Short Principal Ideal Problem in multicubic fields

Abstract

One family of candidates to build a post-quantum cryptosystem upon relies on euclidean lattices. In order to make such cryptosystems more efficient, one can consider special lattices with an additional algebraic structure such as ideal lattices. Ideal lattices can be seen as ideals in a number field. However recent progress in both quantum and classical computing showed that such cryptosystems can be cryptanalysed efficiently over some number fields. It is therefore important to study the security of such cryptosystems for other number fields in order to have a better understanding of the complexity of the underlying mathematical problems. We study in this paper the case of multicubic fields.

Norm relations and computational problems in number fields

“…• For every ℤ[𝐺]-module 𝑀 and 𝑥 ∈ 𝑀, we have 𝑁 𝐻 𝑥 ∈ 𝑀 𝐻 This is the relation used by Parry [44] and by Lesavourey, Plantard and Susilo [35].…”

“…Then we have the norm relation $$\begin{equation*} 3 = N_{\langle u\rangle } + N_{\langle v\rangle } + N_{\langle uv\rangle } - (u+uv)N_{\langle u^2v\rangle }. \end{equation*}$$This is the relation used by Parry [44] and by Lesavourey, Plantard and Susilo [35].…”

“…Recent work of Bauch, Bernstein, de Valence, Lange and van Vredendaal [10] describes how to reduce the computation of principal ideal generators in multiquadratic fields (that is, $$G = C_2^n$$) to quadratic subfields, thus for the first time improving (both in theory and practice) upon classical algorithms by exploiting subfields. This was then generalized to the computation of $$S$$‐units by Biasse and van Vredendaal [13] and to multicubic fields (that is, $$G = C_3^n$$) by Lesavourey, Plantard and Susilo [35].…”

“…They generalize the classical relations () where the coefficient of the trivial group is nonzero, which are exactly the relations one needs to determine invariants of $$K$$ from those of its subfields. Although our systematic treatment is new, these norm relations have been used in an ad hoc way for $$G = C_2 \times C_2$$ by Wada [50], Bauch, Bernstein, de Valence, Lange and van Vredendaal [10] and Biasse and van Vredendaal [13], and for $$G = C_3 \times C_3$$ by Parry [44] and Lesavourey, Plantard and Susilo [35]. We give a systematic study of these relations; we link them to fixed point free unitary representations and, from a theorem of Wolf's [51, 55], we obtain the following classification of groups admitting a norm relation (see Theorem 2.11).…”

Log-$$\mathcal {S}$$-unit Lattices Using Explicit Stickelberger Generators to Solve Approx Ideal-SVP

“…, g n ) such that g = n i=1 g i ω i . As explained in [LPS20], it is possible to mutualize the computation of B κ and reuse the unitary transformation to hasten computations when increasing κ is required.…”