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Title:
A search to distinguish reduction for the isomorphism problem on direct sum lattices
Authors: Daniël van Gent, Wessel van Woerden
Abstract:At Eurocrypt 2003, Szydlo presented a search to distinguish reduction for the Lattice Isomorphism Problem (LIP) on the integer lattice \mathbb{Z}^n. Here the search problem asks to find an isometry between \mathbb{Z}^n and an isomorphic lattice, while the distinguish variant asks to distinguish between a list of auxiliary lattices related to \mathbb{Z}^n.
In this work we generalize Szydlo’s search to distinguish reduction in two ways. Firstly, we generalize the reduction to any lattice isomorphic to \Gamma^n, where \Gamma is a fixed base lattice. Secondly, we allow \Gamma to be a module lattice over any number field. Assuming the base lattice \Gamma and the number field K are fixed, our reduction is polynomial in n.
As a special case we consider the module lattice \mathcal{O}_K^2 used in the module-LIP based signature scheme HAWK, and we show that one can solve the search problem, leading to a full key recovery, with less than 2d^2 distinguishing calls on two lattices each, where d is the degree of the power-of-two cyclotomic number field and \mathcal{O}_K its ring of integers.
ePrint: https://eprint.iacr.org/2025/1204
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