# (Integer) Lattice definition of volume

I am heavily invested into lattice-based cryptography, but I do have one confusing issue dealing with the definition of the "volume", I have found at least two, which seems conflicting to me,

The volume vol(L) (or determinant) of the lattice L is the square root of the determinant of the Gram matrix of any basis of L... i.e.:

$$vol\,L = \sqrt{\det G(b_1, \dots, b_n)}$$

and

Let L be a lattice of dimension n and let F be a fundamental domain for L. Then the n-dimensional volume of F is called the determinant of L (or sometimes the covolume of L). It is denoted by $\det L$, i.e.:

$$vol\, L = \vert \det F(b_1,\dots,b_n) \vert$$

The very first one is from the research paper, the second one is from John Silverman's book called Introduction to mathematical cryptography.

Am I missing something? Are these definitions equivalent? It seems like the second one is computationally less complex due to $n^2$ of inner products that occurs in Gram matrix.

They are the same. If the columns of matrix $F$ are the edges of a fundamental parallelepiped then $$G = F^t F$$