Hard lattice problems

This section is not complete. Help is needed with relevance + examples in cryptography, algorithms + hardness, relations between problems.

Also needs review from more experienced people.

Introduction

Now that we are comfortable with lattices we shall study why are they important to cryptography.

Like we said, when we construct cryptosystems we usually are looking for hard problems to base them on. The lattice world provides us with such problems such as the shortest vector problem or the closest vector problem.

What makes lattices even more special is that some cryptographic problems (which we will study in the next chapter) can be reduced to worst-case lattice problems which makes them crazy secure. Moreover, some problems are even secure against quantum computers.

But enough talk, let's get right into it!

Shortest vector problem + GapSVP

Before we go into any problems we must first define the concept of distance in a lattice.

Let:

  • L=L= Lattice

  • B=\mathcal B = the basis of the lattice

  • n=n = the dimension of the lattice

Distance function

Given some distance function (Example: Euclidean norm) the distance from a vector tt to the lattice LL is the distance from the vector to the closest point in the in lattice.

μ(t,L)=minvLtv\mu(t, L) = \underset{v \in \mathcal{L}}{\min}{\|t-v\|}

We will denote the length of the shortest vector with v=λ1(L)\|v\| = \lambda_1(L)and the length of the next independent vectors in order with λi(L)λ1(L)λ2(L)...λn(L)\lambda_i(L) \Rightarrow\lambda_1({L}) \leq \lambda_2({L}) \leq ... \leq \lambda_n({L})

Shortest vector problem

Given a lattice LL and an arbitrary basis B\mathcal{B} for it our task is to find the shortest vector vLv \in L.

Approximate SVP

We relax the SVP problem a bit. Given an arbitrary basis B\mathcal{B}find a shortest nonzero lattice vector vLv \in Lsuch that v<γ(n)λ1(L)v < \gamma(n)\cdot \lambda_1(L). Here γ(n)>1\gamma(n) > 1is some approximation factor.

Decision SVP (GapSVP)

Given a lattice LLwith a basis B\mathcal B we must distinguish if λ1(L)1\lambda_1(L) \leq 1 or λ>γ(n)\lambda > \gamma(n)

Sage example

# We can find the shortest vector using the LLL algorithm
M = matrix([[-1, 2], [-2, 3]])
B = M.LLL()
print(B[0])
# (0, -1)
# Or we can use the Integer Lattice class
L = IntegerLattice(M)
L.shortest_vector()
# (-1, 0)

Closest Vector problem + GapCVP

Closest vector problem

Given a lattice LL with an arbitrary basis B\mathcal B and a vector wRnw \in \mathbb{R}^n find the closest lattice vector to ww vL,vwμv \in {L}, \|v-w\| \leq \mu

Approximate CVP

Given a lattice LL with an arbitrary basis B\mathcal B and a vector wRnw \in \mathbb{R}^n find the closest lattice vector to ww vL,vw<γ(n)μv \in {L}, \|v-w\| < \gamma(n) \cdot \mu

Decision CVP (GapCVP)

Given a lattice LLwith a basis B\mathcal B and a vector ww we must decide if

  • There exists vLv \in Ls.t vw1\| v - w\| \leq 1

  • vL:vw>γ(n)\forall v \in L: \|v - w\| > \gamma(n)

Sage example

M = matrix([[-1, 2], [-2, 3]])
L = IntegerLattice(M)
w = vector([1.8, 1.5])
L.closest_vector(w)
# (2.00000000000000, 2.00000000000000)

Bounded distance decoding

Given a lattice LL with an arbitrary basis BB, a vector wRnw \in \mathbb{R}^n and a real number dRd \in \mathbb{R} find a lattice vector vLv \in {L} s.t wv<dλ1(L)\|w-v\| < d \cdot \lambda_1({L})

Remark

  • If we have d<12d < \dfrac 12 the solution to the BDD problem is guaranteed to be unique.

Shortest independent vectors (SIVP)

Given a full rank lattice LL with an arbitrary basis B\mathcal Bfind nn linearly independent lattice vectors of length at most λn(L)maxiviλn(L)\lambda_n(L) \Rightarrow \max_i\|v_i\| \leq \lambda_n(L) or maxiviγ(n)λn(L)\max_i|v_i| \leq \gamma(n) \lambda_n(L)for the approximate version.

Hardness of lattice problems

Resources