An Analysis of a Generalization of Loidreau's Encryption Scheme

We generalize the Gabidulin codes based encryption scheme presented by Loidreau in 2017, by combining the original idea with an idea proposed by Gabidulin, Rashwan and Honary in 2009. We then adapt the combinatorial attack proposed by Briaud and Loidreau in 2023 to evaluate the state of the art complexity of an algorithm recovering a decoder from the public-key. This enables to design, for a same security, schemes with smaller parameters than for the original scheme and to analyse the security of another modification of Loidreau's encryption scheme already published.