Vector form of the finite fields $GF(p^m)$

Specially defined multiplication operation in the $m$-dimensional vector space (VS) over a ground finite field (FF) imparts properties of the extension FF to the VS. Conditions of the vector FF (VFF) formation are derived theoretically for cases $m=2$ and $m=3$. It has been experimentally demonstrated that under the same conditions VFF are formed for cases $m=4$, $m=5$, and $m=7$. Generalization of these results leads to the following hypotheses: for each dimension value $m$ the VS defined over a ground field $GF(p)$, where $p$ is a prime and $m|p-1$, can be transformed into a VFF introducing special type of the vector multiplication operations that are defined using the basis-vector multiplication tables containing structural coefficients. The VFF are formed in the case when the structural coefficients that could not be represented as the $m$th power of some elements of the ground field are used. The VFF can be also formed in VS defined over extension FF represented by polynomials. The VFF present interest for cryptographic application.