The Research Of Hidden Subgroup Problems Over Semidirect Product Groups Z_p^r（?）_ΦZ_p²

We discover that qubits have a unique phenomenon called quantum superposition compared with the classical bits,which leads to a kind of quantum algorithms based on quantum Fourier transform.Shor’s factoring algorithm is one of those algorithms,and its polynomial-time complexity has serious threat to the existing public key cryptosystem.The general framework of Shor’s algorithm can be classified into hidden subgroup problem by using the language of the group theory.The Shor’s algorithm also directly solves the hidden subgroup problem of cyclic groups.Several fruitful results have been obtained in hidden subgroup problem of finite abelian cases.The current researches focus on the hidden subgroup problem of non-abelian groups.In this paper,we first review some effective algorithms for any hidden subgroup problems,including the well-known Shor’s factoring algorithm and the discrete logarithm algorithm.Then we explain how these quantum algorithm can effectively solve the hidden subgroup problem of the finite abelian group.In the end,we focus on the study of a class of semi-direct product groups(?),where p is an odd prime and integer r ≥ 3.The solution of this question can be divided into three steps.First,we obtain that r≥ 3 under the assumption of φ is an embedding map.In this condition,we calculate the structure of φ(1)(1)= Tpr-2 +1 in Zpr for 0≤T<p2.Second,we show all the subgroups of the(?),then we connect subgroups with the abelian hidden subgroup problem by using the group(xp2,y)and reduce part of subgroups to the abelian hidden subgroup problem.Third,we construct efficient quantum algorithm applying the rest cases.