Rank error-correcting code

Rank metric

Let X^n be an n-dimensional vector space over the finite field GF\left( {q^N } \right), where q is a power of a prime and N is a positive integer. Let \left(u_1, u_2, \dots, u_N\right), with u_i \in GF(q^N), be a base of GF\left( {q^N } \right) as a vector space over the field GF\left( {q} \right).

Every element x_i \in GF\left( {q^N } \right) can be represented as x_i = a_{1i}u_1 + a_{2i}u_2 + \dots + a_{Ni}u_N. Hence, every vector \vec x = \left( {x_1, x_2, \dots, x_n } \right) over GF\left( {q^N } \right) can be written as matrix:

: \vec x = \left| {\begin{array}{*{20}c} a_{1,1} & a_{1,2} & \ldots & a_{1,n} \ a_{2,1} & a_{2,2} & \ldots & a_{2,n} \ \ldots & \ldots & \ldots & \ldots \ a_{N,1} & a_{N,2} & \ldots & a_{N,n} \end{array}} \right|

Rank of the vector \vec x over the field GF\left( {q^N} \right) is a rank of the corresponding matrix A\left( {\vec x} \right) over the field GF\left( {q} \right) denoted by r\left( {\vec x; q} \right).

The set of all vectors \vec x is a space X^n = A_N^n. The map \vec x \to r\left( \vec x; q \right)) defines a norm over X^n and a rank metric:

: d\left( {\vec x;\vec y} \right) = r\left( {\vec x - \vec y;q} \right)

Rank code

A set {x_1, x_2, \dots, x_n} of vectors from X^n is called a code with code distance d = \min d\left( x_i ,x_j \right). If the set also forms a k-dimensional subspace of X^n, then it is called a linear (n, k)-code with distance d. Such a linear rank metric code always satisfies the Singleton bound d \leq n - k + 1 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 (and Kshevetskiy ).

Let's define a Frobenius power [i] of the element x \in GF(q^N) as

: x^{[i]} = x^{q^{i \mod N}}. ,

Then, every vector \vec g = (g_1, g_2, \dots, g_n), ~ g_i \in GF(q^N), ~ n \leq N, linearly independent over GF(q), defines a generating matrix of the MRD (n, k, d = n − k + 1)-code.

: G = \left| {\begin{array}{*{20}c} g_1 & g_2 & \dots & g_n \ g_1^{[m]} & g_2^{[m]} & \dots & g_n^{[m]} \ g_1^{[2 m]} & g_2^{[2 m]} & \dots & g_n^{[2 m]} \ \dots & \dots & \dots & \dots \ g_1^{[(k-1) m]} & g_2^{[(k-1) m]} & \dots & g_n^{[(k-1) m]} \end{array}} \right|,

where \gcd(m,N) = 1.

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).

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

Notes

References

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• {{Cite book|first1= Alexander
• {{Cite book|first1= Ernst M.

References

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.
3. (4–9 September 2005). "Proceedings. International Symposium on Information Theory, 2005. ISIT 2005".
4. (2008). "Structural Attacks for Public Key Cryptosystems based on Gabidulin Codes". Journal of Cryptology.