By David Kohel, Robert Rolland (ed.)
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Extra resources for Arithmetic, Geometry, Cryptography and Coding Theory 2009
The point addition in this special case only costs 5M +2S. It is even faster than the general point doubling in Jacobian coordinates. In this state, the algorithm is not very useful because it is unlikely for both P1 and P2 to have the same Z-coordinate. Meloni noticed that, while computing the addition, one can easily modify the entry point P1 so that P1 and P1 + P2 have the same Z-coordinate at the end of the addition. He calls this algorithm NewAdd(P1 , P2 ) → (P˜1 , P1 + P2 ). NewAdd. Let P1 = (X1 , Y1 , Z), P2 = (X2 , Y2 , Z) both unequal to ∞ and P2 = ±P1 .
We use a sieving argument to show that there exists a choice of t(x) such that pt (x) is squarefree. Note we have q g+2 choices for t(x). Suppose that pt is not squarefree. Then pt (x) is divisible by the square of a monic irreducible polynomial v(x) of degree m ≤ g + 1. But note that if v 2 | pt1 and v 2 | pt2 for two choices t1 , t2 , then subtracting we have v 2 | r(t1 + t2 ) + t21 + t22 = (t1 + t2 )(r + t1 + t2 ). Moreover, if v divides each of these two factors then in fact v | r. We are then led to consider two cases.
51–102 in S. ), The eightfold way: the beauty of Klein’s quartic curve, MSRI Publication Series 35, Cambridge University Press, 352 pp. (1999)  A. Enge, How to distinguish hyperelliptic curves in even characteristic, proceedings of Public– key Cryptography and Computational Number Theory (Warsaw 2000), de Gruyter, Berlin, pp. 49–58 (2001) ´ndez Encinas, J. Mun ˜oz Masqu´  J. Espinosa Garc´ıa, L. Herna e, A review on the isomorphism classes of hyperelliptic curves of genus 2 over ﬁnite ﬁelds admitting a Weierstrass point, Acta Appl.
Arithmetic, Geometry, Cryptography and Coding Theory 2009 by David Kohel, Robert Rolland (ed.)