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The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard, and was accepted in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard, and is under consideration for inclusion in some other ISO standards. Unlike the(More)
Previously, no general-purpose algorithm was known for the elliptic curve logarithm problem that ran in better than exponential time. In this paper we demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying hit e field. For the class of supersingu-lar elliptic(More)
The fundamental operation in elliptic curve cryptographic schemes is the multiplication of an elliptic curve point by an integer. This paper describes a new method for accelerating this operation on classes of elliptic curves that have efficiently-computable endomorphisms. One advantage of the new method is that it is applicable to a larger class of curves(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)
We present practical and realistic attacks on some standardized elliptic curve key establishment and public-key encryption protocols that are effective if the receiver of an elliptic curve point does not check that the point lies on the appropriate elliptic curve. The attacks combine ideas from the small subgroup attack of Lim and Lee, and the differential(More)