Finding an Integral vector in an Unknown Polyhedral Cone

We present an algorithm to find an integral vector in the polyhedral cone $Γ=\{X | \textbf{A}X \leq \textbf{0}\}$, without assuming the explicit knowledge of $\textbf{A}$. About the polyhedral cone, $Γ$, it is only given that, (i) the elements of \textbf{A} are in $\{-d,-d+1,\...,0,\...,d-1,d\}$, $d \in \mathbb{N}$, and, (ii) $Y=[y(1),y(2),\...,y(n)]$ is a non-zero integral solution to $Γ$. The proposed algorithm finds a non-zero integral vector in $Γ$ such that its maximum element is less than ${(2d)^{2^{n-1}-1}}/{2^{n-1}}$.

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