Hadamard closure of general Reed–Solomon codes and filtration attack on the original Niederreiter cryptosystem

The Shur–Hadamard closure of generalized Reed–Solomon codes is considered. The Shur–Hadamard closure or Hadamard closure is the set of linear codes which can be obtained from a given code by applying the operations of Hadamard product (Shur product or component-wise product) and taking of dual code. The structure of the Shur–Hadamard closure of a generalized Reed–Solomon code is described. In addition, two new polynomial attacks on the original Niederreiter cryptosystem based on generalized Reed–Solomon codes are constructed. One of these attacks relies on the structure of Hadamard closure of the generalized Reed—Solomon code, and the other uses a code filtration procedure.