Method of generating cryptographic key using elliptic curve and expansion in joint sparse form and using same

Abstract

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 rand wand the second user selects integers rand w. Next, the first user generates points R=rP and W=wP and the second user generates points R=rP and W=wP. Next, the first user transmits Rand Wto the second user and the second user transmits Rand Wto the first user. Next, the first user generates cfrom what it possesses while the second user generates cfrom what it possesses in equivalent fashion. Next, the first user generates gas a function of w, r, W, and Rand the second user generates gas a function of w, r, W, and R, in equivalent fashion. Next, the first user binarily expands on cand gin joint sparse form and the second user does the same on cand g. Next, the first user generates a point K by computing cW+gRvia twin multiplication, double-add-subtract method, and the binary expansions in joint sparse form and the second user generates the point K by computing cW+gRin similar fashion. The last step is for each user to derive the cryptographic key from K in the same manner.