An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero

Abstract

Let f and g be polynomials of a bounded Euclidean norm in the ring Z[X]/⟨X+1⟩. Given the polynomial [f/g]q ∈ Zq[X]/⟨X+1⟩, the NTRU problem is to find a, b ∈ Z[X]/⟨X + 1⟩ with a small Euclidean norm such that [a/b]q = [f/g]q. We propose an algorithm to solve the NTRU problem, which runs in 2 2 λ) time when ∥g∥, ∥f∥, and ∥g−1∥ are within some range. The main technique of our algorithm is the reduction of a problem on a field to one in a subfield. Recently, the GGH scheme, the first candidate of a (approximate) multilinear map, was found to be insecure by the Hu–Jia attack using low-level encodings of zero, but no polynomial-time attack was known without them. In the GGH scheme without low-level encodings of zero, our algorithm can be directly applied to attack this scheme if we have some top-level encodings of zero and a known pair of plaintext and ciphertext. Using our algorithm, we can construct a level-0 encoding of zero and utilize it to attack a security ground of this scheme in the quasi-polynomial time of its security parameter using the parameters suggested by [GGH13].