An Algebraic Attack Against McEliece-like Cryptosystems Based on BCH codes

We present an algebraic attack on a McEliece-like scheme based on BCH codes (BCH-McEliece), where the Goppa code is replaced by a suitably permuted BCH code. Our attack continues the line of work devising attacks against McEliece-like schemes with Goppa-like codes, with the goal of getting a better understanding of why Goppa codes are so intractable. Our starting point is the work of Faugère, Perret and Portzamparc (Asiacrypt 2014). We take their algebraic model and adapt and improve their attack algorithm so that it can handle BCH-McEliece. We demonstrate experimentally that our attack is practical for high rate codes over non-prime fields for parameters where generic attacks suggest cryptographic security.