In 2022, Ducas et al. introduced the signature scheme Hawk, based of the presumed hardness of a new problem in lattice-based cryptography: the Lattice Isomorphism Problem for the module-lattice O_L^2, where L is a cyclotomic number field. Last year we presented a polynomial time algorithm solving this problem when L is a totally real number field (thus not affecting the security of Hawk). More recently, we provided a reduction of the same problem when L is now a CM field (thus containing Hawk's instance) to the problem of finding a generator of a principal quaternionic ideal. 

In this talk we give a framework containing both the totally real and the CM case, and we will discuss the differences. This is based on a joint work with C. Chevignard, P-A. Fouque, A. Pellet-Mary, H. Pliatsok and A. Wallet.