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Chapter 2: Mathematical Preliminaries 28
defined as 〈v 1 ,v 2 〉 = a 1 b 1 +a 2 b 2 +···+a m b m =
m∑
a i b i .
Definition 2.12 (Euclidean norm). Euclidean norm of a vector v 1 is denoted by
||v 1 || and defined as ||v 1 || = √ 〈v 1 ,v 1 〉.
Definition 2.13 (Lattice). Let v 1 ,...,v n ∈ Z m (m ≥ n) be n linearly independent
vectors. A lattice L spanned by {v 1 ,...,v n } is the set of all integer linear
combinations of v 1 ,...,v n . That is,
L =
{
v ∈ Z m |v =
i=1
}
n∑
a i v i with a i ∈ Z .
i=1
We often say that L is the lattice spanned by the rows of the matrix M whose
rows are v 1 ,...,v n . We say m is the rank of the lattice L. The set of vectors
B = {v 1 ,...,v n } is called a basis for L. The dimension of L is the number of
linearly independent vectors in B, that is, dim(L) = n. If m = n, then lattice L is
called full rank lattice.
We say that two vectors v 1 ,v 2 are mutually orthogonal if 〈v 1 ,v 2 〉 = 0. Given a
basisB = {v 1 ,...,v n }foralatticeL, aquestionofinterestiswhetheritispossible
to produce another basis B ∗ of L such that all the new basis vectors are orthogonal
to each other. This question had given rise to orthogonalization techniques in
lattices, and the main algorithm used for lattice basis orthogonalization is the
Gram-Schmidt orthogonalization process, as stated in Definition 2.14.
Definition 2.14 (Gram-Schmidt Orthogonalization). The Gram-Schmidt orthogonalization
of the set of vectors {v 1 ,...,v n } in Z m is denoted by {v 1 ∗ ,...,v n ∗ }
where
∑i−1
v ∗ ∗
i = v i − µ i,j v j
j=1
with µ i,j = 〈v i,v j ∗ 〉
||v j∗ || 2 .
After the Gram-Schmidt orthogonalization is performed on a lattice L, we can
define another invariant of the lattice, called the Determinant.
Definition 2.15 (Determinant). The determinant of L is defined as det(L) =
n∏
||v ∗ i ||, where ||v|| denotes the Euclidean norm of v, and v ∗ i arise from Gram-
i=1
Schmidt orthogonalization algorithm (Definition 2.14) applied to L.