Global InformationMcEliece cryptosystem information
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 algebraic geometry 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.[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.[7] 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.
- ^ McEliece, Robert J. (1978). "A Public-Key Cryptosystem Based on Algebraic Coding Theory" (PDF). DSN Progress Report. 44: 114–116. Bibcode:1978DSNPR..44..114M.
- ^ Dinh, Hang; Moore, Cristopher; Russell, Alexander (2011). Rogaway, Philip (ed.). McEliece and Niederreiter cryptosystems that resist quantum Fourier sampling attacks. Advances in cryptology—CRYPTO 2011. Lecture Notes in Computer Science. Vol. 6841. Heidelberg: Springer. pp. 761–779. doi:10.1007/978-3-642-22792-9_43. ISBN 978-3-642-22791-2. MR 2874885.
- ^ Berlekamp, Elwyn R.; McEliece, Robert J.; Van Tilborg, Henk C.A. (1978). "On the Inherent Intractability of Certain Coding Problems". IEEE Transactions on Information Theory. IT-24 (3): 384–386. doi:10.1109/TIT.1978.1055873. MR 0495180.
- ^ N. J. Patterson (1975). "The algebraic decoding of Goppa codes". IEEE Transactions on Information Theory. IT-21 (2): 203–207. doi:10.1109/TIT.1975.1055350.
- ^ Bernstein, Daniel J.; Lange, Tanja; Peters, Christiane (8 August 2008). "Attacking and Defending the McEliece Cryptosystem". Post-Quantum Cryptography. Lecture Notes in Computer Science. Vol. 5299. pp. 31–46. CiteSeerX 10.1.1.139.3548. doi:10.1007/978-3-540-88403-3_3. ISBN 978-3-540-88402-6.
- ^ Bernstein, Daniel J. (2010). Sendrier, Nicolas (ed.). Grover vs. McEliece (PDF). Post-quantum cryptography 2010. Lecture Notes in Computer Science. Vol. 6061. Berlin: Springer. pp. 73–80. doi:10.1007/978-3-642-12929-2_6. ISBN 978-3-642-12928-5. MR 2776312.
- ^ "eBATS: ECRYPT Benchmarking of Asymmetric Systems". bench.cr.yp.to. 25 August 2018. Retrieved 1 May 2020.