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:

• $L=$ Lattice

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

• $n =$the dimension of the lattice

Distance function

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

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

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

Shortest vector problem

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

Approximate SVP

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

Decision SVP (GapSVP)

Given a lattice $L$with a basis $\mathcal B$ we must distinguish if $\lambda_1(L) \leq 1$ or $\lambda > \gamma(n)$

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

Approximate CVP

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

Decision CVP (GapCVP)

Given a lattice $L$with a basis $\mathcal B$ and a vector $w$ we must decide if

• There exists $v \in L$s.t $\| v - w\| \leq 1$
• $\forall v \in L: \|v - w\| > \gamma(n)$

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 $L$ with an arbitrary basis $B$, a vector $w \in \mathbb{R}^n$ and a real number $d \in \mathbb{R}$ find a lattice vector $v \in {L}$ s.t $\|w-v\| < d \cdot \lambda_1({L})$

Remark

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

Shortest independent vectors (SIVP)

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

Hardness of lattice problems

Resources

• Pictures taken from https://simons.berkeley.edu/sites/default/files/docs/14953/intro.pdf and "Cryptography made simple - Nigel Smart" and edited a bit

• Or generated by me