Rank error-correcting code

In coding theory, rank codes (also called Gabidulin codes) are non-binary[1] linear error-correcting codes over not Hamming but rank metric. They described a systematic way of building codes that could detect and correct multiple random rank errors. By adding redundancy with coding k-symbol word to a n-symbol word, a rank code can correct any errors of rank up to t =  (d  1) / 2 ⌋, where d is a code distance. As an erasure code, it can correct up to d 1 known erasures.

Rank codes
Classification
HierarchyLinear block code
Rank code
Block lengthn
Message lengthk
Distancen k + 1
Alphabet sizeQ = qN  (q prime)
Notation[n, k, d]-code
Algorithms
Berlekamp–Massey
Euclidean
with Frobenius polynomials

A rank code is an algebraic linear code over the finite field similar to Reed–Solomon code.

The rank of the vector over is the maximum number of linearly independent components over . The rank distance between two vectors over is the rank of the difference of these vectors.

The rank code corrects all errors with rank of the error vector not greater than t.

Rank metric

Let be an n-dimensional vector space over the finite field , where is a power of a prime and is a positive integer. Let , with , be a base of as a vector space over the field .

Every element can be represented as . Hence, every vector over can be written as matrix:

Rank of the vector over the field is a rank of the corresponding matrix over the field denoted by .

The set of all vectors is a space . The map ) defines a norm over and a rank metric:

Rank code

A set of vectors from is called a code with code distance . If the set also forms a k-dimensional subspace of , then it is called a linear (n, k)-code with distance . Such a linear rank metric code always satisfies the Singleton bound with equality.

Generating matrix

There are several known constructions of rank codes, which are maximum rank distance (or MRD) codes with d = n  k + 1. The easiest one to construct is known as the (generalized) Gabidulin code, it was discovered first by Delsarte (who called it a Singleton system) and later by Gabidulin [2] (and Kshevetskiy [3] ).

Let's define a Frobenius power of the element as

Then, every vector , linearly independent over , defines a generating matrix of the MRD (n, k, d = n  k + 1)-code.

where .

Applications

There are several proposals for public-key cryptosystems based on rank codes. However, most of them have been proven insecure (see e.g. Journal of Cryptology, April 2008[4]).

Rank codes are also useful for error and erasure correction in network coding.

See also

Notes

  1. Codes for which each input symbol is from a set of size greater than 2.
  2. Gabidulin, Ernst M. (1985). "Theory of codes with maximum rank distance". Problems of Information Transmission. 21 (1): 1–12.
  3. Kshevetskiy, Alexander; Gabidulin, Ernst M. (4–9 September 2005). "The new construction of rank codes". Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. pp. 2105–2108. doi:10.1109/ISIT.2005.1523717. ISBN 978-0-7803-9151-2. S2CID 11679865.
  4. Overbeck, R. (2008). "Structural Attacks for Public Key Cryptosystems based on Gabidulin Codes". Journal of Cryptology. 21 (2): 280–301. doi:10.1007/s00145-007-9003-9. S2CID 2393853.

References

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