This paper provides three improvements over previous work on analyzing CubeHash, based on its classes of symmetric states: (1) We present a detailed analysis of the hierarchy of symmetry classes. (2) We point out some flaws in previously claimed attacks which tried to exploit the symmetry classes. (3) We present and analyze new multicollision and preimage attacks. For the default parameter setting of CubeHash, namely for a message block size of b = 32, the new attacks are slightly faster than 2^384 operations. If one increases the size of a message block by a single byte to b = 33, our multicollision and preimage attacks become much faster – they only require about 2^256 operations. This demonstrates how sensitive the security of CubeHash is, depending on minor changes of the tunable security parameter b.

The operations addition modulo 2^n and exclusive-or have recently been combined to obtain an efficient mechanism for nonlinearity in block cipher design. In this paper, we show that ciphers using this approach may be approximated by pseudo-linear expressions relating groups of contiguous bits of the round key, round input, and round output. The bias of an approximation can be large enough for known plaintext attacks. We demonstrate an application of this concept to a reduced-round version of the Threefish block cipher, a component of the Skein entry in the secure hash function competition.