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 coset 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  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 can be efficiently decoded, thanks to an 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 2008 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 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.

Scheme definition
McEliece consists of three algorithms: a probabilistic key generation algorithm which produces a public and a private key, a probabilistic encryption algorithm, and a deterministic decryption algorithm.

All users in a McEliece deployment share a set of common security parameters: .

Key generation

The principle is that Alice chooses a linear code  from some family of codes for which she knows an efficient decoding algorithm, and to make  public knowledge but keep the decoding algorithm secret. Such a decoding algorithm requires not just knowing , in the sense of knowing an arbitrary generator matrix, but requires one to know the parameters used when specifying  in the chosen family of codes. For instance, for binary Goppa codes, this information would be the Goppa polynomial and the code locators. Therefore, Alice may publish a suitably obfuscated generator matrix of .

More specifically, the steps are as follows:

Alice selects a binary -linear code  capable of (efficiently) correcting  errors from some large family of codes, e.g. binary Goppa codes. This choice should give rise to an efficient decoding algorithm . Let also  be any generator matrix for . Any linear code has many generator matrices, but often there is a natural choice for this family of codes. Knowing this would reveal  so it should be kept secret.
Alice selects a random  binary non-singular matrix .
Alice selects a random  permutation matrix .
Alice computes the  matrix .
Alice's public key is ; her private key is . Note that  could be encoded and stored as the parameters used for selecting .

Message encryption 

Suppose Bob wishes to send a message m to Alice whose public key is :

Bob encodes the message  as a binary string of length .
Bob computes the vector .
Bob generates a random -bit vector  containing exactly  ones (a vector of length  and weight )
Bob computes the ciphertext as .

Message decryption 
Upon receipt of , Alice performs the following steps to decrypt the message:
Alice computes the inverse of  (i.e. ).
Alice computes .
Alice uses the decoding algorithm  to decode  to .
Alice computes .

Proof of message decryption 
Note that ,
and that  is a permutation matrix, thus  has weight .

The Goppa code  can correct up to  errors, and the word  is at distance at most  from . Therefore, the correct code word  is obtained.

Multiplying with the inverse of  gives , which is the plain text message.

Key sizes 
McEliece originally suggested security parameter sizes of , resulting in a public key size of 524*(1024−524) = 262,000 bits. Recent analysis suggests parameter sizes of  for 80 bits of security when using standard algebraic decoding, or  when using list decoding for the Goppa code, giving rise to public key sizes of 520,047 and 460,647 bits respectively. For resiliency against quantum computers, sizes of  with Goppa code were proposed, giving the size of public key of 8,373,911 bits. In its round 3 submission to the NIST post quantum standardization the highest level of security, level 5 is given for parameter sets 6688128, 6960119, and 8192128. The parameters are , ,  respectively.

Attacks 
An attack consists of an adversary, who knows the public key  but not the private key, deducing the plaintext from some intercepted ciphertext . Such attempts should be infeasible.

There are two main branches of attacks for McEliece:

Brute-force / unstructured attacks 
The attacker knows  which is the generator matrix of an  code  which is combinatorially able to correct  errors.
The attacker may ignore the fact that  is really the obfuscation of a structured code chosen from a specific family, and instead just use an algorithm for decoding with any linear code. Several such algorithms exist, such as going through each codeword of the code, syndrome decoding, or information set decoding.

Decoding a general linear code, however, is known to be NP-hard, however, and all of the above-mentioned methods have exponential running time.

In 2008, Bernstein, Lange, and Peters described a practical attack on the original McEliece cryptosystem, using the information set decoding method by Stern.
Using the parameters originally suggested by McEliece, the attack could be carried out in 260.55 bit operations. Since the attack is embarrassingly parallel (no communication between nodes is necessary), it can be carried out in days on modest computer clusters.

Structural attacks 

The attacker may instead attempt to recover the "structure" of , thereby recovering the efficient decoding algorithm  or another sufficiently strong, efficient decoding algorithm.

The family of codes from which  is chosen completely determines whether this is possible for the attacker. Many code families have been proposed for McEliece, and most of them have been completely "broken" in the sense that attacks which recover an efficient decoding algorithm has been found, such as Reed-Solomon codes.

The originally proposed binary Goppa codes remain one of the few suggested families of codes which have largely resisted attempts at devising structural attacks.

Post-quantum encryption candidate 

A variant of this algorithm combined with NTS-KEM was entered into and selected during the second round of the NIST post-quantum encryption competition.

References

External links
 

 

  (Submission to the NIST Post-Quantum Cryptography Standardization Project)

Public-key encryption schemes
Code-based cryptography
Post-quantum cryptography