The cube attack is a powerful cryptanalytic technique and is especially powerful against stream ciphers. Since we need to analyze the complicated structure of a stream cipher in the cube attack, the cube attack basically analyzes it by regarding it as a blackbox. Therefore, the cube attack is an experimental attack, and we cannot evaluate the security when the size of cube exceeds an experimental range, e.g., 40. In this paper, we propose cube attacks on non-blackbox polynomials. Our attacks are developed by using the division property, which is recently applied to various block ciphers. The clear advantage is that we can exploit large cube sizes because it never regards the cipher as a blackbox. We apply the new cube attack to Trivium , Grain128a, ACORN and Kreyvium. As a result, the secret keys of 832-round Trivium , 183-round Grain128a, 704-round ACORN and 872-round Kreyvium are recovered. These attacks are the current best key-recovery attack against these ciphers.