Arithmetic, Geometry, Cryptography and Coding Theory 2009 by David Kohel, Robert Rolland (ed.)

By David Kohel, Robert Rolland (ed.)

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One of the benefits of this type of countermeasure is that there is no use of dummy operations, hence fault analysis techniques cannot be used. FASTER SIDE-CHANNEL RESISTANT ELLIPTIC CURVE SCALAR MULTIPLICATION 33 5 We can also mention the NAF-based multiplication algorithms [JY00, OT04]. The non-adjacent (NAF) form is a unique signed digit representation of an integer using the digits {−1, 0, 1}, such that no two adjacent digits are both non-zeros. NAF algorithms take advantage of the fact that negating a point on an elliptic curve simply requires a change in the sign of the Y -coordinate, substractions are cheap operations.

Lorsque m est un entier naturel pair, on a cherch´e des fonctions courbes de la forme x −→ χ(G(x)) o` u G est un polynˆome sur k. Par exemple, les fonctions de r Gold x −→ χ(x2 +1 ) sont des fonctions courbes (voir [8]). Dans cet article, nous consid´erons, lorsque m est un entier naturel pair, les fonctions bool´eennes de la forme x −→ χ(G(x)) o` u G est un polynˆ ome `a coefficients dans k de la forme s i G(x) = a7 x7 + bi x2 +1 i=0 o` u a7 = 0 et s est un entier naturel. Nous verrons que de ces fonctions bool´eennes ne sont pas courbes, mais qu’elles ont des propri´et´es de non-lin´earit´e plutˆot bonnes.

Point addition. Let P1 = (X1 , Y1 , Z1 ), P2 = (X2 , Y2 , Z2 ) both unequal to ∞ and P2 = ±P1 . Let P3 = P1 + P2 = (X3 , Y3 , Z3 ). A = Z12 , B = Z22 , G = D − C, C = X1 B, H = (2G)2 , D = X2 A, I = GH, E = Y1 Z2 B, J = 2(F − E), F = Y2 Z1 A, K = CH FASTER SIDE-CHANNEL RESISTANT ELLIPTIC CURVE SCALAR MULTIPLICATION 31 3 ⎧ 2 ⎪ ⎨X3 = J − I − 2K, Y3 = J(K − X3 ) − 2EI, ⎪ ⎩ Z3 = ((Z1 + Z2 )2 − A − B)G. A general point addition costs 11M + 5S. We use in our point scalar multiplication algorithm the simplified addition formula found by Meloni [Mel07].

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