Hyperelliptic Curves Of Genus Research Articles

Models and integral differentials of hyperelliptic curves

Abstract Let $C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$ , defined over a complete discretely valued field $K$ , with ring of integers $O_K$ . Under certain conditions on $C$ , mild when residue characteristic is not $2$ , we explicitly construct the minimal regular model with normal crossings $\mathcal{C}/O_K$ of $C$ . In the same setting we determine a basis of integral differentials of $C$ , that is an $O_K$ -basis for the global sections of the relative dualising sheaf $\omega _{\mathcal{C}/O_K}$ .

Integrable maps in 4D and modified Volterra lattices

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus $g$ algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted ${\cal V}_g$. The original motivation behind our work was the fact that, in the particular case $g=2$, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the $g=2$ Volterra map ${\cal V}_2$. We dedicate this work to a dear friend and colleague, Decio Levi.

Efficient Search for Superspecial Hyperelliptic Curves of Genus Four with Automorphism Group Containing C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{C}}_6$$\\end{document}

In arithmetic and algebraic geometry, superspecial (s.sp. for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If g≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g \\ge 4$$\\end{document}, it is not even known whether there exists such a curve of genus g in general characteristic p>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p > 0$$\\end{document}, and in the case of g=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=4$$\\end{document}, several computational approaches to search for those curves have been proposed. In the genus-4 hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp. curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group as a subgroup. In this paper, we propose an algorithm with complexity O~(p4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{O}}(p^4)$$\\end{document} in theory but O~(p3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ilde{O}}(p^3)$$\\end{document} in practice to enumerate s.sp. hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order 6. By executing the algorithm over Magma, we enumerate those curves for p up to 1000. We also succeeded in finding a s.sp. hyperelliptic curve of genus 4 in every p with p≡2(mod3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\equiv 2 \\pmod {3}$$\\end{document}.

ON THE NUMBER OF POINTS OF GIVEN ORDER ON ODD-DEGREE HYPERELLIPTIC CURVES

For integers N≥2 and g≥1, we study bounds on the cardinality of the set of points of order dividing N lying on a hyperelliptic curve of genus g embedded in its jacobian using a Weierstrass point as base point. This leads us to revisit division polynomials introduced by Cantor in 1995 and strengthen a divisibility result proved by him. Several examples are discussed.

Équation de Pell–Abel et applications

In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r>g. This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive k-differential on an hyperelliptic curve of genus g with a unique zero of order k(2g-2) for every (k,g)≠(2,2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n>2g.

Hyperelliptic Covers of Different Degree for Elliptic Curves

In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyperelliptic curves to those on elliptic curves. We can also use the cover map as a kind of cover attacks. Given an elliptic curve over an extension field of degree n , one might try to use the cover attack to reduce the discrete logarithm problem (DLP) in the group of rational points of the elliptic curve to DLPs in the Jacobian of a curve of genus g ≥ n over the base field. An algorithm has been proposed for finding genus 3 hyperelliptic covers as a cover attack for elliptic curves with cofactor 2. Our algorithms are about the cover map from hyperelliptic curves of genus 2 to elliptic curves of prime order. As an application, an example of an elliptic curve whose order is a 256-bit prime vulnerable to our algorithms is given.

Torsion points of small order on hyperelliptic curves

Let $$\mathscr {C}$$ be a hyperelliptic curve of genus $$g>1$$ over an algebraically closed field K of characteristic zero and $$\mathscr {O}$$ one of the $$(2g{+}2)$$ Weierstrass points in $$\mathscr {C}(K)$$ . Let J be the Jacobian of $$\mathscr {C}$$ , which is a g-dimensional abelian variety over K. Let us consider the canonical embedding of $$\mathscr {C}$$ into J that sends $$\mathscr {O}$$ to the zero of the group law on J. This embedding allows us to identify $$\mathscr {C}(K)$$ with a certain subset of the commutative group J(K). A special case of the famous theorem of Raynaud (Manin–Mumford conjecture) asserts that the set of torsion points in $$\mathscr {C}(K)$$ is finite. It is well known that the points of order 2 in $$\mathscr {C}(K)$$ are exactly the “remaining” $$(2g{+}1)$$ Weierstrass points. One of the authors (Zarhin in Izv Math 83:501–520, 2019) proved that there are no torsion points of order n in $$\mathscr {C}(K)$$ if $$3\leqslant n\leqslant 2g$$ . So, it is natural to study torsion points of order $$2g+1$$ (notice that the number of such points in $$\mathscr {C}(K)$$ is always even). Recently, the authors proved that there are infinitely many (for a given g) mutually non-isomorphic pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least four points of order $$2g+1$$ . In the present paper we prove that (for a given g) there are at most finitely many (up to an isomorphism) pairs $$(\mathscr {C},\mathscr {O})$$ such that $$\mathscr {C}(K)$$ contains at least six points of order $$2g+1$$ .

