On the order of the frobenius EN-domorphism action on L-torsion subgroup of abelian sur | Applied Discrete Mathematics. Supplement. 2019. № 12. DOI: 10.17223/2226308X/12/2

On the order of the frobenius EN-domorphism action on L-torsion subgroup of abelian sur

FACES. One of the most powerful tools for point-counting on elliptic curves over finite fields is the Schoof - Elkies - Atkin algorithm. Its main idea is to construct characteristic polynomials for the action of the Frobenius endomorphim on l-torsion group. In this work, we study a probabilistic approach to find these characteristic polynomials in case of abelian surface. To do this, we introduce a random variable £ that takes values from a list r_i,..., r_n, where r is a possible order of Frobenius action on l-torsion subgroup. As soon as we have an explicit distribution of orders, we can obtain a characteristic polynomial in more effective way than in a classical Schoof-like algorithm. In this work, we derive formulas for calculating variance and standard deviation of the random variable £: where ф(1) = 2lⁱ⁰ + 56l⁹ - 316l⁸ +13441⁷ - 19481⁶ - 17701⁵ + 66601⁴ - 35161³ - 38311² + 46841 - 1369.

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Keywords

абелевы поверхности, гиперэллиптические кривые, подсчёт числа точек, многочлен Фробениуса, Abelian surfaces, hyperelliptic curves, point-counting, Frobenius endomor-phism, l-torsion

Authors

Name	Organization	E-mail
Kolesnikov N. S.	I. Kant Baltic Federal University	NiKolesnikov@stud.kantiana.ru
Novoselov S.A.	I. Kant Baltic Federal University	snovoselov@kantiana.ru

Всего: 2

References

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