Fast Practical Lattice Reduction through Iterated Compression

Abstract: We introduce a new lattice basis reduction algorithm with approximation guarantees analogous to the LLL algorithm and practical performance that far exceeds the current state of the art. We achieve these results by iteratively applying precision management techniques within a recursive algorithm structure and show the stability of this approach. We analyze the asymptotic behavior of our algorithm, and show that the heuristic running time is $O(n^{\omega}(C+n)^{1+\varepsilon})$ for lattices of dimension $n$, $\omega\in (2,3]$ bounding the cost of size reduction, matrix multiplication, and QR factorization, and $C$ bounding the log of the condition number of the input basis $B$. This yields a running time of $O\left(n^\omega (p + n)^{1 + \varepsilon}\right)$ for precision $p = O(\log \|B\|_{max})$ in common applications. Our algorithm is fully practical, and we have published our implementation. We experimentally validate our heuristic, give extensive benchmarks against numerous classes of cryptographic lattices, and show that our algorithm significantly outperforms existing implementations.

cryptography and securityMathematics

Audience: researchers in the discipline

Comments: Keegan Ryan is a 4th year PhD student advised by Prof. Nadia Heninger at the University of California, San Diego. His research interests include practical cryptanalysis of real-world systems, particularly problems involving lattice reduction.

Florida Atlantic University Crypto Café

Series comments: A seminar series of the FAU crypto group in the mathematics department. We welcome speakers, both online or in person, to join us and discuss their research or job-related opportunities. Beach lovers - come and believe!