The key agreement scheme is an important part of the cryptography theory. The first study in this field belongs to Diffie–Hellman and Merkle. We present a new key agreement scheme using a group action of special orthogonal group of 2 × 2 matrices with real entries on the complex projective line.

In this paper, we consider the Diophantine equation P n − 3 a = ν and find all ν having at least two representations. In the proof of the main theorem, we use a version of the Baker-Davenport reduction method.

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Reduction algorithms are an important tool for understanding structural properties of groups. They play an important role in algorithms designed to investigate matrix groups over a finite field. One such algorithm was designed by Brooksbank et al. for members of the class C6 in Aschbacher's theorem, namely groups N that are normalizers in GL(d, q) of certain absolutely irreducible symplectic-type r -groups R , where r is a prime and d = r n with n > 2 .However, the analysis of this algorithm has only been completed when d = r 2 and when d = r n and n > 2 , in the latter case under the condition that G/RZ(G) ∼ = N/RZ(N ) . We prove that the algorithm runs successfully for some groups in the case of d = r 3 without any assumption.