We present an algorithm allowing the rapid identification of low order nonlinear Boolean functions. An extension of the method allowing the identification of good low order approximations (if they exist) is then described. We discuss the application of the method to cryptanalysis of black-box cipher functions. We present results indicating that the method can be expected to perform better than random search in locating good low order approximating Boolean functions. An expression for the effectiveness of the attack is derived, and it is shown that highly nonlinear balanced Boolean functions constructed as modified low order bent functions are particularly vulnerable to the attack. The required tradeoff in resisting both linear and quadratic approximation is also discussed.