Some mini-sized Shortest Vector Problem (SVP) instances for practice:
Problem 1 (2D - Easy)
Find the shortest non-zero vector in the lattice generated by:
b₁ = (3, 1)
b₂ = (1, 3)
Problem 2 (2D - Medium)
Find the shortest non-zero vector in the lattice generated by:
b₁ = (5, 2)
b₂ = (2, 5)
Problem 3 (3D - Easy)
Find the shortest non-zero vector in the lattice generated by:
b₁ = (2, 0, 0)
b₂ = (1, 2, 0)
b₃ = (1, 1, 2)
Problem 4 (2D - Harder)
Find the shortest non-zero vector in the lattice generated by:
b₁ = (7, 3)
b₂ = (5, 8)
Problem 5 (3D - Medium)
Find the shortest non-zero vector in the lattice generated by:
b₁ = (4, 1, 0)
b₂ = (1, 4, 1)
b₃ = (0, 1, 4)
Hints:
- Any lattice vector can be written as v = c₁b₁ + c₂b₂ + ... where cᵢ are integers
- Try small integer combinations first (0, ±1, ±2)
- Calculate ||v||² = v·v for each candidate
- The shortest vector has minimal Euclidean norm
Some mini-sized Closest Vector Problem (CVP) instances:
Problem 1 (2D - Easy)
Given the lattice generated by:
b₁ = (3, 0)
b₂ = (0, 3)
Find the closest lattice vector to target: t = (5, 7)
Problem 2 (2D - Medium)
Given the lattice generated by:
b₁ = (4, 1)
b₂ = (1, 4)
Find the closest lattice vector to target: t = (6, 8)
Problem 3 (2D - Harder)
Given the lattice generated by:
b₁ = (5, 2)
b₂ = (2, 5)
Find the closest lattice vector to target: t = (7.5, 9.3)
Problem 4 (3D - Easy)
Given the lattice generated by:
b₁ = (2, 0, 0)
b₂ = (0, 2, 0)
b₃ = (0, 0, 2)
Find the closest lattice vector to target: t = (3, 5, 7)
Problem 5 (3D - Medium)
Given the lattice generated by:
b₁ = (3, 1, 0)
b₂ = (1, 3, 1)
b₃ = (0, 1, 3)
Find the closest lattice vector to target: t = (5, 6, 4)
Problem 6 (2D - Tricky)
Given the lattice generated by:
b₁ = (6, 2)
b₂ = (3, 7)
Find the closest lattice vector to target: t = (10, 10)
Hints:
- Lattice vectors have form: v = c₁b₁ + c₂b₂ + ... (cᵢ integers)
- Calculate distance: ||t - v||² for candidate vectors
- Start with Babai's nearest plane algorithm (round coefficients in basis representation)
- Check neighboring integer combinations to ensure you found the minimum