ((log | H ⋆ | ) 2 ). If B ⋆ is the basis of H ⋆ computed b y BASIS, then we write BASIS( B , x, ( p k , δ 1 , . . . , δ n )) = B ⋆ . 3 Pro of of Theorem 1 In this section w e dev elop an algorithm for finding a basis of a finite ab elian group G . If A and B are t w o subsets of G , then w e recall that A \ B denotes the set of elemen ts of x whic h do not b elong t o B . As w e hav e noted in t he In tr o duction, for ev ery prime p dividing | G | , w e ha v e to compute a basis for the p -primary componen t G ( p ) of G . BASIS1 Input: An ab elian group ( G, +) with | G | = N and the prime factorization N = p a 1 1 · · · p a k k of N . Output: F or i = 1 , . . . , k , ( y 1 ,i , n 1 ,i ) , . . . , ( y l ( i ) ,i , n l ( i ) ,i ), where y j,i ∈ G with ord( y j,i ) = n j,i , suc h that the p i -primary comp o nen t o f G is G ( p i ) = < y 1 ,i > ⊕ · · · ⊕ < y l ( i ) ,i > . (1) Set G 0 ( p i ) = { 0 } , B 0 ,i = Ø, ( i = 1 , . . . , k ) and G 0 = { 0 } . (2) Compute DIVISORS( N = p a 1 1 · · · p a k k ) = D ( N ). (3) F or j = 1 , 2 , 3 . . . (a) Cho ose x j ∈ G \ G j − 1 . (b) Compute ORDER( x j , D ( N )) = [ m j , m j, 1 , . . . , m j,k ]. (c) F or i = 1 , . . . , k (i) Compute x j,i = m j,i x j . (ii) Compute SOL VE-EDLP( x j,i , B j − 1 ,i ) = ( z j,i , z j,i, 1 , . . . , z j,i,n ). (iii) Compute the bigg er in teger k j,i ≥ 0 suc h that p k j,i i divides z j,i , s j,i = z j,i /p k j,i i and h j,i = s j,i x j,i . 6 (iv) Compute BASIS( B j − 1 , h j,i , ( p k j,i i , z j,i, 1 , . . . , z j,i,n )) = B j,i . (v) Set G j ( p i ) = < B j,i > and G j = G j ( p 1 ) + · · · + G j ( p k ). (d) Compute the elemen ts of G j \ G j − 1 . (e) If | G j | = N , then stop. (4) Output the couples ( y 1 ,i , n 1 ,i ) , . . . , ( y l ( i ) ,i , n l ( i ) ,i ), where y 1 ,i , . . . , y l ( i ) ,i are the eleme n ts of B j,i and o rd( y j,i ) = n j,i ( i = 1 , . . . , k ). Pro of of correctness of BASIS1 Since in eve ry step j w e choose an elemen t x j ∈ G \ G j − 1 , at least o ne of the elemen ts x j,i do es no t b elongs to G j − 1 and so G j is strictly bigger tha n G j − 1 . F or j = 1 , 2 , . . . , BASIS1 constructs a basis for the group < { x j,i } ∪ B j − 1 ,i > and, a fter a finite n umber of steps, we obtain j = r with G r ( p i ) = G ( p i ) ( i = 1 , . . . , k ). Hence, the elemen ts of the sets B r,i ( i = 1 , . . . , k ) form a basis for G ( p i ). So, G r = G and the elemen ts of the sets B r,i ( i = 1 , . . . , k ) form a basis for G . Time Complexit y of BASIS1 Step 2 uses O ( N ǫ (log N ) 2 ) n umerical op erations. Supp ose that G r = G . A b ound fo r r is giv en b y τ ( N ), the n um b er of divisors of , and so r = O ( N ǫ ), where ǫ is arbitr a ry small. In Step 3(b), w e r ep eat r times the pro cedure ORDER and so, the time complexit y of Step 3(b) is O ( N ǫ (log N ) 2 ). The com- putation of eac h x j,i requires O (lo g m j,i ) group o p erations [9, pag e 69]. Since m j,i ≤ p a i i the time complexit y of Step 3 c(i) is O ( r k X i =1 log p a i i ) = O ( N ǫ log N ) group op erat