  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.
