By Jean Chaumine, James Hirschfeld, Robert Rolland
This quantity covers many subject matters together with quantity thought, Boolean services, combinatorial geometry, and algorithms over finite fields. This booklet comprises many attention-grabbing theoretical and applicated new effects and surveys provided by way of the simplest experts in those parts, resembling new effects on Serre's questions, answering a question in his letter to most sensible; new effects on cryptographic purposes of the discrete logarithm challenge relating to elliptic curves and hyperellyptic curves, together with computation of the discrete logarithm; new effects on functionality box towers; the development of recent periods of Boolean cryptographic services; and algorithmic purposes of algebraic geometry.
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Extra info for Algebraic Geometry and Its Applications: Dedicated to Gilles Lachaud on His 60th Birthday (Series on Number Theory and Its Applications)
February 11, 2008 15:6 WSPC - Proceedings Trim Size: 9in x 6in saga˙master Fast addition on non-hyperelliptic genus 3 curves 13 can assume d = 1. Finally P4∞ ∈ E implies that E has no x3 term. This gives the form of the cubic. As for the cubic, the conic Q must have a tangent line at P1∞ equal to l∞ . This gives directly the desired form. 2]]. Note that all the computations are carried over k. 1 (Algorithm for Addition). Input: D1 = [u1 , v1 ] and D2 = [u2 , v2 ] Output: D1 + D2 = [uD1 +D2 , vD1 +D2 ] 1.
The main point is to find the possible Frobenius elements by finding generators of certain principal ideals (Step √ 2) with absolute value equal to p (Step 4a). 1. Let K be a primitive quartic CM field and K ∗ the reflex of K. The following algorithm takes as input the field K, a prime p that splits completely in K and splits completely into principal ideals in K ∗ , and a curve C defined over the finite field Fp . The algorithm returns true or false according to whether End(J) is an order in OK , where J = Jac(C).
For each (i, j) with ℓij = p, let kij be an integer such that π kij − 1 ∈ ℓij OK . Suppose p > 3. Then the following set generates OK over Z[π, π]: ni αi : ℓ2ij | [OK : Z[π]] d ℓijij ∪ π kij − 1 2 : ℓij ∤ [OK : Z[π]], ℓij = p . 9. 8 shows that if p > 3 and the index [OK : Z[π, π]] is square-free, then OK can be generated over Z[π, π] by a collection k of elements of the form π ℓ−1 . This answers a question raised by Eisentr¨ager and Lauter [11, Remark 5]. 8 can still be used to determine a generating set for OK .
Algebraic Geometry and Its Applications: Dedicated to Gilles Lachaud on His 60th Birthday (Series on Number Theory and Its Applications) by Jean Chaumine, James Hirschfeld, Robert Rolland