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Let $k$ and $n$ be positive integers. Define $R(n,k)$ to be the minimumpositive value of $$ | e_i \sqrt{s_1} + e_2 \sqrt{s_2} + ... + e_k \sqrt{s_k}-t | $$ where $ s_1, s_2, ..., s_k$ are positive integers no larger than $n$,$t$ is an integer and $e_i\in \{1,0, -1\}$ for all $1\leq i\leq k$. It isimportant in computational geometry to determine a good lower and upper boundof $ R(n,k)$. In this paper we show that this problem is closely related to theshortest vector problem in certain integral lattices and present an algorithmto find lower bounds based on lattice reduction algorithms. Although we canonly prove an exponential time upper bound for the algorithm, it is efficientfor large $k$ when an exhaustive search for the minimum value is clearlyinfeasible. It produces lower bounds much better than the root separationtechnique does. Based on numerical data, we formulate a conjecture on thelength of the shortest nonzero vector in the lattice, whose validation impliesthat our algorithm runs in polynomial time and the problem of comparing twosums of square roots of small integers can be solved in polynomial time. As aside result, we obtain constructive upper bounds for $R(n,k)$ when $ n$ is muchsmaller than $2^{2k}$.

Bounding the Sum of Square Roots via Lattice Reduction