Method of elliptic curve cryptographic key exchange using reduced base tau expansion in non-adjacent form

A method of cryptographic key exchange, comprising the steps of:

• a) agreeing upon an elliptic curve between a first user and a second user, where the elliptic curve is of the form y2+xy=x3+ax2+1, where “a” is a member of a field F2, where the elliptic curve is defined over a field F2 m, where m is an integer;
• b) agreeing upon a point G as the base point of the elliptic curve, where the point G is of order q, and where q is an integer;
• c) generating a private integer xA by the first user;
• d) reducing xA by mod (τm−1) in the form of w+zτ by the first user;
• e) generating a base tau expansion, in non-adjacent form, of the result of step (d) by the first user;
• f) multiplying G by the result of step (e) by the first user;
• g) transmitting, by the first user, the result of step (f) to the second user;
• h) receiving, by the first user, a value yB from the second user that is based on a private integer xB of the second user, where the received value is computed in a similar manner as was the result of step (f);
• i) multiplying, by the first user, xA by the value received fiom the second user in order to form a common secret point;
• j) multiplying, by the second user, xB by the value received from the first user in order to form the common secret point;
• and
• k) deriving the cryptographic key from the common secret point by the first user and the second user.