A Hybrid of Dual and Meet-in-the-Middle Attack on Sparse and Ternary Secret LWE

Abstract

The dual attack is one of the most efficient attack algorithms for learning with errors (LWE) problem. Recently, an efficient variant of the dual attack for sparse and small secret LWE was reported by Albrecht (Eurocrypt 2017), which forces some LWE-based cryptosystems, especially fully homomorphic encryptions (FHE), to change parameters. In this paper, we propose a new hybrid of dual and meet-in-themiddle (MITM) attack, which outperforms the improved variant on the same LWE parameter regime. To this end, we adapt the MITM attack for NTRU due to Odlyzko to LWE and give a rigorous analysis for it. The performance of our MITM attack depends on the relative size of error and modulus, and hence, for a large modulus LWE samples, our MITM attack works well for quite large error. We then combine our MITM attack with Albrecht's observation that understands the dual attack as a dimension-error tradeoff, which finally yields our hybrid attack. We also implement a sage module that estimates the attack complexity of our algorithm upon LWE-estimator, and our attack shows significant performance improvement for the LWE parameter for FHE. For example, for the LWE problem with dimension n = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">15</sup> , modulus q = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">628</sup> , and ternary secret key with Hamming weight 64 which is one parameter set used for HEAAN bootstrapping (Eurocrypt 2018), our attack takes 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">112.5</sup> operations and 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">70.6</sup> bit memory, while the previous best attack requires 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">127.2</sup> operations as reported by the LWE-estimator.