In the framework of the Blum-Shub-Smale real number model \cite{BSS}, we study the {\em algebraic complexity} of the integer linear programming problem
(ILP_{\bf R}) : Given a matrix A \in {\bf R}^{m \times n} and vectors
b \in {\bf R}^m, d \in {\bf R}^n, decide if there is x \in {\bf Z}^n such that
Ax \leq b, where {\bf 0} \leq x \leq d.

We show that there is an O\left( m\log ||d||\right) algorithm for ILP _{\bf R} , when the value of n is fixed.
As a consequence, we obtain a tight algebraic complexity bound \Theta \left( \log \frac{1}{a_{\min}}\right), a_{\min} = \min \{ a_1,\dots,a_n \} for the Knapsack problem
(KP_{\bf R}) : Given a \in {\bf R}_{+}^n, decide if there is x \in {\bf Z}^n such that a^T x = 1, when the dimension n is fixed.
We achieve these results in particular through a careful analysis of the algebraic complexity of the Lov\'asz' basis reduction algorithm and Kannan-Bachem's Hermite normal form algorithm, which may be of interest in its own.

We also obtain an O\left( mn^{5}\log n\left( n+\log ||d||\right) \right) depth {\em algebraic
decision tree} for ILP _{\bf R} , for every m and n .