Abstract: A method of generating and verifying a digital signature by selecting an elliptic curve; selecting a point G; generating x and M; reducing x; generating a base tau expansion, in non-adjacent form, of the reduced x; multiplying G by the expansion; computing h=Hash(M); generating k; reducing k; generating a base tau expansion, in non-adjacent form, of the reduced k; multiplying G by the expansion of k to form K=(Kx,Ky); computing R=(Kx mod q); returning to the step of generating k if R=0, otherwise computing S=(k{circumflex over ( )}−1)(h+xR); returning to the step of generating k if S=0, otherwise transmitting y, q, M, R, and S; receiving y, q, M, R, and S; proceeding with the next step if 0<R<q and 0<S<q, otherwise not verifying the digital signature and stopping; forming h=Hash(M); computing f=((S{circumflex over ( )}−1) mod q), b=(hf mod q), and t=(Rf mod q); reducing b and t; generating a base tau expansion,

Abstract: A method of cryptographic key exchange by two users agreeing on an elliptic curve 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 F2m, where m is an integer; agreeing on 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; generating a private integer xA by a first user; reducing xA by mod (&tgr;m−1) in the form of w+z&tgr; by the first user; generating a base tau expansion, in non-adjacent form, of the modular reduced xA by the first user; multiplying G by the base tau expansion of the modular reduced xA; transmitting the product to the second user; 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 product transmitted to the second user; multiplying, by the first user, xA by the value received from the seco