Paper 2023/582

New NTRU Records with Improved Lattice Bases

Abstract

The original NTRU cryptosystem from 1998 can be considered the starting point of the great success story of lattice-based cryptography. Modern NTRU versions like NTRU-HPS and NTRU-HRSS are round-3 finalists in NIST's selection process, and also Crystals-Kyber and especially Falcon are heavily influenced by NTRU. Coppersmith and Shamir proposed to attack NTRU via lattice basis reduction, and variations of the Coppersmith-Shamir lattice have been successfully applied to solve official NTRU challenges by Security Innovations, Inc. up to dimension $n=173$. In our work, we provide the tools to attack modern NTRU versions, both by the design of a proper lattice basis, as well as by tuning the modern BKZ with lattice sieving algorithm from the G6K library to NTRU needs. Let $n$ be prime, $\Phi_n := (X^n-1)/(X-1)$, and let $\mathbb{Z}_q[X]/(\Phi_n)$ be the cyclotomic ring. As opposed to the common belief, we show that switching from the Coppersmith-Shamir lattice to a basis for the cyclotomic ring provides benefits. To this end, we slightly enhance the LWE with Hints framework by Dachman-Soled, Ducas, Gong, Rossi with the concept of projections against almost-parallel hints. Using our new lattice bases, we set the first cryptanalysis landmarks for NTRU-HPS with $n \in [101,171]$ and for NTRU-HRSS with $n \in [101,211]$. As a numerical example, we break our largest HPS-171 instance using the cyclotomic ring basis within $83$ core days, whereas the Coppersmith-Shamir basis requires $172$ core days. We also break one more official NTRU challenges by Security Innovation, Inc., originally worth 1000\$, in dimension $n=181$ in $20$ core years.