Linear complexity of some sequences derived from hyperelliptic curves of genus 2

For a given hyperelliptic curve C over a finite field with Jacobian JC, we consider the hyperelliptic analogue of the congruential generator defined by Wn = Wn− 1 + D for n ≥ 1 and D, W0 ∈ JC. We show that curves of genus 2 produce sequences with large linear complexity.

Tropical sequences associated with Somos sequences

Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-k sequence for k = 4, 5, 6, 7 was proved. In the works of Hone, representations for Somos-4 and 5 sequences were found via the Weierstrass sigma function on elliptic curves, and for k = 6 via the Klein sigma function on hyperelliptic curve of genus It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For k = 4, 5, 6, 7, the Somos sequences are Laurent polynomials of k initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables. Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free  terms are written explicitly

Pairs of rational triangles with equal symmetric invariants

The semiperimeter s, inradius r and circumradius R are the symmetric invariants of a triangle. In this short note, we complement previous work of Hirakawa and Matsumura by determining all pairs (up to similitude) consisting of a rational right angled triangle and a rational isosceles triangle having two corresponding symmetric invariants equal. In the proof of our main theorems, we determine the set of rational points on two hyperelliptic curves of genus 3. The first of these can be settled via a classical descent argument, whereas the second one is proved using ‘‘elliptic curve Chabauty’’.

THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

AbstractLet C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

An intriguing hyperelliptic Shimura curve quotient of genus 16

Let $F$ be the maximal totally real subfield of $\mathbf{Q}(\zeta_{32})$, the cyclotomic field of $32$nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptional. Second, there are $34$ Weierstrass points on $C$, and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension $E/F$ of class number $17$ contained in $\mathbf{Q}(\zeta_{64})$, the cyclotomic field of $64$th roots of unity. Third, the normal closure of the field of $2$-torsion of the Jacobian of $C$ is the Harbater field $N$, the unique Galois number field $N/\mathbf{Q}$ unramified outside $2$ and $\infty$, with Galois group $\mathrm{Gal}(N/\mathbf{Q})\simeq F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. In fact, the Jacobian $\mathrm{Jac}(X_0^D(1))$ has the remarkable property that each of its simple factors has a $2$-torsion field whose normal closure is the field $N$. Finally, and perhaps the most striking fact about $C$, is that it is also hyperelliptic over $\mathbf{Q}$.

Continued Fractions and Hankel Determinants from Hyperelliptic Curves

AbstractFollowing van der Poorten, we consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of J‐fractions and orthogonal polynomials, we show that in the simplest case g = 1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos‐4 sequence, which were found in a particular form by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus 2 satisfy a Somos‐8 relation. Moreover, for all g we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system. © 2020 the Authors. Communications on Pure and Applied Mathematics is published by Wiley Periodicals LLC.

Периодические элементы $\sqrt f$ в эллиптических полях с полем констант нулевой характеристики

Исследование проблемы периодичности функциональных непрерывных дробей элементов эллиптических и гиперэллиптических полей было начато около 200 лет назад в классических работах Н.~Абеля и П.~Л.~Чебышева. В 2014 году В.~П.~Платоновым был предложен общий концептуальный метод, базирующийся на глубокой связи трех классических проблем: проблема существования и построения фундаментальных $S$-единиц в гиперэллиптических полях, проблема кручения в якобианах гиперэллиптических кривых и проблема периодичности непрерывных дробей элементов в гиперэллиптических полях. В 2015-2019 годах в работах В. П. Платонова с соавторами был достигнут большой прогресс в исследовании проблемы периодичности элементов в гиперэллиптических полях, в особенности в эффективной классификации таких периодических элементов. Так, например, в указанных работах В.~П.~Платонова с соавторами были найдены все эллиптические поля $\mathbb{Q}(x)(\sqrt{f})$ такие, что $\sqrt{f}$ разлагается в периодическую непрерывную дробь в $\mathbb{Q}((x))$, а также были получены дальнейшие продвижения в обобщении указанного результата, как на другие числовые поля констант, так и на гиперэллиптические кривые рода $2$ и выше. В настоящей статье мы приводим полное доказательство анонсированного нами в 2019 году результата о конечности числа эллиптических полей $k(x)(\sqrt{f})$ над произвольным числовым полем $k$ с периодическим разложением $\sqrt{f}$, для которых соответствующая эллиптическая кривая содержит $k$-точку четного порядка не превосходящего $18$ или $k$-точку нечетного порядка не превосходящего $11$. Для произвольного поля $k$ являющегося квадратичным расширением $\mathbb Q$ найдены все такие эллиптические поля, а для поля $k=\mathbb Q$ было получено новое доказательство конечности числа периодических $\sqrt{f}$, не использующее параметризацию эллиптических кривых и точек конечного порядка на них.

