Quantum Key Search for Ternary LWE

Ternary LWE, i.e., LWE with coefficients of the secret and the error vectors taken from {−1, 0, 1}, is a popular choice among NTRUtype cryptosystems and some signatures schemes like BLISS and GLP. In this work we consider quantum combinatorial attacks on ternary LWE. Our algorithms are based on the quantum walk framework of MagniezNayak-Roland-Santha. At the heart of our algorithms is a combinatorial tool called the representation technique that appears in algorithms for the subset sum problem. This technique can also be applied to ternary LWE resulting in faster attacks. The focus of this work is quantum speed-ups for such representation-based attacks on LWE. When expressed in terms of the search space S for LWE keys, the asymptotic complexity of the representation attack drops from S (classical) down to S (quantum). This translates into noticeable attack’s speedups for concrete NTRU instantiations like NTRU-HRSS [CHES’17] and NTRU Prime [SAC’17]. Our algorithms do not undermine current security claims for NTRU or other ternary LWE based schemes, yet they can lay ground for improvements of the combinatorial subroutines inside hybrid attacks on LWE.