On the quantum attacks against schemes relying on the hardness of finding a short generator of an ideal in ℚ(𝜁<sub>2<sup>𝑠</sup> </sub>)

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℚ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:msup> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{Q}(\zeta_{2^{s}})} . In this paper, we present a polynomial-time quantum algorithm for recovering a generator of a principal ideal in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℚ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:msup> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{Q}(\zeta_{2^{s}})} , and we recall how this can be used to attack the schemes relying on the short-PIP in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℚ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:msup> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℚ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ζ</m:mi> <m:msup> <m:mn>2</m:mn> <m:mi>s</m:mi> </m:msup> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\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].