Universal secure rank-metric coding schemes with optimal communication overheads

Abstract

We study the problem of reducing the communication overhead from a noisy wire-tap channel or storage system where data is encoded as a matrix, when more columns (or their linear combinations) are available. We present its applications to reducing communication overheads in universal secure linear network coding and secure distributed storage with crisscross errors and erasures and in the presence of a wire-tapper. Our main contribution is a method to transform coding schemes based on linear rank-metric codes, with certain properties, to schemes with lower communication overheads. By applying this method to pairs of Gabidulin codes, we obtain coding schemes with optimal information rate with respect to their security and rank error correction capability, and with universally optimal communication overheads, when n ≤ m, being n and m the number of columns and number of rows, respectively. Moreover, our method can be applied to other families of maximum rank distance codes when n > m. The downside of the method is generally expanding the packet length, but some practical instances come at no cost.

Appendix A: Proof of Lemma 3

We start with an auxiliary result, which is a particular case of [18, Th. 3]:

Lemma 4 ([ 18 ])

Let$B \in \mathbb {F}_{q}^{\mu \times n}$and let$\mathcal {C}_{2} \subsetneqq \mathcal {C}_{1} \subseteq \mathbb {F}_{q^{m}}^{n}$be$\mathbb {F}_{q^{m}}$-linearcodes. It holds that

$$\begin{array}{lll} d_{R}(\mathcal{C}_{1}, \mathcal{C}_{2}) = \min \{ & \text{Rk}(A) \mid A \in \mathbb{F}_{q}^{\nu \times n}, \nu \in \mathbb{N}, \text{ and} \\ & \dim \left( \mathcal{C}_{1} \cap \text{Row}(A) / \mathcal{C}_{2} \cap \text{Row}(A) \right) \geq 1 \}, \end{array} $$

where$\text {Row}(A) \subseteq \mathbb {F}_{q^{m}}^{n}$denotes the$\mathbb {F}_{q^{m}}$-linear vector space generated by the rows of the matrix$A \in \mathbb {F}_{q}^{\nu \times n}$.

With this auxiliary result, we may now prove Lemma 3, which we now recall:

Lemma 3

Let$B \in \mathbb {F}_{q}^{\mu \times n}$andlet$\mathcal {C}_{2} \subsetneqq \mathcal {C}_{1} \subseteq \mathbb {F}_{q^{m}}^{n}$be$\mathbb {F}_{q^{m}}$-linearcodes. If$\text {Rk}(B) < d_{R} \left (\mathcal {C}_{2}^{\perp }, \mathcal {C}_{1}^{\perp } \right )$,then

$$\mathcal{C}_{2} B^{T} = \mathcal{C}_{1} B^{T}, $$

where$\mathcal {C} B^{T} = \left \lbrace \mathbf {c} B^{T} \mid \mathbf {c} \in \mathcal {C} \right \rbrace \subseteq \mathbb {F}_{q^{m}}^{\mu }$, for a code$\mathcal {C} \subseteq \mathbb {F}_{q^{m}}^{n}$.

Proof

Given an $\mathbb {F}_{q^{m}}$-linear code $\mathcal {C} \subseteq \mathbb {F}_{q^{m}}^{n}$, consider the map $\mathcal {C} \longrightarrow \mathcal {C} B^{T}$ defined by c ↦ cB^T, for $\mathbf {c} \in \mathcal {C}$. It is surjective and its kernel is $\mathcal {C} \cap \left (\mathcal {V}^{\perp } \right )$, where $\mathcal {V} = \text {Row}(B)$. Therefore

$$\dim(\mathcal{C}) = \dim \left( \mathcal{C} B^{T} \right) + \dim \left( \mathcal{C} \cap \left( \mathcal{V}^{\perp} \right) \right) . $$

Using this equation and computing dimensions, it follows that

$$ \dim \left( \mathcal{C}_{1} B^{T} / \mathcal{C}_{2} B^{T} \right) = \dim \left( \mathcal{C}_{2}^{\perp} \cap \mathcal{V} / \mathcal{C}_{1}^{\perp} \cap \mathcal{V} \right). $$

(19)

Now, using that $\text {Rk}(B) < d_{R} \left (\mathcal {C}_{2}^{\perp }, \mathcal {C}_{1}^{\perp } \right )$and thep revious lemma, it holds that $\mathcal {C}_{2}^{\perp } \cap \mathcal {V} = \mathcal {C}_{1}^{\perp } \cap \mathcal {V}$. Hence the result follows by (19). □