A new class of public key agreement (PKA) algorithm was introduced, and its breaking complexity was discussed essentially in the previous paper. The algorithm is constructed by non–commutative algebra, for example, finite dimensional matrices and vectors. The asymmetry helps to increase its security and appends flexibility of computational requirements between the sender and the receiver. In the case of creating secret shared keys, for instance, between a server and a smartphone, the computational abilities are difference. The algorithm to create key for the smartphone is lighter for calculation and memory usage than for the server. The strongly asymmetric PKA with a larger key length is one of efficient algorithms for such biased environments. Moreover, we evaluated the breaking complexity mathematically assuming that the computational cost for the discrete logarithm is zero. In this study we review its mathematical description and examples of implementation. Moreover, we discuss its performance comparing the other standard PKAs.