The strongest well-known measure for the quality of a universal hash-function family H is its being e-strongly universal, which measures, for randomly chosen h ∈ H, one's inability to guess h(m') even if h(m) is known for some m ≠ m' We give example applications in which this measure is too weak, and we introduce a stronger measure for the quality of a hash-function family, e-variationally universal, which measures one's inability to distinguish h(m') from a random value even if h(m) is known for some m ≠ m'. We explain the utility of this notion and provide an approach for constructing efficiently compu table e-VU hash-function families.