In at least one implementation, described herein, P and Q.sub.1, . . . , Q.sub.n are public points on an elliptic curve over a finite field, but the ratios of Q.sub.i to P are private. Those ratios are the components (.alpha..sub.1, . . . , .alpha..sub.n) of a private key, where Q.sub.i=.alpha..sub.i P. This implementation generates short digital ciphers (i.e., signatures), at least in part, by mapping a message M to a point T on the elliptic curve and then scaling that point T based upon the private key .alpha. to get S. At least one other implementation, described herein, verifies those ciphers by comparing pairing values of two pairs, where one pair is the public point P and the scaled point S and another pair is public Q and the point T. This implementation tests whether log(Q)/log(P)=log(S)/log(T), without computing any elliptic curve discrete logarithm directly.

 
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