Coggia-Couvreur Attack on Loidreau’s Rank-metric Cryptosystem for any $\lambda$

Coggia and Couvreur’s attack [DC19] gives a polynomial time key recovery on Loidreau’s scheme [Loi17], when the dimension of the secret subspace is precisely $\lambda = 2$. Their approach gives the possibility for cryptanalysis of rank-metric schemes for any $\lambda$. Recently, the extending Coggia Couveur ’s attack [Gha20] claimed the attack for the case $\lambda = 3$. In this article we continue extending the Coggia-Couvreur key-recovery attack on Loidreau’s cryptosystem for any $\lambda$, especially complete the attack for $\lambda = 3$.

[DC19]  A. Couvreur D. Coggia. On the security of a loidreau rank metric code based encryption scheme. 2019.
[Gha20] A. Ghatak. Extending coggia-couvreur attack on loidreau’s rank-metric cryptosystem. 2020
[Loi17] P. Loidreau. A New rank metric codes based encryption scheme. 8th International Conference on Post-Quantum Cryptography, PQCrypto 2017