Constructing SVK_Lattices from Cyclic Bases

SVK-lattices, in which the shortest vector is known, are proposed for the first time. Two theorems on the relationship between cyclic lattices and SVK-lattices are proposed and proved. By these constructive theorems, SVK-lattices can be simply generated. Pseudo-cyclic lattices, whose random properties are better than cyclic lattice, are also investigated for the first time. Two algorithms are designed for generating random SVK-lattices from pseudo-cyclic lattices. A general algorithm for randomizing a lattice basis is presented at the end of this paper.