Belief Propagation Meets Lattice Reduction: Security Estimates for Error-Tolerant Key Recovery from Decryption Errors

In LWE-based KEMs, observed decryption errors leak information about the secret key in the form of equations or inequalities. Several practical fault attacks have already exploited such leakage by either directly applying a fault or enabling a chosen-ciphertext attack using a fault. When the leaked information is in the form of inequalities, the recovery of the secret key is not trivial. Recent methods use either statistical or algebraic methods (but not both), with some being able to handle incorrect information. Having in mind that integration of the side-channel information is a crucial part of several classes of implementation attacks on LWEbased schemes, it is an important question whether statistically processed information can be successfully integrated in lattice reduction algorithms.
We answer this question positively by proposing an error-tolerant combination of statistical and algebraic methods that make use of the advantages of both approaches. The combination enables us to improve upon existing methods – we use both fewer inequalities and are more resistant to errors. We further provide precise security estimates based on the number of available inequalities.
Our recovery method applies to several types of implementation attacks in which decryption errors are used in a chosen-ciphertext attack. We practically demonstrate the improved performance of our approach in a key-recovery attack against Kyber with fault-induced decryption errors.