🔒 Exploring Lattice Cryptography & Why GGH Fails 🔒
Recently, I dove into lattice-based cryptography — a fascinating field that underpins many post-quantum encryption schemes. Two core problems define its security:
Shortest Vector Problem (SVP): Find the shortest vector in a lattice.
Closest Vector Problem (CVP): Given a target point, find the closest lattice vector.
One of the earliest lattice schemes, GGH encryption, works like this:
Alice has a private “good” basis (nearly perpendicular vectors) and publishes a public “bad” basis (skewed vectors).
Bob encodes a message by picking lattice coordinates in the public basis, adds a small error, and sends the ciphertext to Alice.
Alice decrypts by solving the closest vector problem using her private basis.
✅ Sounds secure in theory — but in practice, GGH is breakable:
LLL / BKZ basis reduction: Attackers can transform the public “bad” basis into a nearly orthogonal one.
Babai’s nearest-plane algorithm: With a reduced basis, the closest lattice point (the original message) is easily recovered.
Embedding trick (CVP → SVP): CVP can be converted to a shortest vector problem in a slightly higher-dimensional lattice, allowing attackers to use SVP heuristics.
💡 Key Insight: A “bad” public basis still generates the same lattice. Modern lattice reduction algorithms can often reconstruct enough of the private structure to decode messages — which is why GGH is considered insecure today.
Tiny example in 2D:
Private basis: [[7,1],[1,6]][ [7,1],[1,6] ][[7,1],[1,6]]
Public basis: [[7,15],[1,8]][ [7,15],[1,8] ][[7,15],[1,8]]
Encrypt message [1,2][1,2][1,2] with small error → ciphertext [38,16][38,16][38,16]
LLL reduces the public basis back to nearly the private one, Babai’s rounding recovers the message effortlessly.
✨ Takeaway: Lattice cryptography is powerful, but parameter choice and structure matter. Modern schemes like LWE and NTRU fix these weaknesses and are the foundation for post-quantum security.