AMS Subj. Classification: Primary 20N05, Secondary 94A60 The intention of this research is to justify deployment of quasigroups in cryptography, especially with new quasigroup based cryptographic hash function NaSHA as a runner in the First round of the ongoing NIST SHA-3 competition. We present new method for fast generation of huge quasigroup operations, based on the so-called extended Feistel networks and modification of the Sade’s diagonal method. We give new design of quasigroup based family of cryptographic hash functions - NaSHA, which deploy the new method and with a novel approach - different quasigroups for every application of component quasigroup transformations in every iteration of the compression function and, much more, the used quasigroups are functions of the processed message block.