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Our purpose is to describe elliptic curves with complex multiplication which in characteristic 2 have the following useful properties for constructing Diffie-Hellman type cryptosystems: (1) they are nonsupersingular (so that one cannot use the Menezes-Okamoto-Vanstone reduction of discrete log from elliptic curves to finite fields); (2) the order of the group has a large prime factor (so that discrete logs cannot be computed by giant-step/baby-step or the Pollard rho method); (3) doubling of points can be carried out almost as efficiently as in the case of the supersingular curves used by Vanstone; (4) the curves are easy to find.