Cryptanalysis of the CLT13 Multilinear Map
In this paper, we describe a polynomial time cryptanalysis of the (approximate) multilinear map proposed by Coron, Lepoint, and Tibouchi in Crypto13 (CLT13). This scheme includes a zero-testing functionality that determines whether the message of a given encoding is zero or not. This functionality is useful for designing several of its applications, but it leaks unexpected values, such as linear combinations of the secret elements. By collecting the outputs of the zero-testing algorithm, we construct a matrix containing the hidden information as eigenvalues, and then recover all the secret elements of the CLT13 scheme via diagonalization of the matrix. In addition, we provide polynomial time algorithms to directly break the security assumptions of many applications based on the CLT13 scheme. These algorithms include solving subgroup membership, decision linear, and graded external Diffie–Hellman problems. These algorithms mainly rely on the computation of the determinants of the matrices and their greatest common divisor, instead of performing their diagonalization.