Paper 2025/1435 Original: eprint.iacr.org/2025/1435 ↗

Weak keys in QC-MDPC-based cryptosystems via the extended Euclidean algorithm

Abstract

In this work we analyze a problem strictly linked with the Rational Reconstruction, which forms the foundation of some post-quantum Quasi-Cyclic Moderate-Density Parity-Check and Quasi-Cyclic Low-Density Parity-Check code-based schemes such as LEDAkem and BIKE. Given a polynomial in a cyclic ring as input, our aim is to recover two polynomials, with specific properties, whose ratio is the input one. The starting point of this work is the paper of Bardet, Dragoi, Luque, and Otmani, which describes some approaches, based on the Extended Euclidean Algorithm, that solves this problem in some specific cases. In comparison to previous work, we define an additional setting in which the problem can be solved. We also provide an alternative approach to estimate the probability of success, by taking into account a requirement that was not considered in the original paper, thus getting a more precise estimation. Finally, we present a key-recovery attack on BIKE, evaluate its computational cost, and compare it with that of the most efficient known attacks. Although this last step is performed specifically on BIKE, the methodology can be extended to other schemes as well.