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 andâ€”more generallyâ€”measuring coset states using Fourier sampling. A recent improvement of an information-set decoding algorithm for quantum computing, however, requires key sizes to be increased by a factor of four.

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 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 is perturbated by two randomly selected invertible matrices and (see below).

Variants of this cryptosystem exist, using different types of codes. Most of them were proven less secure; they were broken by structural decoding.

McEliece with Goppa codes has resisted cryptanalysis so far. The most effective attacks known use information set decoding algorithms. A recent paper describes both an attack and a fix. Another paper shows that for quantum computing key sizes must be increased by a factor of four due to improvements in information set decoding.

The McEliece cryptosystem has some advantages over, for example, RSA. The encryption and decryption are faster (for comparative benchmarks see the eBATS benchmarking project at bench.cr.yp.to) and with the growth of the key size, the security grows much faster. For a long time it was thought that McEliece could not be used to produce signatures. However, a signature scheme can be constructed based on the Niederreiter scheme, the dual variant of the McEliece scheme. One of the main disadvantages of McEliece is that the private and public keys are large matrices. For a standard selection of parameters, the public key is 512 kilobits long. This is why the algorithm is rarely used in practice. One exceptional case that used McEliece for encryption is the Freenet-like application Entropy.

Read more about McEliece Cryptosystem: Scheme Definition, Proof of Message Decryption, Key Sizes, Attacks

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