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We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over(More)
In this paper we discuss a source of finite abelian groups suitable for cryptosystems based on the presumed intractability of the discrete logarithm problem for these groups. They are the jacobians of hyperelliptic curves defined over finite fields. Special attention is given to curves defined over the field of two elements. Explicit formulas and examples(More)
Our purpose is to describe elliptic curves with complex multiplication which in characteristic 2 have the following useful properties for constructing Diffie-HeUman 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(More)
Since the introduction of public-key cryptography by Diffie and Hellman in 1976, the potential for the use of the discrete logarithm problem in public-key cryptosystems has been recognized. Although the discrete logarithm problem as first employed by Diffie and Hellman was defined explicitly as the problem of finding logarithms with respect to a generator(More)
The security of elliptic curve cryptosystems is based on the presumed intractability of the discrete logarithm problem on the curve. Other than algorithms that work in an arbitrary group and are exponential in the general case, the only general-purpose algorithm that has ever been proposed for the elliptic curve discrete logarithm is that of(More)
We give an informal analysis and critique of several typical "provable security" results. In some cases there are intuitive but convincing arguments for rejecting the conclusions suggested by the formal terminology and "proofs," whereas in other cases the formalism seems to be consistent with common sense. We discuss the reasons why the search for(More)
Over a period of sixteen years elliptic curve cryptography went from being an approach that many people mistrusted or misunderstood to being a public key technology that enjoys almost unquestioned acceptance. We describe the sometimes surprising twists and turns in this paradigm shift, and compare this story with the commonly accepted Ideal Model of how(More)