On the Support Splitting Algorithm for Induced Codes

Abstract

As shown by N. Sendrier in 2000, if a \([n{\text{,}}\,k{\text{,}}\,d]\)-linear code \(C( \subseteq \mathbb{F}_{q}^{n})\) with length \(n\), dimensionality \(k\) and code distance \(d\) has a trivial group of automorphisms \({\text{PAut}}(C)\), it allows one to construct a determined support splitting algorithm in order to find a permutation \(\sigma \) for a code \(D\), being permutation-equivalent to the code \(C\), such that \(\sigma (C) = D\). This algorithm can be used for attacking the McEliece cryptosystem based on the code\(C\). This work aims the construction and analysis of the support splitting algorithm for the code \(\mathbb{F}_{q}^{l} \otimes C\), induced by the code \(C\), \(l \in \mathbb{N}\). Since the group of automorphisms PAut\((\mathbb{F}_{q}^{l} \otimes C)\) is nontrivial even in the case of that trivial for the base code \(C\), it enables one to assume a potentially high resistance of the McEliece cryptosystem on the code \(\mathbb{F}_{q}^{l} \otimes C\) to the attack based on a carrier split. The support splitting algorithm is being constructed for the code \(\mathbb{F}_{q}^{l} \otimes C\) and its efficiency is compared with the attack to a McEliece cryptosystem based on the code \(\mathbb{F}_{q}^{l} \otimes C.\)