On using expansions to the base of −2

Lange, Tanja; Avanzi, Roberto; Frey, Gerhard; Oyono, Roger

doi:10.1080/00207160410001661311

Cited by 4 publications

(8 citation statements)

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“…http://www.exp-math.uni-essen.de/~oyono Remark 3. As explained in [4], one can try to use −2-adic expansion rather than usual 2-adic expansion, in order to save time for scalar multiplication. But this is only worthwile if the computation of −(D 1 + D 2 ) is easier than that of D 1 + D 2 .…”

Section: Algebraic Descriptionmentioning

confidence: 99%

“…The curveC has a rational flex. After a linear transformation, and by denoting new coordinates still by x, y, z, we havẽ C : y 3 z + y 2 (5057xz + 22616z 2 ) + y(6567x 3 + 18877x 2 z + 162xz 2 + 14333z 3 ) = 8673x 4 + 24517x 3 z + 20295x 2 z 2 + 17815xz 3 + 3799z 4 Choosing a random rational divisor, and computing its order, we may check that this curve has the correct cardinality in 0.14 seconds.…”

Section: -Dimensional Factors Of J Newmentioning

confidence: 99%

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Fast addition on non-hyperelliptic genus 3 curves

Flon¹,

Oyono²,

Ritzenthaler³

2008

Algebraic Geometry and Its Applications

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We present a fast addition algorithm in the Jacobian of a genus 3 non-hyperelliptic curve over a field of any characteristic. When the curve has a rational flex and char(k) > 5, the computational cost for addition is 148M + 15SQ + 2I and 165M + 20SQ + 2I for doubling. An appendix focuses on the computation of flexes in all characteristics. For large odd q, we also show that the set of rational points of a nonhyperelliptic curve of genus 3 can not be an arc.

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Section: Algebraic Descriptionmentioning

confidence: 99%

Section: -Dimensional Factors Of J Newmentioning

confidence: 99%

Fast addition on non-hyperelliptic genus 3 curves

Flon¹,

Oyono²,

Ritzenthaler³

2008

Algebraic Geometry and Its Applications

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“…subject to the relation (1). To obtain a basis of W N , we restrict ourselves to monomials with exponent pairs (i, j) with j ≤ 2, or alternatively to pairs (i, j) with i ≤ 3; this takes equation (1) into account.…”

Section: Overview Of Our Algorithmsmentioning

confidence: 99%

“…In practice, most of the use of Jacobian arithmetic will be to find a multiple m · ξ with m ∈ Z. In that case, we can use the "base −2 expansion" of [1] and only find the addflips ξ ′′ in the intermediate steps without any need for further negations.…”

Section: Overview Of Our Algorithmsmentioning

confidence: 99%

Fast Jacobian Group Operations for C_3,4 Curves over a Large Finite Field

Salem¹,

Khuri-Makdisi²

2007

LMS J. Comput. Math.

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Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Mathematics of Computation). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g 2.376 ) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C3,4 curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.Remark (added August 22, 2007): A revised version of this article has been published as LMS J. Comput. Math. 10 (2007) 307-328 with an appendix of sample Magma code of our algorithms. The URL for the published version is:

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“…On the other hand, for cryptographic purposes, scalar multiplication is the main topic. In that respect, our algorithm benefits from the two following remarks, which should approximately halve the complexity: one can speed up scalar multiplication using the fast automorphism σ defined p. 57, (see [6]), and rather use −2-adic expansions instead of 2-adic usual expansions (see [1]). …”

Section: Remarks and Outlookmentioning

confidence: 99%