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ECC: Do We Need to Count?
, 1999
"... A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming. ..."
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A prohibitive barrier faced by elliptic curve users is the difficulty of computing the curves' cardinalities. Despite recent theoretical breakthroughs, point counting still remains very cumbersome and intensively time consuming.
Optimizing the MenezesOkamotoVanstone (MOV) Algorithm for NonSupersingular Elliptic Curves
, 1999
"... . We address the MenezesOkamotoVanstone (MOV) algorithm for attacking elliptic curve cryptosystems which is completed in subexponential time for supersingular elliptic curves. There exist two hurdles to clear, from an algorithmic point of view, in applying the MOV reduction to general elliptic cur ..."
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. We address the MenezesOkamotoVanstone (MOV) algorithm for attacking elliptic curve cryptosystems which is completed in subexponential time for supersingular elliptic curves. There exist two hurdles to clear, from an algorithmic point of view, in applying the MOV reduction to general elliptic curves: the problem of explicitly determining the minimum extension degree k such that E[n] E(F q^k) and that of efficiently finding an ntorsion point needed to evaluate the Weil pairing, where n is the order of a cyclic group of the elliptic curve discrete logarithm problem. We can find an answer to the first problem in a recent paper by Balasubramanian and Koblitz. On the other hand, the second problem is important as well, since the reduction might require exponential time even for small k. In this paper, we actually construct a method of efficiently finding an ntorsion point, which leads to a solution of the second problem. In addition, our contribution allows us to draw the conclusion that the ...