ions. Let µ ( p i ) b e the num b er of cyclic groups whic h are direct factors of G ( p i ). The Step 3c(ii) uses the pro cedure SOL VE-EDLP and so, its time complex it y is O ( r X i =1 k X i =1 ⌈ p 1 / 2 i ⌉ | B j − 1 ,i | e ( p i )) = O ( N ǫ k X i =1 ⌈ p 1 / 2 i ⌉ µ ( p i ) e ( p i )) = O ( N ) . The Step 3c(iii) has time comple xit y O ( r k X i =1 (log s j,i + a i )) = O ( N ǫ log N ) . The Step 3c(iv) uses the pro cedure BASIS a nd so, its time complexit y is O ( r X j =1 k X i =1 (log | G j ( p i ) | ) 2 ) = O ( N ǫ (log N ) 2 ) . 7 Since | G j − 1 | < | G j | and | G j − 1 | divid es | G j | , w e ha v e P r j =1 | G j | ≤ 2 N , and so, the time complexit y of Step3(d) is O ( N ). Therefore, the time complexit y of BASIS1 is O ( N ). 4 Pro of of Theorem 2 In this section w e dev elop our algorithm for finding a basis of a finite ab elian group G in case where a generating system of G is kno wn. W e denote b y µ ( p ) the num b er of cyclic subgroups whic h are direct factors of G ( p ) and b y p e ( p ) the exponen t of G ( p ). BASIS2 Input: An ab elian group ( G, +) with | G | = N , a generating system { g 1 , . . . , g M } for G and the prime f a ctorization N = p a 1 1 · · · p a k k of N . Output: F or i = 1 , . . . , k , ( y 1 ,i , n 1 ,i ) , . . . , ( y l ( i ) ,i , n l ( i ) ,i ), where y j,i ∈ G with ord( y j,i ) = n j,i , suc h that the p i -primary comp o nen t o f G is G ( p i ) = < y 1 ,i > ⊕ · · · ⊕ < y l ( i ) ,i > . (1) Compute DIVISORS( N = p a 1 1 · · · p a k k ) = D ( N ). (2) F or j = 1 , . . . , M , compute ORDER( g j , D ( N )) = [ m j , m j, 1 , . . . , m j,k ]. (3) F or i = 1 , . . . , k and j = 1 , . . . , M , compute g j,i = m j,i g j . (4) F or i = 1 , . . . , k , Set B 1 ,i = { g 1 ,i } . F or j = 2 , . . . , M , If | < B j,i > | 6 = p a i i , (a) Compute SOL VE-EDLP( g j,i , B j − 1 ,i ) = ( z j,i , z j,i, 1 , . . . , z j,i,n ). (b) Compute the bigger in teger k j,i ≥ 0 suc h that p k j,i i divides z j,i , s j,i = z j,i /p k j,i i and h j,i = s j,i g j,i . (c) Compute BASIS( B j − 1 , h j,i , ( p k j,i i , z j,i, 1 , . . . , z j,i,n )) = B j,i . (5) Output the couples ( y 1 ,i , n 1 ,i ) , . . . , ( y l ( i ) ,i , n l ( i ) ,i ), where y 1 ,i , . . . , y l ( i ) ,i are the eleme n ts of B M ,i and ord( y j,i ) = n j,i ( i = 1 , . . . , k ). Pro of of correctness of BASIS2 F or j = 1 , . . . M the algorithm constructs a basis of the group < g j,i , B j − 1 ,i > un t il a basis of G ( p i ) is obta ined. Time Complexit y of BASIS2 Step 1 requires O ( N ǫ (log N ) 2 ) bit op erations. The complexit y o f Step 2 is O ( M N ǫ (log N ) 2 ), where ǫ is a rbitrary small. F or the computation o f ev- ery g j,i = m j,i g j w e need O ( log m j,i ) group op erations. Since m j,i ≤ p a i i , 8 Step 3 requires O ( M log N ) gro up op erations. F or i = 1 , . . . , k , the use of SOL VE-EDLP , in Step 4(a) , requires O ( e ( p i ) M ⌈ p 1 / 2 ⌉ µ ( p i ) ) o p erations, Step 4(b) O ( M log p a i i ) op erations and the use of BASIS, in Step 4(c), O ( M (log p a i i ) 2 ) op erations. Hence, the time complexit y o f BASIS2 is O ( M k X i =1 e ( p i ) ⌈ p 1 / 2 ⌉ µ ( p i ) ) . Moreo v er, using the lo wer b ound for the time complexit y of SOL VE -EDLP , w e hav e Ω( k X i =1 p ( µ ( p i ) − 1) / 2 i ) . 