I am doing a cryptography course via long distance and we have been given an assignment which is based on lattice-based cryptography. I have spent the majority of the past week sifting through papers and videos in an attempt to build my understanding of the subject, but due to the intensely technical manner in which information on the subject is presented, I have not been able to answer many questions in my mind. Due to the structure of my long distance course I have no access to professors or extensive libraries so my question is one which merely seeks to increase my understanding, please bear with me.
Thus far I have understood that:
- Lattices are a collection of regularly ordered points in euclidean space (its terms like this which have caused me to be searching for answers for days)
- A lattice may be defined through n vectors called a basis where n = the dimension (for now I am working in dimension 2), and any other basis can be found by applying positive or negative multiples of each vector in the basis to another vector in the basis
- Defining a lattice L1 as a set of points allows one to multiply the whole set by some co-efficient which will essentially transform it into a new lattice L2
- The determinant of the vectors of the basis essentially tells us the volume (area in 2 dimensions) of the parallelpiped which will be repeated to form the lattice, the lattice points being the vertices.
Having said this, I am now stuck in understanding how I move all of this to cryptography. I am on a question which asks me to compute the output for 3 inputs to a function:
We will see that q-ary lattices give provably collision-resistent hashing. We choose integers q; a and b. Our hash function (presented by Mikl´os Ajtai in a breakthrough paper in 1996) is a 2-variable function: h(x, y) = ax + by mod q. (a) For q = 5; a = 14; b = 13; compute h(17, 8), h(21, 16) and h((17, 8) - (21, 16))
Having found the answers…
h(17, 8) = 14*17 + 13*8 = 342 mod 5 = 2; h(21, 16) = 14*21 + 13*16 = 502 mod 5 = 2; h((17, 8)-(21, 16)) -> (h(-4, -8) = 14*-4 + 13*-8 = 160 mod 5 = 0
I believe this means that with the first two points the lattice has been moved by some multiple of 5 plus two, which has displaced it by two and thus formed a new lattice. While in the last case, the lattice has been moved by a multiple of its determinant and thus is the same lattice.
However, I just cannot understand the link to hashing. In an application would the points be the cleartext and the
answer modulo 5 the output of the hash function? If so would'nt there be too many collisions as we have already seen two examples of which give the answer
2? How exactly is this linked to the shortest vector problem or short integer solution if they are not the same thing? Are q-ary lattices aonly an attempt to store the basic pattern needed to reproduce the lattice? and if so wouldn't the lattice itself be the cleartext and the basic pattern (given by the basis and the determinant) the answer from the hash function? How exactly would you encrypt a message using a lattice?
I just wish I could find some text which explained this in English rather than symbols… like this http://www.wisdom.weizmann.ac.il/~odedg/COL/cfh.pdf . I really hope I have not repeated any questions through my lack of understanding. Thanks in advance.