CrPT🔑: Lattice Problem
Published:
Welcome to the second episode of the CrPT🔑 series. In this episode, I discuss lattices and how they are connected to cryptography. As we discussed in the introduction episode, since quantum computers are expected to break many current asymmetric algorithms, lattice-based methods have attracted growing interest as a promising post-quantum alternative. Lattice-based cryptography relies on the hardness of solving lattice problems, let’s look at the fundamental idea behind them.
What is a Lattice?
A lattice in mathematics is a regular, repeating arrangement of points in an n-dimensional space. Formally, a lattice Λ in \(\mathbb{R}^n\) is defined as the set of all integer linear combinations of n linearly independent basis vectors v₁, v₂, …, vₙ:
\[\Lambda = \left\{ a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \dots + a_n \mathbf{v}_n \;\bigg|\; a_i \in \mathbb{Z} \right\}\]Here, each vector \(\mathbf{v}_i\) serves as a basis vector, and the coefficients \(a_i\) are integers. This structure forms a discrete subgroup of \(\mathbb{R}^n\), providing the foundation for various applications in cryptography, such as constructing secure cryptographic schemes based on the hardness of lattice problems. The geometric properties of lattices, including their symmetry and densest packing arrangements, play a crucial role in their cryptographic strength and efficiency.
Let’s Simplify Things
Instead of n-dimensional space, lets narrow down it to a two-dimensional space. Our basis vectors can be denoted as \(\mathbf{v}_1\) and \(\mathbf{v}_2\). So in this case, every point in this two-dimentional lattice can be expressed as an integer linear combination of these basis vectors:
\[\Lambda = \left\{ a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 \;\bigg|\; a_1, a_2 \in \mathbb{Z} \right\}\]This means that by scaling and adding the basis vectors with integer coefficients, you can generate all the points that make up the lattice. The choice of basis vectors determines the shape and structure of the lattice.
In the interactive lattice visualization above, you can drag the blue and green basis vectors to different positions. As you move these vectors, the entire lattice adjusts in real-time to reflect the new basis. This dynamic behavior demonstrates how altering the basis vectors directly affects the arrangement and density of the lattice points.
Observe that any combination of basis vectors points to a point on the lattice it self. You can change the coffeicients fo the \(a_1\) and \(a_2\) to see how the result falls back again on the lattice. Try entering different coefficients in the input boxes and click the “Update Vector” button to observe how linear combinations of the basis vectors generate new lattice points.
Lattice to Cryptography
Lattice-based cryptography relies on the hardness of lattice problems such as,
- Shortest Vector Problem (SVP)
- Closest Vector Problem (CVP)
which are believed to be resistant to both classical and quantum attacks. Additionally, lattice-based schemes often offer advanced functionalities like fully homomorphic encryption, which allows computations to be performed directly on encrypted data, and secure multi-party computation, enhancing their versatility and security in various applications.
Shortest Vector Problem (SVP)
The Shortest Vector Problem (SVP) is a fundamental challenge in lattice theory and cryptography. Given a lattice Λ generated by a set of basis vectors, SVP seeks to find the shortest non-zero vector within the lattice. Formally, SVP can be defined as:
\[\text{Find } \mathbf{v} \in \Lambda \setminus \{\mathbf{0}\} \text{ such that } \|\mathbf{v}\| = \min \{ \|\mathbf{w}\| : \mathbf{w} \in \Lambda \setminus \{\mathbf{0}\} \}\]Closest Vector Problem (CVP)
The Closest Vector Problem (CVP) is another challenge in lattice theory and cryptography. Given a lattice Λ generated by a set of basis vectors and a target point t in \(\mathbb{R}^n\), CVP seeks to find the lattice vector v in Λ that is closest to t in terms of Euclidean distance. Formally, CVP can be defined as:
\[\text{Find } \mathbf{v} \in \Lambda \text{ such that } \|\mathbf{v} - \mathbf{t}\| = \min \{ \|\mathbf{w} - \mathbf{t}\| : \mathbf{w} \in \Lambda \}\]SVP and CVP are widely recognized as computationally hard problems, especially in high-dimensional lattices. There are lattice-based cryptographic schemes that rely on the challenge of finding the closest lattice vector to a given target. In the subsequent episodes, we will explore why these problems are so challenging and how they are effectively utilized in cryptographic algorithms.
Conclusion
In this episode of the CrPT🔑 series, we explored the basics of lattices. You had the opportunity to select two basis vectors to create your own two-dimensional lattice using the interactive visualization tool. Additionally, we briefly discussed the Shortest Vector Problem (SVP) and Closest Vector Problem (CVP), which are foundational to lattice-based cryptography. In the upcoming episodes, we’ll look deeper into these problems and their applications in securing cryptographic systems.