This technology is a combination of three patented methods for generating a crypto-graphic key between two users. The patents include the groundbreaking methods of identifying a digital signature using an efficient elliptic curve but also builds on previous technologies by adding a binary expansion in joint sparse form and a method of generating and verifying a cryptographic digital signature using coefficient splitting. Further innovation includes a method of generating a cryptographic key using coefficient splitting on two different classes of elliptic curves. This can be implemented in both a non-authenticated key exchange method and an authenticated key exchange method, minimizing the number of elliptic curve operations.