Summary: Let a finite Abelian multiplicative group \(G\) be specified by the basis \(B = \{a_1, a_2, \dots, a_q\}\), that is, the group \(G\) is decomposed into a direct product of cyclic subgroups generated by the elements of the set \(B: G = \langle a_1\rangle \times \langle a_2\rangle \times \dots \times \langle a_q\rangle\). The complexity \(L(g;B)\) of an element \(g\) of the group \(G\) in the basis \(B\) is defined as the minimum number of multiplication operations required to compute the element \(g\) given the basis \(B\) (it is allowed to use the results of intermediate computations many times). Let \(L(G,B) = \mathop{\max}\limits_{g \in G} L(g;B)\), \(LM(G) = \mathop{\max}\limits_{B} L(G,B)\), \(Lm(G)=\mathop{\min}\limits_{B} L(G,B)\), \(M(n) = \mathop{\max}\limits_{G:|G| \leq n} LM(G)\), \(m(n) = \mathop{\max}\limits_{G : |G| \leq n} Lm(G)\), \(M_{\mathrm{av}}(n) = \left( \sum\limits_{G : |G|= n} LM(G)\right)/A(n)\), \(m_{\mathrm{av}}(n) = \left( \sum\limits_{G : |G|= n} Lm(G)\right)/A(n)\), where \(A(n)\) is the number of Abelian groups of order \(n\). In this work the asymptotic estimates for the quantities \(L(G,B)\), \(M(n)\), \(m(n)\), \(M_{\mathrm{av}}(n)\), and \(m_{\mathrm{av}}(n)\) are established.