“McEliece cryptosystem”的意思、由来-开放百科全书

In cryptography, the McEliece cryptosystem is an asymmetric encryption algorithm developed in 1978 by Robert McEliece.^[1] 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.^[2]

The algorithm is based on the hardness of decoding a general linear code (which is known to be NP-hard^[3]). 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.^[4] 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.^[5] Another paper shows that for quantum computing, key sizes must be increased by a factor of four due to improvements in information set decoding.^[6]

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). 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.^[7]

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.

Key generation

Message encryption

Message decryption

Proof of message decryption

The Goppa code can correct up to

errors, and the word

is at distance at most

from

. Therefore, the correct code word

is obtained.

Key sizes

McEliece originally suggested security parameter sizes of

,^[1] resulting in a public key size of 524*(1024-524) = 262,000 bits{{clarify |date=May 2016 |reason= 524*1024 = 536576. The numbers do match the "standard form" from the generator matrix article. If that is the case, please clarify.}}. 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.^[5] For resiliency against quantum computers, sizes of

with Goppa code were proposed, giving the size of public key of 8,373,911 bits.^[8]

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.

Exhaustive Search

An attacker may try to find out what

is, and so be able to use the Sardinas–Patterson algorithm.{{Citation needed|date=June 2014}} 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, {{harvnb|Canteaut|Sendrier|1998|Ref=none}}^[9]). In 2008, Bernstein, Lange and Peters^[5] described a practical attack on the original McEliece cryptosystem, based on finding low-weight code words using an algorithm published by Jacques Stern in 1989.^[10] Using the parameters originally suggested by McEliece, the attack could be carried out in 2^60.55 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.

Implementations

The McEliece cryptosystem has been implemented in the secure Instant Messenger and E-Mail Client http://goldbug.sf.net since the version 3.1 release (McEliece Release).

References

1. ^¹²{{cite journal|url=http://ipnpr.jpl.nasa.gov/progress_report2/42-44/44N.PDF|last=McEliece|first=Robert J.|bibcode=1978DSNPR..44..114M|title=A Public-Key Cryptosystem Based On Algebraic Coding Theory|journal=DSN Progress Report|volume=44|pages=114–116|year=1978}}
2. ^{{cite conference |year=2011 |title=McEliece and Niederreiter cryptosystems that resist quantum Fourier sampling attacks |first1=Hang | last1=Dinh | first2=Cristopher | last2=Moore | first3=Alexander | last3=Russell |pages=761–779 |location=Heidelberg |publisher=Springer |mr=2874885 |doi=10.1007/978-3-642-22792-9_43 |editor1-first=Philip | editor1-last=Rogaway |series=Lecture Notes in Computer Science |volume=6841 |isbn=978-3-642-22791-2 |conference=Advances in cryptology—CRYPTO 2011}}
3. ^{{cite journal|last1=Berlekamp |first1= Elwyn R.|last2=McEliece |first2=Robert J.|last3=Van Tilborg |first3=Henk C.A.|year=1978|title=On the Inherent Intractability of Certain Coding Problems|journal=IEEE Transactions on Information Theory|volume=IT-24|issue= 3|pages=384–386| doi = 10.1109/TIT.1978.1055873| mr = 0495180}}
4. ^{{cite journal|author=N. J. Patterson|year=1975|title=The algebraic decoding of Goppa codes|journal=IEEE Transactions on Information Theory|volume=IT-21|issue=2|pages=203–207 |doi=10.1109/TIT.1975.1055350}}
5. ^¹²{{cite book|last1=Bernstein |first1=Daniel J.|last2=Lange |first2=Tanja|author2-link=Tanja Lange|last3=Peters |first3=Christiane|date=8 August 2008|title=Attacking and defending the McEliece cryptosystem|journal=Proc. 2nd International Workshop on Post-Quantum Cryptography|series=Lecture Notes in Computer Science|volume=5299 |pages=31–46|doi=10.1007/978-3-540-88403-3_3|url=http://eprint.iacr.org/2008/318|isbn=978-3-540-88402-6|citeseerx=10.1.1.139.3548}}
6. ^{{cite conference |first=Daniel J. | last=Bernstein |year=2010 |title=Grover vs. McEliece |url=https://cr.yp.to/codes/grovercode-20091123.pdf |publisher=Springer |location=Berlin |series=Lecture Notes in Computer Science |volume=6061 |conference=Post-quantum cryptography 2010 |mr=2776312 |pages=73–80 |doi=10.1007/978-3-642-12929-2_6 |editor1-first=Nicolas | editor1-last=Sendrier |isbn=978-3-642-12928-5}}
7. ^{{cite news |title=1978 Cryptosystem Resists Quantum Attack |url=https://www.technologyreview.com/s/420287/1978-cryptosystem-resists-quantum-attack/ |date=2010-08-18 |publisher=MIT Technology Review}}
8. ^{{cite web| url=https://pqcrypto.eu.org/docs/initial-recommendations.pdf| title=Initial recommendations of long-term secure post-quantum systems| author=Daniel Augot| date=2015-09-07| work=PQCRYPTO: Post-Quantum Cryptography for Long-Term Security|display-authors=etal}}
9. ^{{cite book|last1=Ohta |first1=Kazuo|last2=Pei |first2=Dingyi|year=1998|editor1-last=Ohta|editor1-first=Kazuo|editor2-first=Dingyi|title=Cryptanalysis of the Original McEliece Cryptosystem|journal=Advances in Cryptology — ASIACRYPT'98|series=Lecture Notes in Computer Science|volume=1514 |pages=187–199|doi=10.1007/3-540-49649-1|editor2-last=Pei|isbn=978-3-540-65109-3}}
10. ^{{cite book|author=Jacques Stern|year=1989|title=A method for finding codewords of small weight|journal=Coding Theory and Applications|series=Lecture Notes in Computer Science|publisher=Springer Verlag|volume=388 |pages=106–113|doi=10.1007/BFb0019850|isbn=978-3-540-51643-9}}