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3.2. Algebraic curves 107Let us begin our survey of such algorithms with the simplest case: elliptic curves. This isindeed the best mastered situation and there are several different algorithms to compute thecardinalities of such curves.The most famous one is without any doubt the so-called SEA algorithm. It is named afterSchoof [231] who developed in 1985 the first deterministic polynomial time algorithm for pointcounting, and Elkies and Atkin who subsequently proposed practical improvements [233, 81]. Theoverall idea of such l-adic algorithms is to compute the trace of the Frobenius endomorphismmodulo small primes different from the characteristic of the field, and to gather this informationback using the Chinese Remainder Theorem. Overviews of these algorithms, and in particularof additional practical improvements, are given in the theses of Müller [214] and Lercier [171].These l-adic algorithms were subsequently extended to higher genera by Pila [220] and madepractical, at least in genus 2, by Gaudry, Harley and Schost [111, 113].In small characteristic, and especially in even characteristic that will be our principal interestin Chapter 4, more efficient algorithms have been developed. The first breakthrough is dueto Satoh [228] who proposed in 1999 to compute the trace of the Frobenius endomorphismon a canonical lift of the curve over the p-adics for p ≥ 5. This is the so-called canonicallift method. This method was extended to characteristic 2 and 3, and improved, by differentauthors [98, 247, 276, 200, 229, 150, 109, 172]. The main result we need relies on the AGMmethod described by Mestre [200] and has been given by Lercier and Lubicz [172] and furtherimproved by Harley [126].Theorem 3.2.15 ([126]). Let E be an elliptic curve defined over F 2 m. There exists an algorithmto compute the cardinality of E in O(m 2 (log m) 2 log log m) time and O(m 2 ) space.Mestre [201] extended the AGM method to higher genera and a quasi-quadratic algorithmwas described by Lercier and Lubicz [173].Theorem 3.2.16. Let H be a hyperelliptic curve of genus g defined over F 2 m. There exists analgorithm to compute the cardinality of H inbit operations and O(2 3g+o(1) m 2 ) memory.O(2 4g+o(1) g 3 m 2+o(1) )It should be remarked that the complexity of this algorithm is exponential in the genus of thecurve.There also exist other efficient algorithms in small and medium characteristic computing thetrace of the Frobenius endomorphism:1. on Dwork cohomology groups [78] in the approach of Lauder and Wan [165, 163, 164];2. on Monsky–Washnitzer cohomology groups [211, 208, 210, 209] in the approach initiated byKedlaya [146, 147] and extended to even characteristic by Denef and Vercauteren [66, 67];3. using deformation theory [79], for example in the work of Lauder [162] and Hubrechts [137,136].A complete description of many of the existing p-adic algorithms can be found in the theses ofVercauteren [275] and Hubrechts [135], or in different articles [274, 174]. Such algorithms havethe advantage to extend naturally to higher genera in any characteristic.We now state more precisely a result of Denef and Vercauteren [66, 67, 275] about hyperellipticcurves defined over a finite field of even characteristic.