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 $\mathbb{F}_{q}^{l} \otimes C.$