Rankin's Constant and Blockwise Lattice Reduction

Abstract Lattice reduction is a hard problem of interest to both publickey cryptography and cryptanalysis. Despite its importance, extremely few algorithms are known. The best algorithm known in high dimension is due to Schnorr, proposed in 1987 as a block generalization of the famous LLL algorithm. This paper deals with Schnorr's algorithm and potential improvements. We prove that Schnorr's algorithm outputs better bases than what was previously known: namely, we decrease all former bounds on Schnorr's approximation factors to their (ln 2)-th power. On the other hand, we also show that the output quality may have intrinsic limitations, even if an improved reduction strategy was used for each block, thereby strengthening recent results by Ajtai. This is done by making a connection between Schnorr's algorithm and a mathematical constant introduced by Rankin more than 50 years ago as a generalization of Hermite's constant. Rankin's constant leads us to intro...

• Type: Conference
• Year: 2006
• Where: CRYPTO
• Authors: Nicolas Gama, Nick Howgrave-Graham, Henrik Koy, Phong Q. Nguyen
• Keywords: Algorithm, CRYPTO 2006, Cryptology, Schnorr's Algorithm, Smallest Volume Problem