5 Pro of of Theorem 3 In this section, w e supp ose that the ab elian group G is cyclic. W e prop ose a simple algorithm in case where a generating system of G is kno wn. BASIS3 Input: A cyclic g r oup ( G, +) with | G | = N , a generating sy stem { g 1 , . . . , g M } for G and the prime f a ctorization N = p a 1 1 · · · p a k k of N . Output: ( y 1 , . . . , y k ), where y i ∈ G with ord( y i ) = p a i i , and so, the p i -primary comp onen t of G is G ( p i ) = < y i > . (1) Compute DIVISORS( N = p a 1 1 · · · p a k k ) = D ( N ). (2) F or j = 1 , . . . , M , compute ORDER( g j , D ( N )) = [ m j , m j, 1 , . . . , m j,k ]. (3) F or i = 1 , . . . , k and j = 1 , . . . , M , compute g j,i = m j,i g j . (4) F or i = 1 , . . . , k , find s ∈ { 1 , . . . , M } , with ord( g s,i ) = max 1 ≤ t ≤ M { ord( g t,i ) } , and set y i = g s,i . (5) Output ( y 1 , . . . , y k ). Pro of of correctness of BASIS3 W e remark that G ( p i ) = < g 1 ,i , . . . , g M ,i > . On the other hand, since G is a cyclic group, we ha v e G ( p i ) ∼ = Z p a i i . It follows that, the elemen t ha ving the maxim um order among g 1 ,i , . . . , g M ,i , is the generator of G ( p i ). Time Complexit y of BASIS3 The Step 1 and 2, require O ( N ǫ (log N ) 2 ) and O ( M N ǫ (log N ) 2 ) group op era- tions, resp ectiv ely . The Step 3 needs O ( M log N ) group op eratio ns. The Step 4 needs O ( k M ) opera t ions, and since k = O (log log N ) [10, page 359], it follo ws 9 that the time complexit y of this step is O ( M log log N ) op erat io ns. Therefore, the time complexit y of BASIS3, is O ( M N ǫ ). Ac kno wledgmen ts The first author gratefully ac kno wledges supp ort of the pro j ect Autonomic Net work Arhitecture (ANA), under con tract n umber IST-27489, whic h is funded b y the IST FET Program of the Europ ean Commision. References [1] W. M. Beynon and C . S. Iliop oulos, Computing a b asis f or a finite ab elian p -group, Information Pr o c essing L e tters, 20 (1985) 161-163 . [2] M. Borges-Quin tana, M. A. Borges-T renard and E . Martinez-Moro, On th e use of Gr obner Bases for C omp uting the s tructure of fin ite ab elian groups. Computer Algebra in S cien tific Computing 8th Inte rnational W orkshop, CASC 2005, Kalamata, Greece, September 12-16, 2005, Pro ceedings Series: Lecture Notes in Compu ter Science , V ol. 37 18, 52 -64. [3] J. Buc hmann and A. Schmidt, Computing the structure of a finite ab elian group, Mathematics of Computation, 74 (20 05) 20 17-2026. [4] L. Chen, Algorithms and their complexit y analysis for some prob lems in finite group, Journal of Sandong Normal University , in Chin ese 2 (1984) 27-33. [5] L. Chen and B. F u, Linear and Sublinear Time Algorithms f or the Basis of Ab elian groups, Ele ctr onic Col lo quium on Comp utational Complexity, R evision 1 of R ep ort No 