Interpolation attack was presented by Jakobsen and Knudsen at FSE’97. Interpolation attack is effective against ciphers that have a certain algebraic structure like the $ \mathcal{P}\mathcal{U}\mathcal{R}\mathcal{E} $ \mathcal{P}\mathcal{U}\mathcal{R}\mathcal{E} cipher which is a prototype cipher, but it is difficult to apply the attack to real-world ciphers. This difficulty is due to the difficulty of deriving a low degree polynomial relation between ciphertexts and plaintexts. In other words, it is difficult to evaluate the security against interpolation attack. This paper generalizes the interpolation attack. The generalization makes easier to evaluate the security against interpolation attack. We call the generalized interpolation attack linear sum attack. We present an algorithm that efficiently evaluates the security of byte-oriented ciphers against linear sum attack. Moreover, we show the relationship between linear sum attack and higher order differential attack. In addition, we show the security of CRYPTON, E2, and RIJNDAEL against linear sum attack using the algorithm.