The notion of alpha-equivalence between lambda-terms is commonly used to identify terms that are considered equal. However, due to the primitive treatment of free variables, this notion falls short when comparing subterms occurring within a larger context. Depending on the usage of the Barendregt convention (choosing different variable names for all involved binders), it will equate either too few or too many subterms. We introduce a formal notion of context-sensitive alpha-equivalence, where two open terms can be compared within a context that resolves their free variables. We show that this equivalence coincides exactly with the notion of bisimulation equivalence. Furthermore, we present an efficient O(n log n) runtime hashing scheme that identifies lambda-terms modulo context-sensitive alpha-equivalence, generalizing over traditional bisimulation partitioning algorithms and improving upon a previously established O(n log^2 n) bound for a hashing modulo ordinary alpha-equivalence by Maziarz et al. Hashing lambda-terms is useful in many applications that require common subterm elimination and structure sharing. We have employed the algorithm to obtain a large-scale, densely packed, interconnected graph of mathematical knowledge from the Coq proof assistant for machine learning purposes.