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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), where ∆ K denotes 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).

Lecture Notes in Computer Science 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2017) https://hal.archives-ouvertes.fr/hal-01518438 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT 2017), Apr 2017, Paris, France. Lecture Notes in Computer Science, 10210, pp.60--88, 2017, Advances in Cryptology – EUROCRYPT 2017 36th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Paris, France, April 30 – May 4, 2017, Proceedings, Part I. <10.1007/978-3-319-56620-7_3>ARRAY(0x7f5471253390) 2017-04-30