Characteristic polynomials of the curve y2 = x7 + ax4 + bx over finite fields

Novoselov, S. A.; Boltnev, Y. F.

doi:10.17223/2226308x/12/12

Cited by 2 publications

(1 citation statement)

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“…Another way for point-counting in genus 3 case is to derive an explicit formulae for the #J C (F q ) as it is done in [2] for genus 2. A complete list of the characteristic polynomials for the genus 3 curves to appear in [25].…”

Section: P P+1mentioning

confidence: 99%

Counting points on hyperelliptic curves of type y2 = x2g+1 + axg+1 + bx

Novoselov¹

2018

Applied Discrete Mathematics. Supplement

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In this work, we investigate hyperelliptic curves of type C : y 2 = x 2g+1 + ax g+1 + bx over the finite field Fq, q = p n , p > 2. For the case of g = 3 and 4 we propose algorithms to compute the number of points on the Jacobian of the curve with complexityÕ(log 4 p) andÕ(log 8 p). For curves of genus 2 − 7 we give a complete list of the characteristic polynomials of Frobenius endomorphism modulo p. * The reported study was funded by RFBR according to the research project 18-31-00244. 1The polynomial L X,q (T ) has a formwhere a i ∈ Z and a i ≤ 2g i q i 2 . Coefficients of the polynomial L X,q k (T ) are denoted by a i,k .Let J X (F q ) be the Jacobian of the curve over finite field F q and let χ X,q (T ) be the characteristic polynomial of the Frobenius endomorphism on J X . Then we have L X (T ) = T 2g χ X,q ( 1 T ) and #J X (F q ) = L X,q (1) = χ X,q (1). Because of that, point-counting on the Jacobian is equivalent to determining of χ X,q (T ) (or L X,q (T )).The order of the Jacobian satisfies Hasse-Weil bounds

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Section: P P+1mentioning

confidence: 99%