Elements of high multiplicative order in extended finite fields on a base of modified GAO approach

The Gao approach to construction of high order elements in arbitrary finite fields is to choose a convenient polynomial, which defines an extension of an initial prime field. This choice depends on one polynomial-parameter. That is why the mentioned approach can be considered as using of a finite field description with one degree of freedom. We explore in the paper the possibility of improvement of lower bound on element orders in finite fields of general form with using of two degrees of freedom. We have performed computer calculations in Maple environment, that would show possible winnings in this case, and given the correspondent results. Elements of high multiplicative order are used in a series of cryptographic primitives (Diffie-Hellman protocol, El-Gamal public