On the Security of Subspace Subcodes of Reed–Solomon Codes for Public Key Encryption

This article discusses the security of McEliece-like encryption schemes using subspace subcodes of Reed–Solomon codes, i.e. subcodes of Reed–Solomon codes over ${\mathbb {F}_{q^{m}}}$ whose entries lie in a fixed collection of ${\mathbb {F}_{q}}$ –subspaces of ${\mathbb {F}_{q^{m}}}$ . These codes appear to be a natural generalisation of Goppa and alternant codes and provide a broader flexibility in designing code based encryption schemes. For the security analysis, we introduce a new operation on codes called the twisted product which yields a polynomial time distinguisher on such subspace subcodes as soon as the chosen ${\mathbb {F}_{q}}$ –subspaces have dimension larger than $m/2$ . From this distinguisher, we build an efficient attack which in particular breaks some parameters of a recent proposal due to Khathuria, Rosenthal and Weger.