In mathematics, the term lattice is used for two very different kinds of mathematical objects, arising respectively in order theory and number theory. Here, by lattice, we always mean number-theoretical lattices. Lattice theory [3, 7] is called Geometry of numbers, a name due to its founder Hermann Minkowski [5.
A lattice can be defined in many equivalent ways. Informally speaking, a lattice is a regular arrangement of points in n-dimensional space. To be more formal, we need to recall a few definitions. Let x, y ∈ ℝn denote two vectors \( (x_1,\dots,x_n) \) and \( (y_1,\dots,y_n) \) , where the x i S and y i S are real numbers. Let 〈x, y〉 denote the Euclidean inner product of x with y: 
. Let \( \|\vec{x}\| \) denote the Euclidean norm of x: ∥x∥=〈x, x〉1/2. A set of vectors \( \{ \vec{b}_1, \dots, \vec{b}_d \} \) are said to be ℝ-linearly independent if and only if any equality of the form \( \mu_1 \vec{b}_1 + \cdots + \mu_d \vec{b}_d=0 \) , where the μ i S are real numbers, implies...
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Nguyen, P.Q. (2005). LATTICE. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_225
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