Distinguishing Goppa Codes Using Higher-Order Vanishing

We present a new algorithm to distinguish alternant and Goppa codes from general linear codes. Our approach is based on higher-order vanishing of polynomials and can be applied to a wide set of code parameters. It also applies to Goppa code parameters used in the Classic McEliece key encapsulation mechanism. While these parameters are out of reach in practice, we analyse the behaviour of our distinguisher and estimate its complexity for them, indicating that it is more efficient than previous distinguishing approaches. This is supported by concrete experiments that distinguish codes with larger parameters in practice.