We introduce large groups of quadratic transformations of a vector space over the finite fields defined via symbolic computations with the usage of algebraic constructions of Extremal Graph Theory. They can serve as platforms for the protocols of Noncommutative Cryptography. The modifications of these symbolic computations in the case of large fields of characteristic two allow us to define quadratic bijective multivariate public keys such that the inverses of public maps has a large polynomial degree. We suggest the usage of constructed protocols for the private delivery of quadratic encryption maps instead of the public usage of these transformations.