Abstract
Since the group of an elliptic curve defined over a finite field Fq, was proposed for Diffie-Hellman type cryptosystems in [7] and [15], some work on implementation has been done using special types of elliptic curves for which the order of the group is trivial to compute ([2], [13]). A consideration which discourages the use of an arbitrary elliptic curve is that one needs Schoof’s algorithm [16] to count the order of the corresponding group, and this algorithm, in addition to being rather complicated, has running time O(log9 q) for an elliptic curve defined over Fq. Thus, in applications of elliptic curves where one needs extremely large q — for example, the original version of the elliptic curve primality test ([4], [11]) — this algorithm is too time-consuming.
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Koblitz, N. (1991). Constructing Elliptic Curve Cryptosystems in Characteristic 2. In: Menezes, A.J., Vanstone, S.A. (eds) Advances in Cryptology-CRYPTO’ 90. CRYPTO 1990. Lecture Notes in Computer Science, vol 537. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-38424-3_11
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DOI: https://doi.org/10.1007/3-540-38424-3_11
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