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By David Kohel, Robert Rolland (ed.)

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Extra info for Arithmetic, Geometry, Cryptography and Coding Theory 2009

Example text

L’indice de somme des carr´ees d’une fonction bool´eenne f `a m variables, introduit par Zhang et Zheng [21], est σf = Notons que f 2 ≤ f 4 ≤ f 1 q f (x)4 = f 4 4. x∈Vm ∞. 3. Les fonctions bool´ ennes x −→ Tr G(x) o` u G est un polynˆ ome de degr´ e binaire 3 Soit m un entier pair. Soit k un corps ﬁni `a q = 2m ´el´ements. Soit f la fonction bool´eenne `a m variables d´eﬁnie par f (x) = χ(G(x)) o` u G un polynˆome `a coeﬃcients k de la forme s i G(x) = a7 x7 + bi x2 +1 i=0 avec a7 = 0 et s un entier naturel.

A-d. α7 = a−1 a la courbe 7 , alors la courbe Cα est isomorphe ` d’´equation y 2 + y = ax5 + cx + d avec a = λ5 a7 α2 = α−5 et 1/4 1/2 c = λ(a7 α6 + a7 α3/4 + a7 α5/2 + −i (bi α)2 + i bi α2 ) o` u l’on a pos´e λ = 1. Le polynˆome P (x) se factorise dans k[x] sous la forme P (x) = a2 x5 + a = a2 z 4 (x + z)(z −2 x2 + ζz −1 x + 1)(z −2 x2 + ζ 2 z −1 x + e) o` u z = λ−1 α = α et ζ ∈ k est une racine primitive troisi`eme de l’unit´e. −7 . 3. Valeurs prises par Xα . Soit e = a−1 7 α Supposons que e ∈ (k∗ )3 de telle sorte qu’il existe l ∈ k tel que l3 = e.

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 . Let P3 = P1 + P2 = (X3 , Y3 , Z3 ). A = (X2 − X1 )2 , B = X1 A, C = X2 A, D = (Y2 − Y1 )2 , ⎧ ⎪ ⎨X3 = D − B − C, Y3 = (Y2 − Y1 )(B − X3 ) − E, ⎪ ⎩ Z3 = Z(X2 − X1 ), E = Y1 (C − B), ⎧ ⎪ ⎨X1 = B, Y1 = E, ⎪ ⎩ Z = Z3 .