Hyperelliptic curves and its applications

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

Although numerical procedures often supply a required accuracy, closed-form expressions allow one to escape any accumulation of errors. In this paper, we discuss the possibility of obtaining explicit results for a variance gamma process. We derive the exact distribution of the maximum of the variance gamma process over a finite interval of time and establish the prices of path-dependent options including digital barrier, fixed-strike lookback, and lookback options. The obtained formulas are based on values of hypergeometric functions.

При изучении арифметических свойств значений обобщенных гипергеометрических функций часто применяют известный в теории трансцендентных чисел метод Зигеля. Наиболее общие результаты в данной области были получены именно этим методом. Однако возможности метода Зигеля в случае гипергеометрических функций с иррациональнымипараметрами ограничены. Это связано с тем, что такие гипергеометрические функции не являются E-функциями, и по этой причине построить линейную приближающую форму свысоким порядком нуля с помощью принципа Дирихле здесь не удается. При рассмотрении задач, связанных с исследованием арифметической природы значений гипергеометрических функций с иррациональными параметрами, в некоторых случаях можно применить метод, основанный на эффективном построении линейной приближающей формы, но возможности этого метода также ограничены из-за того, что слишком общие эффективные конструкции отсутствуют. Трудности имеются также и в тех случаях, когда такие конструкции известны. Особенности этих конструкций таковы, что часто не удается реализовать арифметическую часть метода.Поэтому представляют интерес ситуации, когда можно провести требуемое исследование, опираясь на особые свойства конкретных гипергеометрических функций. Иногдаудается так подобрать параметры исследуемых функций, что можно преодолеть те трудности, которые возникают в общем случае. В настоящей работе рассматривается гипергеометрическая функция специального вида и ее производные. С помощью эффективной конструкции удалось не только доказать линейную независимость значений этих функций над некоторым мнимым квадратичным полем, но и получить соответствующий количественный результат в виде оценки модуля линейной формы от указанных значений.

A subexponential algorithm for discrete logarithms over hyperelliptic curves of large genus over GF( q)

We propose a reduced equation for hyperelliptic curves of genus 2 over finite fields ? q of q elements with characteristic different from 2 and 5. We determine the number of isomorphism classes of genus-2 hyperelliptic curves having an ? q -rational Weierstrass point. These results have applications to hyperelliptic curve cryptography.

One of the most celebrated applications of Gauss’s 2 F 1 _2F_1 hypergeometric functions is in connection with the rapid convergence of sequences and special values that arise in the theory of arithmetic and geometric means. This theory was the inspiration for a recent paper (see M. J. Griffin, Ken Ono, N. Saikia, and W.-L. Tsai, [AGM and jellyfish swarms of elliptic curves, Amer. Math. Monthly 130 (2023), no. 4, 355-369]) in which a finite field analogue of A G M R \mathrm {AGM}_\mathbb {R} was defined and then studied using finite field hypergeometric functions. Instead of convergent sequences, one gets directed graphs that combine to form disjoint unions of graphs that individually resemble jellyfish. Echoing the connection of hypergeometric functions to periods of elliptic curves, these graphs organize elliptic curves over finite fields. Here we use such “jellyfish swarms” to prove new identities for Gauss’s class numbers of positive definite binary quadratic forms. Moreover, we prove that the sizes of jellyfish are in part dictated by the order of the prime above 2 in certain class groups.

We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ over a finite field $$\mathbb {F}_p$$ . Salerno expresses the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of quotients of p-adic gamma functions under the condition that $$d|p-1$$ . In this paper, we first express the number of points over a finite field $$\mathbb {F}_p$$ on the family $$Z_{\lambda }$$ in terms of McCarthy’s p-adic hypergeometric series for any odd prime p not dividing $$d(d-1)$$ , and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene’s finite field hypergeometric series.

