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[Answers] Help in understanding exactly how lattices used as one way functions for hashing

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(This question is related to homework) 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 only 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 . I really hope I have not repeated any questions through my lack of understanding. Thanks in advance.

Asked By : user2012620

Answered By : Yuval Filmus

You have several confusions regarding cryptography. First, the nature of hash functions. The non-cryptographic use of hash functions is to hash long messages into shorter ones in a "uniform" fashion. So we expect there to be many collisions, by design. Cryptographic hash functions have the additional properties that collisions are hard to find (there are several, inequivalent ways of stating this). Therefore, while it is possible to find collisions even for cryptographic hash functions (simply because the range is smaller than the domain), this should be difficult. Such hash functions can be used in various ways in cryptography, for example in digital signatures: you sign on the (small) hash rather than the (large) message.

Second, encryption is a different primitive from hash functions. Encryption itself comes in two main kinds, symmetric and public key, which are rather different, and there are many other cryptographic constructions out there. There are reductions between some of the cryptographic "primitives" (hash functions, one-way functions and the like), for example you can use them to construct HMACs (message authentication codes) and PRNGs (pseudorandom number generators), but perhaps not encryption schemes.

Regarding the matter at hand: the idea of Ajtai's hash function is to use a random lattice $A$, the hash function being $x \mapsto Ax \pmod{q}$, for some prime $q$. Here $x$ is a binary vector in $\{0,1\}^m$, embedded into $\mathbb{Z}_q^m$. The parameters here are important — the size of $A$ and $q$ — but I'll ignore them here. In order to show that this is cryptographically secure, under one definition we need to show that it is difficult to find a collision, that is two messages $x,y$ with the same hash. Such messages would satisfy $A(x-y) \equiv 0 \pmod{q}$, and so $x-y$ would be in the dual lattice $A^{\perp}$. Now $x-y \in \{-1,0,1\}^m$, so (for the correct choice of parameters) it is "short", and finding it is conjectured to be difficult. This shows the (conditional) security of the scheme.

Lattice-based encryption can also be used for public-key encryption (Regev's scheme) and for homomorphic encryption. These schemes rely on a different problem, LWE (learning with errors), which is similar to CVP (closest vector problem). The details are more complicated &mdash you can have a look at Regev's scheme in Section 5.4 of his survey with Micciancio, though it could be hard to glean from there why exactly the scheme works. Perhaps this is explained later in your course, and if you need more explanations, you can ask another question here.

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