Using fuzzy bits and neural networks to partially invert few rounds of some cryptographic hash functions

We consider fuzzy, or continuous, bits, which take values in [0;1] and (-1;1] instead of {0;1}, and operations on them (NOT, XOR etc.) and on their sequences (ADD), to obtain the generalization of cryptographic hash functions, CHFs, for the messages consisting of fuzzy bits, so that CHFs become smooth and non-constant functions of each bit of the message. We then train the neural networks to predict the message that has a given hash, where the loss function for the hash of predicted message and given true hash is backpropagatable. The results of the trainings for the standard CHFs - MD5, SHA1, SHA2-256, and SHA3/Keccak - with small number of (optionally weakened) rounds are presented and compared.