Generating cryptographically-strong random lattice bases and recognizing rotations of $\mathbb{Z}^n$
This work addresses vulnerabilities in lattice-based cryptography for secure communication, revealing that some widely used random basis generation methods are insecure, which is incremental but critical for practical security.
The paper tackled the problem of generating cryptographically-strong random lattice bases by comparing methods for sampling random elements in GL(n,ℤ), finding that standard algorithms like Magma's RandomSLnZ and a method from a NIST submission (DRS) can be efficiently broken even at high dimensions (e.g., nearing 1,500) or 256-bit security settings, while other algorithms appear stronger.
Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in $GL(n,\mathbb{Z})$. We compare the strengths of various methods to sample random elements of $GL(n,\mathbb{Z})$, finding some are stronger than others with respect to the problem of recognizing rotations of the $\mathbb{Z}^n$ lattice. In particular, the standard algorithm of multiplying unipotent generators together (as implemented in Magma's RandomSLnZ command) generates instances of this last problem which can be efficiently broken, even in dimensions nearing 1,500. Likewise, we find that the random basis generation method in one of the NIST Post-Quantum Cryptography competition submissions (DRS) generates instances which can be efficiently broken, even at its 256-bit security settings. Other random basis generation algorithms (some older, some newer) are described which appear to be much stronger.