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.