Hyperelliptic Curves, Cartier — Manin Matrices and Legendre Polynomials

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

“…This work is an extended and refined version of preliminary results presented by author on the conference SibeCrypt '18 [31]. Table 1: Characteristic polynomials modulo p for hyperelliptic curves of the form C : y 2 = x 2g+1 + ax g+1 + bx over a prime finite field F p , p > 2, p | g, P m := 1 + c 2 a 2 1 + c 0 = 0. c 0 =(128q 4 − 128a 1,2 q 3 + 32a 2 1,2 q 2 + 128a 3,2 q − 64a 1,2 a 2,2 q+ + 16a 3 1,2 q − 64a 4,2 + 16a 2 2,2 − 8a 2 1,2 a 2,2 + a 4 1,2 ) 2 , c 2 = − 131072q 7 + 163840a 1,2 q 6 − 32768a 2,2 q 5 − 65536a 2 1,2 q 5 − 81920a 3,2 q 4 + + 45056a 1,2 a 2,2 q 4 + 5120a 3 1,2 q 4 + 65536a 4,2 q 3 + 49152a 1,2 a 3,2 q 3 + − 16384a 2 2,2 q 3 − 12288a 2 1,2 a 2,2 q 3 + 2048a 4 1,2 q 3 − 49152a 1,2 a 4,2 q 2 + + 4096a 2 1,2 a 3,2 q 2 + 8192a 1,2 a 2 2,2 q 2 − 5120a 3 1,2 a 2,2 q 2 + 768a 5 1,2 q 2 + + 16384a 2,2 a 4,2 q − 16384a 2 3,2 q + 4096a 1,2 a 2,2 a 3,2 q − 1024a 3 1,2 a 3,2 q+ − 4096a 3 2,2 q + 4096a 2 1,2 a 2 2,2 q − 1280a 4 1,2 a 2,2 q + 128a 6 1,2 q+ + 8192a 3,2 a 4,2 − 6144a 1,2 a 2,2 a 4,2 + 1536a 3 1,2 a 4,2 + 2048a 2 2,2 a 3,2 + − 1024a 2 1,2 a 2,2 a 3,2 + 128a 4 1,2 a 3,2 − 512a 1,2 a 3 2,2 + 384a 3 1,2 a 2 2,2 + − 96a 5 1,2 a 2,2 + 8a 7 1,2 , c 4 =253952q 6 − 233472a 1,2 q 5 + 47104a 2,2 q 4 + 65024a 2 1,2 q 4 + 57344a 3,2 q 3 + − 26624a 1,2 a 2,2 q 3 − 5632a 3 1,2 q 3 − 61440a 4,2 q 2 − 12288a 1,2 a 3,2 q 2 + + 7168a 2 2,2 q 2 − 2048a 2 1,2 a 2,2 q 2 + 1344a 4 1,2 q 2 + 22528a 1,2 a 4,2 q+ − 2048a 2,2 a 3,2 q − 2560a 2 1,2 a 3,2 q + 3584a 1,2 a 2 2,2 q − 1280a 3 1,2 a 2,2 q+ + 96a 5 1,2 q − 7168a 2,2 a 4,2 + 1280a 2 1,2 a 4,2 + 4096a 2 3,2 + − 2048a 1,2 a 2,2 a 3,2 + 512a 3 1,2 a 3,2 − 256a 3 2,2 + 576a 2 1,2 a 2 2,2 + − 240a 4 1,2 a 2,2 + 28a 6 1,2 , c 6 = − 204800q 5 + 136192a 1,2 q 4 − 16384a 2,2 q 3 − 29696a 2 1,2 q 3 + − 12288a 3,2 q 2 + 1024a 1,2 a 2,2 q 2 + 4096a 3 1,2 q 2 + 20480a 4,2 q+ − 1024a 1,2 a 3,2 q + 1024a 2 2,2 q − 320a 4 1,2 q − 2560a 1,2 a 4,2 + − 1024a 2,2 a 3,2 + 768a 2 1,2 a 3,2 + 384a 1,2 a 2 2,2 − 320a 3 1,2 a 2,2 + 56a 5 1,2 , c 8 =79104q 4 − 38144a 1,2 q 3 + 512a 2,2 q 2 + 7104a 2 1,2 q 2 + 256a 3,2 q+ + 640a 1,2 a 2,2 q − 800a 3 1,2 q − 2176a 4,2 + 512a 1,2 a 3,2 + 96a 2 2,2 + − 240a 2 1,2 a 2,2 + 70a 4 1,2 , c 10 = − 15360q 3 + 5376a 1,2 q 2 + 256a 2,2 q − 768a 2 1,2 q + 128a 3,2 + − 96a 1,2 a 2,2 + 56a 3 1,2 , c 12 =1472q 2 − 352a 1,2 q − 16a 2,2 + 28a 2 1,2 , c 14 =8a 1,2 − 64q.…”

“…The purpose of this note is to generalize results for g = 2 to the higher genera (g ≥ 3) and derive algorithms for counting points on J C in this case. To speed up point counting, where it is possible, we use connection of the Cartier-Manin matrix of the curve C with Legendre polynomials from [3]. On the other side, this connection is also used to obtain new congruences for Legendre polynomials P p−1 8 , P 3p−3 8 extending the results from the works [4,5,6,7].…”

“…This is done by using results and methods from [9,10,11,12]. Section 2 is devoted to the study of Cartier-Manin matrix of the curve C. We obtain a complete list of possible polynomials χ C,p (T ) mod p. This fills missing cases in our previous work [3]. We use this list to derive a pointcounting algorithm in genus 3 case and to find new congruences for Legendre polynomials connected with genus 4 curves.…”

“…For example, the polynomial P p−1 2 appears in formulae for g = 5, 7 in [3, Table 1,2]. Also, we can obtain new congruences for Legendre polynomials using tables from [3] and decomposition of J C ′ from Section 1.…”

“…The curves C and C ′ are isomorphic over the field F q [ 4g √ b], but the theorem works even in the case when the curves are not isomorphic over F q . Now, we can derive all possibilities for χ C (T ) (mod p) using the same method as in [3]. First, it follows from Theorem 4 that Cartier-Manin matrix W of C is (generalized) permutation matrix with permutation σ defined by…”