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Definition
Schemes based on rank codes appear in many areas of communications, cryptography, and information theory.
Background
The rank function defined on the set of matrices (or vectors) is in fact the norm function. The well-known inequalities for sums of matrices \(\vert \mathrm{Rk}(A) -\mathrm{Rk}(B)\vert \leq \mathrm{Rk}(A+B) \leq \mathrm{Rk}(A) + \mathrm{Rk}(B)\) are known from the very beginning of theory of matrices. They define implicitly the rank distance relations on the space of all matrices of identical size. Nevertheless, the definition and applications of rank-metric-based codes are of active interest only for last decades.
Theory
Optimal Linear Rank Codes
Let \({\mathbb{F}}_{q}\) be a base field of q elements and \({\mathbb{F}}_{{q}^{N}}\) be an extension field of degree N. In vector representation, rank codes are defined as subsets of a normed n-dimensional space \(\left...
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Recommended Reading
Gabidulin EM (1985) Theory of codes with maximum rank distance. Probl Inf Transm 21(1):1–12
Gabidulin EM, Paramonov AV, Tretjakov OV (1992) Rank errors and rank erasures correction. In: Proceedings of the 4th international colloquium on coding theory, 30 September–7 October 1991, Dilijan, Armenia, pp 11–19, Yerevan, 1992
Gabidulin EM, Pilipchuk NI (2008) Error and erasure correcting algorithms for rank codes. Designs Codes and Cryptogr 49:105–122. DOI 10.1007/s10623-008-9185-7
Silva D, Kschischang FR, Koetter R (2008) A rank-metric approach to error control in random network coding. IEEE Trans Inf Theory 54(9):3951–3967
Tarokh V, Jafarkhani H, Calderbank AR (1998) Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans Inf Theory 44(2):744–765
Gabidulin EM, Lusina P, Bossert M (2003) Maximum rank codes as space-time codes. IEEE Trans Inf Theory 46(10):2757–2760
Gabidulin EM, Bossert M (2009) Algebraic codes in network coding. Probl. Inf Transm 45(4):3–17
McEliece RJ (1978) A public key cryptosystem based on algebraic coding theory. JPL DSN progress report 42–44, Pasadena, pp 114–116
Gabidulin EM, Paramonov AV, Tretjakov OV (1991) Ideals over a non-commutative ring and their application in cryptology. In: Davies DW (ed) Advances in cryptology | Eurocrypt ’91. Lecture notes in computer science, vol 547. Springer, Berlin and Heidelberg, pp 482–489
Gabidulin EM (2008) Attacks and counter-attacks on the GPT public key cryptosystem. Designs Codes Cryptogr 48(2):171–177. DOI 10.1007/s10623-007-9160-8
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Gabidulin, E.M. (2011). Schemes Based on Rank Codes. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_388
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DOI: https://doi.org/10.1007/978-1-4419-5906-5_388
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