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McEliece cryptosystem based on Goppa codes and Sidelnikov-Shestakov attack

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Hello I am trying to understand why the Sidelnikov-Shestakov attack doesn't work for McEliece cryptosystem based on Goppa codes whereas it works when based of GRS codes.

I understood how the attack works but I don't see why it fails with Goppa codes since they are quite similar than GRS codes.

On the 2007 paper https://www.degruyter.com/document/doi/10.1515/JMC.2007.009/pdf by D. Engelbert, R. Overbeck, and A. Schmidt, it is written at remark 3.1 page 14 that the parity check matrix for Goppa codes can be viewed as a matrix over the binary field but this doesn't enlight me very much.

There is a similar explanation in https://eprint.iacr.org/2009/452.pdf by Christian Wieschebrink but as well without further details.

Thank you for your ideas

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