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Jacobians of Hyperelliptic Curves over ℤn and Factorization of n

Dryło Robert, Pomykała Jacek

Fundamenta Informaticae

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2019

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Vol. 169, nr 4

275--283

EN

E. Bach showed that factorization of an integer n can be reduced in probabilistic polynomial time to the problem of computing exponents of elements in ℤn* (in particular the group order of ℤ*n). It is also known that factorization of square-free integer n can be reduced to the problem of computing the group order of an elliptic curve E/ℤn. In this paper we describe the analogous reduction for computing the orders of Jacobians over ℤn of hyperelliptic curves C over ℤn using the Mumford representation of divisor classes and Cantor’s algorithm for addition. These reductions are based on the group structure of the Jacobian. We also propose other reduction of factorization to the problem of determining the number of points |C(ℤn)|, which makes use of elementary properties of twists of hyperelliptic curves.

2

The analytic rank of a family of jacobians of Fermat curves

Jędrzejak T.

Bulletin of the Polish Academy of Sciences. Mathematics

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2008

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Vol. 56, no 3-4

199-206

EN

We study the family of curves Fm(p) : xp + yp = m, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves Fm(p). As a corollary we conclude that the Jacobians of the curves Fm(5) with even analytic rank and those with odd analytic rank are equally distributed.

3

Rational points on certain hyperelliptic curves over finite fields

Ulas M.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 2

97-104

EN

Let K be a field, a, b ∈ K and ab ≠ 0. Consider the polynomials g1 (x) = xn + ax + b, g2(x) = xn + ax2 + bx, where n is a fixed positive integer. We show that for each k ≥ 2 the hypersurface given by the equation S[...], i = 1, 2 contains a rational curve. Using the above and van de Woestijne's recent results we show how to construct a rational point different from the point at infinity on the curves C1 : y2 = gi(x), (i = 1,2) defined over a finite field, in polynomial time.

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Algorytm znajdowania logarytmu dyskretnego na krzywych hipereliptycznych

Gawinecki J., Kijko T.

Biuletyn Wojskowej Akademii Technicznej

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2002

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Vol. 51, nr 4

25-39

PL

W pracy tej przedstawione są pojęcia związane z krzywymi hipereliptycznymi i kryptosystemami na nich opartymi. Pokazany jest algorytm wyznaczania logarytmu dyskretnego w jakobianie krzywej hipereliptycznej. Dla tego algorytmu przedstawiona jest analiza złożoności obliczeniowej i przykłady wykorzystania do złamania kryptosystemów opartych na krzywych hipereliptycznych.

EN

This paper contains basic definitions and theorems connected to the hyperelliptic curves and the cryptosystems based on those curves. There is presented the liptic curves defined over finite fields. For this algorithm the analysis of complexity is presented and the examples of using it to break cryptosystems based on hyperelliptic curves are given.