Provable secure lightweight multiple shared key agreement based on hyper elliptic curve Diffie–Hellman for wireless sensor networks

ABSTRACTIn this paper, we propose a new protocol in which session keys can be generated when public keys are exchanged in a one run of the aforesaid protocol based on the difficulty of the hyper elliptic curve discrete logarithm problem. In this work, we are implementing the proposed protocol using an hyper elliptic curve of genus 2 over a finite field and the results are very encouraging. The main advantages of the proposed protocol are a considerable decrease in keys exchange overhead, a further increase in the safety of the keys and broadening of applicability. A relative study among the proposed and existing protocols is made and acceptable results have been obtained. The proposed protocol is immune to known-key attack, replay attack, key compromise impersonation, forgery attack and key control. Further, an upper limit for the quantity of shared keys is obtained in terms of the number of keys exchanged and, in fact, we find a formula for the minimal required number of keys to be exchanged for a given number shared keys. Further we can easily provide session key authentication for various wireless users/sessions on proposed protocol by integrating lightweight three-factor authentication scheme in the back ground.

Quantum algorithm for solving hyperelliptic curve discrete logarithm problem

The discrete logarithm problem (DLP) plays an important role in modern cryptography since it cannot be efficiently solved on a classical computer. Currently, the DLP based on the hyperelliptic curve of genus 2 (HCDLP) is widely used in industry and also a research field of hot interest. At the same time, quantum computing, a new paradigm for computing based on quantum mechanics, provides the ability to solve certain hard problems that cannot be efficiently solved on classical computers. In this paper, we consider the problem of solving the HCDLP in the paradigm of quantum computing. We propose a quantum algorithm for solving the HCDLP by applying the framework of quantum algorithm designed by Shor. The key of the algorithm is the realization of divisor addition. We solve the key problem and get analytical results for divisor addition by geometric meaning of the group addition. Therefore, the procedure can be efficiently realized on a quantum computer using the basic modular arithmetic operations. Finally, we conclude that the HCDLP defined over an n-bit prime field $$\mathbb {F}_{p}$$ can be computed on a quantum computer with at most $$13n+2\lfloor \log _{2}n\rfloor +10$$ qubits using $$2624n^{3} \log _{2}n-2209.2n^{3} + 1792n^{2} \log _{2}n-3012.8n^{2}$$ Toffoli gates. For current parameters at comparable classical security levels, there are fewer qubits and Toffoli gates to solve the HCDLP than the ones to solve the DLP based on elliptic curves.

Statistics for products of traces of high powers of the Frobenius class of hyperelliptic curves in even characteristic

We study the averages of products of traces of high powers of the Frobenius class of real hyperelliptic curves of genus [Formula: see text] over a fixed finite field [Formula: see text] in both odd and even characteristic cases.

Ultraelliptic Integrals and Two-Dimensional Sigma Functions

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u) = 0}, where σ(u) is the two-dimensional sigma function. We show that G(z) = F(ξ(z)), where z is a local coordinate in a neighborhood of a point of the smooth curve W and ξ(z) is the smooth function in this neighborhood given by the equation σ(u1, ξ(u1)) = 0. We obtain differential equations for the functions F(z), G(z), and ξ(z), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function G(z) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.

Division by on odd-degree hyperelliptic curves and their Jacobians

Let be an algebraically closed field of characteristic different from , a positive integer, a polynomial of degree with coefficients in and without multiple roots, the corresponding hyperelliptic curve of genus over , and its Jacobian. We identify with the image of its canonical embedding in (the infinite point of goes to the identity element of ). It is well known that for every there are exactly elements such that . Stoll constructed an algorithm that provides the Mumford representations of all such in terms of the Mumford representation of . The aim of this paper is to give explicit formulae for the Mumford representations of all such in terms of the coordinates , where is given by a point . We also prove that if , then does not contain torsion points of orders between and .

Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3

Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3. In Section 8 of our previous paper [Funct. Anal. Appl. 51 (2017), 162-176], there are miscalculations. In appendix of this paper, we correct the errors.