Abstract
A family of ring-based cryptosystems, including the multilinear maps of Garg, Gentry and Halevi
[Candidate multilinear maps from ideal lattices, Advances in Cryptology—EUROCRYPT 2013, Lecture Notes in Comput. Sci. 7881, Springer, Heidelberg 2013, 1–17]
and the fully homomorphic encryption scheme of Smart and Vercauteren
[Fully homomorphic encryption with relatively small key and ciphertext sizes, Public Key Cryptography—PKC 2010, Lecture Notes in Comput. Sci. 6056, Springer, Berlin 2010, 420–443],
are based on the hardness of finding a short generator of a principal ideal (short-PIP) in a number field typically in
{\mathbb{Q}(\zeta_{2^{s}})}
.
In this paper, we present a polynomial-time quantum algorithm for recovering a generator of a principal ideal in
{\mathbb{Q}(\zeta_{2^{s}})}
, and we recall how this can be used to attack the schemes relying on the short-PIP in
{\mathbb{Q}(\zeta_{2^{s}})}
by using the work of Cramer et al.
[R. Cramer, L. Ducas, C. Peikert and O. Regev,
Recovering short generators of principal ideals in cyclotomic rings,
IACR Cryptology ePrint Archive 2015, https://eprint.iacr.org/2015/313],
which is derived from observations of Campbell, Groves and Shepherd
[SOLILOQUY, a cautionary tale].
We put this attack into perspective by reviewing earlier attempts at providing an efficient quantum algorithm for solving the PIP in
{\mathbb{Q}(\zeta_{2^{s}})}
.
The assumption that short-PIP is hard was challenged by Campbell, Groves and Shepherd.
They proposed an approach for solving short-PIP that proceeds in two steps: first they sketched a quantum algorithm for finding an arbitrary generator (not necessarily short) of the input principal ideal.
Then they suggested that it is feasible to compute a short generator efficiently from the generator in step 1.
Cramer et al.
validated step 2 of the approach by giving a detailed analysis.
In this paper, we focus on step 1, and we show that step 1 can run in quantum polynomial time if we use an algorithm for the continuous hidden subgroup problem (HSP) due to Eisenträger et al.
[K. Eisenträger, S. Hallgren, A. Kitaev and F. Song, A quantum algorithm for computing the unit group of an arbitrary degree number field, Proceedings of the 2014 ACM Symposium on Theory of Computing—STOC’14, ACM, New York 2014, 293–302].