 One of the more intriguing types of cryptographic systems that have been invented in recent years is based on mathematically, what is mathematically known as a lattice, and this emerges as follows. Suppose I have a set of n dimensional vectors, a little bit of a review of vector spaces that you may have seen in things like linear algebra. The vector space V is the set of all possible linear combinations of these vectors, and these vectors are said to span our vector space. The subset of vectors, a subset of the vectors is said to be linearly independent if and only if there is no non-trivial linear combination of the vectors equal to the zero vector, which is to say a trivial linear combination zero of these, zero of these, and zero of everything. In fact, well that's a trivial combination, but if I can find some non-trivial linear combination of the vectors, then that's equal to the zero vector, then the set of vectors is not linearly independent. And then finally, a basis of V is a set of linearly independent vectors that span our vector space V. Now if I require that the components of the vectors and the coefficients of all linear combinations be integers, then the vector space V that I obtain is called a lattice. And part of the reason that it's called a lattice is that if we take the viewpoint that any linear combination of these vectors corresponds to some point in our vector space, then the set of points that we're talking about, if everything is scientists and integer, consists of a set of discrete points. Now one of the things we can do is we can construct a change of basis. So my lattice has some initial set of basis vectors, but I can create a new set of basis vectors for the lattice in the following way. What I can do, first of all, is I'll take my matrix B and my rows of this matrix are going to consist of the vectors of our basis. If you remember the term, the vector space corresponds to the row space of this matrix. And then I'm going to take another matrix M and I'm going to require two things happen. I'm going to require that the determinant of M be plus or minus one. It doesn't really matter which. And the matrix product M times B is defined. So again, I have to have the right number of rows and columns in M. And I'll note also one requirement also. We want to make sure that M also has purely integers in its column and row entries. And then finally I will form a new basis by finding the row space of the matrix M times B. In other words, this matrix M times B, the rows of that matrix will consist of my new basis. Well that's the linear algebra. It's always nice to go back to what this actually means. What we've done is equivalent to forming a new set of basis vectors where every basis vector in our new set is some specific linear combination of the old basis vectors. And here's the, here's where everything ties together. The coefficients of our linear combinations are going to be the entries of M. And the requirement that the determinant of M be plus or minus one guarantees that the vector space spanned by the new vectors will be the same as the vector space spanned by the old vectors. Just as a quick observation of why this could be a problem, if every basis vector was twice one of the basis vectors, then I wouldn't be able to get all of the points of my original lattice. I'd only be able to get every other point in the lattice. For example, let's take a vector space V and suppose it has bases, so I don't know, 5, 1 and negative 2, 8. And I want to find a new basis for V. So I'm going to, first of all, form my matrix B, where the rows are going to consist of my basis vectors 5, 1 and negative 2, 8. I'm going to find another matrix M of suitable size where the determinant is plus or minus one. Since this is a 2 by 2 matrix, M also has to be a 2 by 2 matrix, and I'll pick this one. Here's a quick check. It's a 2 by 2 matrix, so I can find the determinant very easily. It's 99 minus 100, suitably negative 1. And so my product is going to be my new basis, 3741, 103, 113. And this gives me my new set of basis vectors, which will be the rows of the product matrix, 3741 and 103, 113.