McEliece cryptosystem
Encyclopedia
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. 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 uses McEliece for encryption is the Freenet
-like application Entropy
.
algorithm, and a deterministic decryption algorithm.
All users in a McEliece deployment share a set of common security parameters:
.
:
, Alice performs the following steps to decrypt the message:
,
and that
is a permutation matrix, thus 
has weight at most 
.
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.
, 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.
but not the private key results in deducing plaintext from some intercepted ciphertext 
. Such attempts must be infeasible. This section discusses attack strategies against the McEliece cryptosystem described in the literature.
is, and so be able to use the Patterson algorithm. This is unlikely to succeed for large values of n and t, since there are just too many possibilities for 
, 
and 
.
A strategy that does not require
is based on the concept of information set decoding. McEliece mentioned a simple form of this attack: selecting k of the n coordinates randomly in hope that none of the k are in error (i.e., for none of the selected coordinates the vector 
has a 1-bit), and under this assumption calculate m. However, if the parameters k, n and t are carefully chosen, the probability of no error in this set of k elements is 
, and thus is negligible.
in 1989. 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.
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
, the McEliece cryptosystem is an asymmetric encryption algorithm developed in 1978 by Robert McEliece
Robert McEliece
Robert J. McEliece is a mathematician and engineering professor at the California Institute of Technology best known for his work in information theory. He was the 2004 recipient of the Claude E. Shannon Award and the 2009 recipient of the IEEE Alexander Graham Bell Medal.Educated at Caltech...
. It was the first such scheme to use randomization
Randomized algorithm
A randomized algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits...
in the encryption process. The algorithm has never gained much acceptance in the cryptographic community but is a candidate for 'post-quantum cryptography
Post-quantum cryptography
Post-quantum cryptography refers to research on cryptographic primitives that are not breakable using quantum computers...
' as it is immune to attacks using Shor's algorithm
Shor's algorithm
Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm for integer factorization formulated in 1994...
and – more generally – 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
Linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although Turbo codes can be seen as a hybrid of these two types. Linear codes allow for...
(which is known to be NP-hard
NP-hard
NP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...
). 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 matrixGenerator matrix
In coding theory, a generator matrix is a basis for a linear code, generating all its possible codewords.If the matrix is G and the linear code is C,where w is a codeword of the linear code C, c is a row vector, and a bijection exists between w and c. A generator matrix for an q-code has...
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
Digital signature
A digital signature or digital signature scheme is a mathematical scheme for demonstrating the authenticity of a digital message or document. A valid digital signature gives a recipient reason to believe that the message was created by a known sender, and that it was not altered in transit...
. However, a signature scheme can be constructed based on the Niederreiter
Niederreiter cryptosystem
In cryptography, the Niederreiter cryptosystem is a variation of the McEliece Cryptosystem developed in 1986 by Harald Niederreiter. It applies the same idea to the parity check matrix H of a linear code....
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 uses McEliece for encryption is the Freenet
Freenet
Freenet is a decentralized, censorship-resistant distributed data store originally designed by Ian Clarke. According to Clarke, Freenet aims to provide freedom of speech through a peer-to-peer network with strong protection of anonymity; as part of supporting its users' freedom, Freenet is free and...
-like application Entropy
Entropy (anonymous data store)
Entropy was a decentralized, peer-to-peer communication network designed to be resistant to censorship, much like Freenet. Entropy was an anonymous data store written in the C programming language. It pooled the contributed bandwidth and storage space of member computers to allowed users to...
.
Scheme definition
McEliece consists of three algorithms: a probabilistic key generation algorithm which produces a public and a private key, a probabilistic encryptionProbabilistic encryption
Probabilistic encryption is the use of randomness in an encryption algorithm, so that when encrypting the same message several times it will, in general, yield different ciphertexts...
algorithm, and a deterministic decryption algorithm.
All users in a McEliece deployment share a set of common security parameters:
.Key generation
- Alice selects a binary
-linear code 
capable of correcting 
errors. This code must possess an efficient decoding algorithm and generates a 
generator matrix 
for the code 
. - Alice selects a random
binary non-singular matrix 
. - Alice selects a random
permutation matrixPermutation matrixIn mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
. - Alice computes the
matrix 
. - Alice’s public key is
; her private key is 
.
Message encryption
Suppose Bob wishes to send a message m to Alice whose public key is:- Bob encodes the message m 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 for the code
to decode 
to 
. - Alice computes
.
Proof of message decryption
Note that,and that
is a permutation matrix, thus has weight at most .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.Attacks
A successful attack of an adversary knowing the public key but not the private key results in deducing plaintext from some intercepted ciphertext . Such attempts must be infeasible. This section discusses attack strategies against the McEliece cryptosystem described in the literature.Brute force
An attacker may try to find out what is, and so be able to use the Patterson algorithm. This is unlikely to succeed for large values of n and t, since there are just too many possibilities for , and .A strategy that does not require
is based on the concept of information set decoding. McEliece mentioned a simple form of this attack: selecting k of the n coordinates randomly in hope that none of the k are in error (i.e., for none of the selected coordinates the vector has a 1-bit), and under this assumption calculate m. However, if the parameters k, n and t are carefully chosen, the probability of no error in this set of k elements is , and thus is negligible.Information set decoding
Information set decoding algorithms have turned out to be the most effective attacks against the McEliece and Niederreiter cryptosystems. Various forms have been introduced. An effective method is based on finding minimum- or low-weight codewords (see, for example). In 2008, Bernstein, Lange and Peters described a practical attack on the original McEliece cryptosystem, based on finding low-weight code words using an algorithm published by Jacques SternJacques Stern
Jacques Stern is a cryptographer, currently a professor at the École Normale Supérieure, where he is Director of the Computer Science Laboratory. He received the 2006 CNRS Gold Medal...
in 1989. Using the parameters originally suggested by McEliece, the attack could be carried out in 260.55 bit operations. Since the attack is embarrassingly parallel
Embarrassingly parallel
In parallel computing, an embarrassingly parallel workload is one for which little or no effort is required to separate the problem into a number of parallel tasks...
(no communication between nodes is necessary), it can be carried out in days on modest computer clusters.