A method of generating a cryptographic key in an authenticated manner using coefficient splitting. Select a prime number p and an elliptic curve of either a first class or a second class. Select a point P. The first user generates r.sub.a, w.sub.a, and R.sub.a=r.sub.aP and W.sub.a=w.sub.aP via coefficient splitting. The second user generates r.sub.b, w.sub.b, and R.sub.b=r.sub.bP and W.sub.b=w.sub.bP via coefficient splitting. After the users have exchanged the points R.sub.a, W.sub.a, R.sub.b, W.sub.b, the first user generates c.sub.a, g.sub.a, and c.sub.aW.sub.b, and g.sub.aR.sub.b via coefficient splitting, and the second user generates c.sub.b, g.sub.b, and c.sub.bW.sub.a and g.sub.bR.sub.a via coefficient splitting. Each user then sums the corresponding results to form K and derives the cryptographic key from K in the same user-definable manner. An unauthenticated key exchange method is also presented.