Hyperelliptic curves, Cartier - Manin matrices and Legendre polynomials

doi:10.17223/20710410/37/2

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Title: Hyperelliptic curves, Cartier - Manin matrices and Legendre polynomials
Creator: Novoselov, S. A.
Date: 2017
DOI: 10.17223/20710410/37/2
Description: Using hyperelliptic curves in cryptography requires the computation of the Jacobian order of a curve. This is equivalent to computing the characteristic polynomial of Frobenius x(A) e Z[A|. By calculating Cartier — Manin matrix, we can recover the polynomial x(A) modulo the characteristic of the base field. This information can further be used for recovering full polynomial in combination with other methods. In this paper, we investigate the hyperelliptic curves of the form C1 : y2 = x2g+1 + + ax9+1 + bx and C2 : y2 = x2g+2 + ax9+1 + b over the finite field Fq, q = pn, p > 2. We transform these curves to the form C1,p : y2 = x2g+1 — 2px9+1 + x and C2,p : y2 = x2g+2 — 2px9+1 +1, where p = —a/(2Vb), and prove that the coefficients of the corresponding Cartier — Manin matrices for the curves in this form are Legendre polynomials. As a consequence, the matrices are centrosymmetric and therefore, for finding the matrix, it’s enough to compute a half of coefficients. Cartier — Manin matrices are determined up to a transformation of the form S(p)WS- 1. It is known that centrosymmetric matrices can be transformed to the block-diagonal form by an orthogonal transformation. We prove that this transformation can be modified to have a form S(p)WS- 1 and be defined over the base field of the curve. Therefore, Cartier — Manin matrices of curves C1,p and C2,p are equivalent to block-diagonal matrices. In the case of gcd(p,g) = 1, Miller and Lubin proved that the matrices of curves C1 and C2 are monomial. We prove that the polynomial x(A) (mod p) can be found in factored form in terms of Legendre polynomials by using permutation attached to the monomial matrix. As an application of our results, we list all possible polynomials x(A) (mod p) in the case of gcd(p,g) = 1, g is from 2 to 7 and the curve C1 is over Fp if /b e Fp and over Fp2 if / b £ Fp.

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collection(s): Журналы ТГУ
Identifier: смотреть в электронном каталоге НБ ТГУ
Type: статьи в журналах
Source: Прикладная дискретная математика. 2017. № 37. С. 20-31
Language: eng

Created: 22-09-2017

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Журналы ТГУ Hyperelliptic curves, Cartier - Manin matrices and Legendre polynomials