Hyperelliptic curves of small genus have the advantage of providing a group of com- parable size as that of elliptic curves, while working over a field of smaller size. Pairing-friendly hyperelliptic curves are those whose order of the Jacobian is divisible by a large prime, whose em- bedding degree is small enough for computations to be feasible, and whose minimal embedding field is large enough for the discrete logarithm problem in it to be difficult. We give a sequence of Fq-isogeny classes for a family of Jacobians of genus two curves over Fq, for q = 2 m , and their corresponding small embedding degrees. We give examples of the parameters for such curves with embedding degree k < (logq) 2 , such as k = 8,13,16,23,26,37,46,52. For secure and efficient implementation of pairing-based cryptography on genus g curves over Fq, it is desirable that the ratio = g log2 q log2 N be approximately 1, where N is the order of the subgroup with embedding degree k. We show that for our family of curves, is often near 1 and never more than 2. We also give a sequence of Fq-isogeny classes for a family of Jacobians of genus 2 curves over Fq whose minimal embedding field is much smaller than the finite field indicated by the embedding degree k. That is, the extension degrees in this example differ by a factor of m, where q = 2 m , demonstrating that the embedding degree can be a far from accurate measure of security. As a result,

We develop algorithms for three problems. Starting with a complex torus of dimension g ≥ 2, isomorphic to a principally polarized, simple abelian variety A/C, the first problem is to find an algorithmic solution of the hyperelliptic Schottky problem: Is there a hvperelliptic curve C of genus g whose jacobian variety Ic is isomorphic to A over C? Our solution is based on [Poor 1994]. If such a hyperelliptic curve C exists, the next problem is the construction of the Rosenhain model C : y2 = X(X−1)(X−λ1)(X−λ2)…(X−λ2g−1) for pairwise distinct numbers λj E ∈ C \\ {0, 1}. Applying the theory of hyperelliptic theta functions we show that these numbers λj can easily be computed by using theta constants with even characteristics. If the abelian variety A is defined over a field k (this field could be the field of rational numbers, an algebraic number field of low degree, or a finite field), we show only in the case k = Q for simplicity, how the method in [Mestre 1991] can be generalized to get a minimal equation over Z[½] for the hyperelliptic curve C with jacobian variety Ic ≅c A. This is our third problem. For some hyperelliptic, principally polarized and simple factors with dimension g = 3, 4, 5 of the jacobian variety J0(N) = IX0(N) of the modular curve X0(N) we compute the corresponding curve equations by applying our algorithms to this special situation.

In this thesis, we work on the characterization of some maximal curves over a finite field F q 2 whose equation is described by Chebyshev polynomials, where q is a power of a prime number p. Being m pxq P F q 2 rxs the Chebyshev polynomial of odd degree m, we describe the conditions on m and the number q for which the hyperelliptic curves y 2 < m pxq and y 2 < px 2 4q m pxq are maximal over F q 2 .These results have already been well studied for the case where m is an even number.With the aim of generalizing the hyperelliptic curve y 2 < m pxq to curves with equation C : y n < m pxq, some results were obtained.In the case where n < 3, that is, the curve C has equation y 3 < m pxq, we describe the conditions for this curve to be maximal over F p 2 .Another maximal curve characterized in this work is the curve with equation y d < d pxq.Furthermore, some other maximal curves were presented throughout the thesis, such as the curve given by y 3 < x 2 px 2m `1q, defined over F p 2 .

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.

With recent developments in hyperelliptic curve cryptographic system (HECC) processing and the proposed work, the myth that HECC is not suited for practical applications in constrained networks is dispelled. This article proposes dynamic authenticated contributory group key agreement (DACGKA) protocol using HEC‐DH. The DACGKA is less costly and ideally appropriate for resource‐restricted networks such as VANETs, MANETs, and WSNs because of its lesser key sizes and advancements in operational processing. The suggested protocol is implemented across a finite field using an HEC of genus 1 (ECC), genus 2, and genus 3, with promising results, including a significant reduction in key size and exchange overhead, an increase in key safety, and a broader range of applicability. According to a comparison of the proposed approach on different genus curves, HECC over genus 2 with a suitable key length can be applied for memory and power restricted devices to increase data secrecy and system efficiency. Further, HECC can outperform ECC for low‐cost, secure group communication applications. As a part of security analysis, first, we proved that each coordinate of HEC‐DH secret value is as hard as the entire DH value, and then concrete security proofs based on the HEC‐DLP were established.

In this article, we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality q with time complexity O(n 2+o(1) ) and space complexity O(n 2 ), where n = log(q). In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level 2 ν p where ν>0 is an integer and . The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

By using Pade approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are obtained. This estimate depends on all the coefficients of the linear form.