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In this paper, we construct a k-RSA algorithm in which the idea of kth power residue theory and RSA algorithm are combined. This algorithm not only inherits the advantage of RSA, whose security depends on the difficulties of factoring large integers and finding discrete logarithms, but also has high flexibility of parameters. It is designed for improved security and to achieve a balance between speed and space. At the same time, it can realize following functions: hierarchical system management, secret sharing and so on.