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I am studying The Mathematics of Lattice-Based Cryptography from Alfred Menezes' Cryptography 101 course. In slide 6 (Ring-SIS and Ring-LWE), page 83, it states that $L(A)$ is a rank $n$ lattice. I understand that a lattice's rank cannot exceed its dimension. I have the following questions:Page 83

  1. Is $A$ a bases for $L$?
  2. $A$ has $m$ columns, where $m = \ell \times n > n$. Since a basis can have at most $n$ columns (full-rank), can we conclude that some rows are linearly dependent on others?
  3. If $A$ is not a basis, what is a basis?
Mahesh S R
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Omid Bodaghi
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1 Answers1

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  1. $A$ is a basis of $L$ (do not get confused with the notion of basis for vector spaces, the lattice $L(A)$ is $\mathbb{Z}$-module, not a vector space)
  2. $A$ has $n$ rows and $nl$ columns and since it is of rank $n$ then there are no linearly dependent rows.
Don Freecs
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