McEliece cryptosystem

In cryptography, the McEliece cryptosystem is an asymmetric encryption algorithm developed in 1978 by Robert McEliece. It was the first such scheme to use randomization in the encryption process. The algorithm has never gained much acceptance in the cryptographic community, but is a candidate for "post-quantum cryptography", as it is immune to attacks using Shor's algorithm and — more generally — measuring cost states using Fourier sampling.

The algorithm is based on the hardness of decoding a general linear code (which is known to be NP-hard). For a description of the private key, an error-correcting code is selected for which an efficient decoding algorithm is known, and which is able to correct ${\displaystyle t}$ errors. The original algorithm uses binary Goppa codes (subfield codes of geometric Goppa codes of a genus-0 curve over finite fields of characteristic 2); these codes are easy to decode, thanks to an efficient algorithm due to Patterson. The public key is derived from the private key by disguising the selected code as a general linear code. For this, the code's generator matrix ${\displaystyle G}$ is perturbated by two randomly selected invertible matrices ${\displaystyle S}$ and ${\displaystyle P}$ (see below).

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