An algorithm for hiding and recovering data using matrices

Abstract

We present an algorithm for the recovery of a matrix $\mathbb{M}$ % (non-singular $\in $ $\mathbb{C}^{N\times N}$) by only being aware of two of its powers, $\mathbb{M}_{k_{1}}:=\mathbb{M}^{k_{1}}$ and $\mathbb{M}% _{k_{2}}:=\mathbb{M}^{k_{2}}$ ($k_{1}>k_{2}$) whose exponents are positive coprime numbers. The knowledge of the exponents is the key to retrieve matrix $\mathbb{M}$ out from the two matrices $\mathbb{M}_{k_{i}}$. The procedure combines products and inversions of matrices, and a few computational steps are needed to get $\mathbb{M}$, almost independently of the exponents magnitudes. Guessing the matrix $\mathbb{M}$ from the two matrices $\mathbb{M}_{k_{i}}$, without the knowledge of $k_{1}$ and $k_{2}$, is comparatively highly consuming in terms of number of operations. If a private message, contained in $\mathbb{M}$, has to be conveyed, the exponents can be encrypted and then distributed through a public key method as, for instance, the DF (Diffie-Hellman), the RSA (Rivest-Shamir-Adleman), or any other.