Let $\mathbf{f}$ and $\mathbf{g}$ be polynomials of a bounded Euclidean norm in the ring $\mathbb{Z}[X]/\langle X^{n}+1\rangle$ . Given the polynomial $[\mathbf{f}/\mathbf{g}]_{q}\in \mathbb{Z}_{q}[X]/\langle X^{n}+1\rangle$ , the NTRU problem is to find $\mathbf{a},\mathbf{b}\in \mathbb{Z}[X]/\langle X^{n}+1\rangle$ with a small Euclidean norm such that $[\mathbf{a}/\mathbf{b}]_{q}=[\mathbf{f}/\mathbf{g}]_{q}$ . We propose an algorithm to solve the NTRU problem, which runs in $2^{O(\log ^{2}\unicode[STIX]{x1D706})}$ time when $\Vert \mathbf{g}\Vert ,\Vert \mathbf{f}\Vert$ , and $\Vert \mathbf{g}^{-1}\Vert$ are within some range. The main technique of our algorithm is the reduction of a problem on a field to one on a subfield. The GGH scheme, the first candidate of an (approximate) multilinear map, was recently 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 Garg, Gentry and Halevi [‘Candidate multilinear maps from ideal lattices’, Advances in cryptology — EUROCRYPT 2013 (Springer, 2013) 1–17].

The aim of the discrete logarithm problem with auxiliary inputs is to solve for ${\it\alpha}$ , given the elements $g,g^{{\it\alpha}},\ldots ,g^{{\it\alpha}^{d}}$ of a cyclic group $G=\langle g\rangle$ , of prime order  $p$ . The best-known algorithm, proposed by Cheon in 2006, solves for ${\it\alpha}$ in the case where $d\mid (p\pm 1)$ , with a running time of $O(\sqrt{p/d}+d^{i})$ group exponentiations ( $i=1$ or $1/2$ depending on the sign). There have been several attempts to generalize this algorithm to the case of ${\rm\Phi}_{k}(p)$ where $k\geqslant 3$ . However, it has been shown by Kim, Cheon and Lee that a better complexity cannot be achieved than that of the usual square root algorithms.

We propose a new algorithm for solving the DLPwAI. We show that this algorithm has a running time of $\widetilde{O}(\sqrt{p/{\it\tau}_{f}}+d)$ group exponentiations, where  ${\it\tau}_{f}$ is the number of absolutely irreducible factors of $f(x)-f(y)$ . We note that this number is always smaller than $\widetilde{O}(p^{1/2})$ .

In addition, we present an analysis of a non-uniform birthday problem.