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**2024/1147**

**Title:**

A reduction from Hawk to the principal ideal problem in a quaternion algebra

**Authors:**
Clémence Chevignard, Pierre-Alain Fouque, Guilhem Mureau, Alice Pellet-Mary, Alexandre Wallet

**Abstract:**

In this article we present a non-uniform reduction from rank-2 module-LIP over Complex Multiplication fields, to a variant of the Principal Ideal Problem, in some fitting quaternion algebra. This reduction is classical deterministic polynomial-time in the size of the inputs. The quaternion algebra in which we need to solve the variant of the principal ideal problem depends on the parameters of the module-LIP problem, but not on the problem’s instance. Our reduction requires the knowledge of some special elements of this quaternion algebras, which is why it is non-uniform.

In some particular cases, these elements can be computed in polynomial time, making the reduction uniform. This is in particular the case for the Hawk signature scheme: we show that breaking Hawk is no harder than solving a variant of the principal ideal problem in a fixed quaternion algebra (and this reduction is uniform).

**ePrint:**
https://eprint.iacr.org/2024/1147

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