The problem of exact-repair regenerating codes against eavesdropping attack is studied. The eavesdropping model we consider is that the eavesdropper has the capability to observe the data involved in the repair of a subset of $ell$ nodes. Under this security constraint, it has been shown that the optimal tradeoff curve has a single corner point for some $(n,k,d,ell)$. The focus of this paper is on finding parameters $(n,k,d,ell)$ whose associated tradeoff curve has this behavior. For $k = d = n-1$, we prove that the tradeoff curve has a single corner point if and only if $ell geq leftlceil frac{1}{4}(d-1) rightrceil$. Previously, it was known that the tradeoff curve has a single corner point if $ell geq leftlceil (sqrt{d}-1)^2 rightrceil$.