Cryptology ePrint Archive: Report 2017/142

Computing generator in cyclotomic integer rings, A subfield algorithm for the Principal Ideal Problem in L(1/2) and application to cryptanalysis of a FHE scheme

Jean-François Biasse and Thomas Espitau and Pierre-Alain Fouque and Alexandre Gélin and Paul Kirchner

Abstract: The Principal Ideal Problem (resp. Short Principal Ideal Problem), shorten as PIP (resp. SPIP), consists in finding a generator (resp. short generator) of a principal ideal in the ring of integers of a number field. Several lattice-based cryptosystems rely on the presumed hardness of these two problems. In practice, most of them do not use an arbitrary number field but a power-of-two cyclotomic field. The Smart and Vercauteren fully homomorphic encryption scheme and the multilinear map of Garg, Gentry, and Halevi epitomize this common restriction. Recently, Cramer, Ducas, Peikert, and Regev showed that solving the SPIP in such cyclotomic rings boiled down to solving the PIP. In this paper, we present a heuristic algorithm that solves the PIP in prime-power cyclotomic fields in subexponential time L(1/2) in the discriminant of the number field. This is achieved by descending to its totally real subfield. The implementation of our algorithm allows to recover in practice the secret key of the Smart and Vercauteren scheme, for the smallest proposed parameters (in dimension 256).

Category / Keywords: public-key cryptography / Principal Ideal Problem, cryptanalysis, FHE

Original Publication (in the same form): IACR-EUROCRYPT-2017

Date: received 15 Feb 2017, last revised 11 Jun 2018

Contact author: alexandre gelin at uvsq fr

Available format(s): PDF | BibTeX Citation

Note: This version is a merger decided by EUROCRYPT PC.

Version: 20180611:191840 (All versions of this report)

Short URL:

[ Cryptology ePrint archive ]