A method of generating a cryptographic key between two users. First, the users select an elliptic curve. Next, the users select a point P on the curve. Next, the first user selects integers ra and wa and the second user selects integers rb and wb. Next, the first user generates points Ra=raP and Wa=waP and the second user generates points Rb=rbP and Wb=wbP. Next, the first user transmits Ra and Wa to the second user and the second user transmits Rb and Wb to the first user. Next, the first user generates ca from what it possesses while the second user generates cb from what it possesses in equivalent fashion. Next, the first user generates ga as a function of wa, ra, Wb, and Rb and the second user generates gb as a function of wb, rb, Wa, and Ra, in equivalent fashion. Next, the first user binarily expands on ca and ga in joint sparse form and the second user does the same on cb and gb. Next, the first user generates a point K by computing caWb+gaRb via twin multiplication, double-add-subtract method, and the binary expansions in joint sparse form and the second user generates the point K by computing cbWa+gbRa in similar fashion. The last step is for each user to derive the cryptographic key from K in the same manner.