52 (2007). [6] K. H. Cheun g and Mic h ele Mosca, Decomp osing finite ab elian groups, Quantum Inf. Comput., 1 (2001 ), no. 3, 26-32. [7] I. Din ur, E. Grigo rescu, S. Kopparty and M. Sud an, Deco dabilit y of Group Homomorphisms b eyo nd the John son Bound , Ele ctr onic Col lo quiu m on Computation al Complexity, R ep ort No 20 (2008) and in 40th STOC 2008 (to app ear). [8] M. Garzon and Y. Z alcstein, On isomorphism testing of a class of 2-nilpotent groups, Journal of Com puter and System Scie nc es, 42 (1991) , 237-2 48. [9] J. Gathen and J. Gerhard , M o dern Computer Algebr a, Cam bridge Univ ersit y Press 1999. [10] G. H. Hardy and E. M. W right, An Intr o duction to the The ory o f Numb ers, Fifth edition, Oxford Univ ersit y Pr ess 1979. [11] C. S. Iliop oulos, Analysis of algorithms on p roblems in general ab elian groups, Information Pr o c essing L e tters, 20 (1985) , 215-2 20. 10 [12] C. S. I liop oulos, Compu ting in general ab elian groups is hard, The or etic al Computer Scienc e, 41 (1985), 81-9 3. [13] C. S. I liop oulos, W orst-case complexity b ounds on algorithms f or compu ting the canonical structure of finite ab elian groups and the Hermite and S mith n ormal forms of an inte ger matrix, SIAM J. COMPU T. 18(4), (1 989), 658-669. [14] T. Ka vith a, Effi cien t algorithms for ab elian group isomorphism and related problems. In Pr o c e e dings of F oundations of Softwar e T e chnolo gy and The or etic al Computer Scienc e, L e ctur es Notes in Computer Scienc es , 2914 (200 3), 277-288. [15] T. Kavit ha, Linear time algorithms for ab elian group isomorphism and related problems, Journal of Computer and System Scienc es, 73 (2007) , n o. 6, 986-996. [16] A. Y. Kitaev, Quan tum computations: algorithms and error correction. Russian Math. Surveys, 52 (1997 ), no. 6, 1191-1 249 [17] N. Koblitz and A. J. Menezes, A sur v ey of public-k ey cry p tosystems, SIAM R e v. 46 (2004), no. 4, 59 9-634. [18] W. Ledermann, Introd u ction to group theory , Longman Grou p Limited, London 1973. [19] C. Lomont, The hidden sub group p roblem - review and op en problems. h ttp://arxiv.org/abs/quan tph/0411037 , 200 4. [20] P . W. Sh or, P olynomial-time algorithms for pr im e factorization an d d iscrete logarithms on a quan tum compu ter, SIAM J. Compu t. 26 (1997), no. 5, 1484- 1509. [21] V. Shoup , Lo wer b oun ds for discrete logarithms and related pr oblems. In Adv ances in Cr y p tology-Eurocry p t ’97, LNCS 1233, pp. 256-266, S pringer- V erlag 199 7. [22] E. T esk e, The P ohlig-Hellman Method Generalized for Group Structure Computation, J. Symb olic Computation, 27, (1999), 521-534. [23] E. T eske, A space efficie nt algorithm f or group structure computation, Mathematics of Computation, 67 , n o. 22 4 (1998), 1637-166 3. [24] N. Vik as, An O ( n ) algorithm for ab elian p -group isomorphism and an O ( n log n ) algorithm for ab elian group isomorph ism, Journal of Computer and System Scienc es, 53 (1996), 1-9. [25] S. Y. Y an, N umb er The ory for Comp uting, Springer V erlag 20 02. 11