A mathematical technique called lattice reduction that is used in numerical analysis, computational geometry, and cryptography to work with lattices in high-dimensional settings. A lattice is a Euclidean space grid-like structure composed of integer combinations of a set of basis vectors in mathematics. A reduced lattice's basis vectors are shorter and nearly orthogonal, which simplifies many computer problems.
Lattice Reduction Algorithms search for an "optimal" or "reduced" basis for a lattice to tackle certain problems, such as the ones listed below.
- Programs utilizing linear algebra with integers.
- Cryptanalysis refers to attacks against lattice-based cryptography algorithms like LWE and NTRU.
- The approximate Diophantine method.
- Whether the shortest vector problem (SVP) or the closest vector problem (CVP).
A popular lattice reduction method that finds a reduced basis for any given lattice in polynomial time is the Lenstra-Lenstra-Lovász (LLL) algorithm. The Lenstra-Lenstra-Lovász (LLL) algorithm reduces a lattice's basis in polynomial time. The objective is to change the provided basis into a shorter, more orthogonal basis that is more accessible.
If the input is:
A basic B= {b 1,b 2,...,b n}, where each bi is a vector in Rn
A Constant δ where ¼< δ<1 typically δ=0.75
Then the output is:
A basis vector that is "nearly orthogonal" and has minimal Euclidean norms is a reduced basis B ′.
Key Concepts:
- A lattice results from the linearly distributed combinations with integer coefficients of basis vectors belonging to a point group in n-dimensional space.
- The set of vectors B= {b 1,b 2,...,b n} that yields the lattice L(B) is known as the lattice basis.
- Narrowed Basis: The freshly computed lattice bases possess smaller and more orthogonal basis vectors, which results in a simplification of the structure of the lattice.
- An abbreviated near-orthogonal basis for a lattice of a given dimension may be computed using the well-known LLL lattice reduction algorithm. It finds great use in applications to cryptography because of both its efficiency and efficacy.
Algorithm Pseudocode Gram-Schmidt Orthogonalization:
Algorithm LLL (B, δ):
Input:
- A basis B = {b_1, b_2, ..., b_n} in R^n
- Reduction parameter δ (e.g., δ = 0.75)
1. Compute Gram-Schmidt orthogonalization B* = {b_1^*, b_2^*, ..., b_n^*}
2. Set k = 1
3. While k < n:
a. For j = k-1 to 1:
i. Compute μ_k,j = (b_k ⋅ b_j^*) / (b_j^* ⋅ b_j^*)
ii. Size reduce b_k:
b_k ← b_k - round(μ_k,j) * b_j
b. Recompute Gram-Schmidt orthogonalization for updated B
c. Check Lovász condition:
If ||b_k^*||^2 < (δ - μ_k,k-1^2) * ||b_{k-1}^*||^2:
i. Swap b_k and b_{k-1}
ii. Recompute Gram-Schmidt orthogonalization
iii. Set k = max(k-1, 1)
Else:
i. Set k = k + 1
4. Output reduced basis B
Example:
Let us take an example to illustrate the Lattice Reduction Algorithm in C++ .
#include <iostream>
#include <vector>
#include <cmath>
#include <algorithm>
// Function to compute dot product of two vectors
double dotProduct(const std::vector<double>& v1, const std::vector<double>& v2) {
double dot = 0.0;
for (size_t i = 0; i < v1.size(); ++i) {
dot += v1[i] * v2[i];
}
return dot;
}
// Gram-Schmidt orthogonalization
std::vector<std::vector<double>> gramSchmidt(const std::vector<std::vector<double>>& basis) {
size_t n = basis.size();
std::vector<std::vector<double>> orthoBasis(n, std::vector<double>(basis[0].size(), 0.0));
for (size_t i = 0; i < n; ++i) {
orthoBasis[i] = basis[i]; // Start with original vector
// Subtract projections on previously computed orthogonal vectors
for (size_t j = 0; j < i; ++j) {
double projection = dotProduct(orthoBasis[j], basis[i]) / dotProduct(orthoBasis[j], orthoBasis[j]);
for (size_t k = 0; k < orthoBasis[i].size(); ++k) {
orthoBasis[i][k] -= projection * orthoBasis[j][k];
}
}
}
return orthoBasis;
}
// Function to perform LLL lattice reduction
void LLL(std::vector<std::vector<double>>& basis, double delta = 0.75) {
size_t n = basis.size();
size_t dim = basis[0].size(); // Dimension of each vector
std::vector<std::vector<double>> orthoBasis = gramSchmidt(basis);
size_t k = 1;
while (k < n) {
// Size reduction step
for (size_t j = k - 1; j < k; --j) {
double mu = dotProduct(basis[k], orthoBasis[j]) / dotProduct(orthoBasis[j], orthoBasis[j]);
for (size_t i = 0; i < dim; ++i) {
basis[k][i] -= round(mu) * basis[j][i];
}
}
// Recompute the orthogonalization for the modified vectors
orthoBasis = gramSchmidt(basis);
// Lovász condition check
double lhs = dotProduct(orthoBasis[k], orthoBasis[k]);
double rhs = (delta - pow(dotProduct(orthoBasis[k - 1], orthoBasis[k]) / dotProduct(orthoBasis[k - 1], orthoBasis[k - 1]), 2))
* dotProduct(orthoBasis[k - 1], orthoBasis[k - 1]);
if (lhs >= rhs) {
k++;
} else {
// Swap basis vectors b_k and b_(k-1)
std::swap(basis[k], basis[k - 1]);
orthoBasis = gramSchmidt(basis); // Recompute Gram-Schmidt for the new basis
k = std::max(k - 1, static_cast<size_t>(1)); // Ensure k doesn't go below 1
}
}
}
int main() {
// Example lattice basis (3D space example)
std::vector<std::vector<double>> basis = {
{1, 1, 1},
{1, 0, 2},
{0, 0, 1}
};
LLL(basis);
// Output reduced basis
std::cout << "Reduced basis:" << std::endl;
for (const auto& vec : basis) {
for (double x : vec) {
std::cout << x << " ";
}
std::cout << std::endl;
}
return 0;
}
Output:
Reduced basis:
0 0 1
0 -1 0
1 0 0
Conclusion:
In conclusion, the Lenstra-Lenstra-Lovász (LLL) method from computational number theory is a spectacular algorithm in lattice-based cryptography. The LLL method converts it to a roughly orthogonal form, which allows it to approximately solve difficult issues like the shortest vector problem (SVP) and closest vector problem (CVP). It has been known to be useful for large applications due to its polynomial time. In the implementation of the LLL algorithm in C++, manipulation of vector and matrix operations is necessary. Improper memory management may cause many problems, such as segmentation faults. The correctness and efficiency of the algorithm can be preserved by applying orthogonalization of Gram-Schmidt and size reduction processes repeatedly. LLL reduction remains a useful technique within applied and theoretical mathematics and is important within the ever-blooming field that is lattice-based cryptography.