Algebraic Curves

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh

Cyclic étale coverings of generic curves and ordinariness of dormant opers

We introduce a number of new results in the context of multi-view geometry from general algebraic curves. We start with the derivation of the extended Kruppa's equations which are responsible for describing the epipolar constraint of two projections of a general (non-planar) algebraic curve. As part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus. We then establish new results on the reconstruction of general algebraic curves from multiple views. We address three different representations of curves: (i) the regular point representation for which we show that the reconstruction from two views of a curve of degree d admits two solutions, one of degree d and the other of degree d(d-1), (ii) the dual space representation (tangents) for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus, and (iii) a new representation (to computer vision) based on the set of lines meeting the curve which does not require any curve fitting in image space, for which we also derive lower bounds for the number of views necessary for reconstruction as a function of the curve degree alone.

In this paper it is shown that cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves, and more generally to "separating solutions" of such systems, on general algebraic curves. For example, for separating solutions the analog of Bezout's theorem is valid. The proofs are straightforward. They are based on the version of Rolle's theorem and Bezout's theorem for planar algebraic curves offered below. We note that according to Hilbert's conjecture the total number of limit cycles of a dynamical system with a polynomial vector field is bounded above by the degree of the field. From the validity of the conjecture and the results of the present note it follows that the curve consisting of all the limit cycles of a polynomial dynamical system resembles an algebraic curve. At the present time, even the finiteness of the number of limit cycles has not been proved: Ii'yashenko proved that Dyudaktsproof [i] contains an unfillable gap (cf. [2]) . I. Rolle's Theorem for Dynamical Systems We consider a smooth dynamical system on the plane

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.

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.

This paper discusses parallelization of elliptic curve cryptography hardware accelerators using elliptic curves over binary fields F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sub> . Elliptic curve point multiplication, which is the operation used in every elliptic curve cryptosystem, is hierarchical in nature, and parallelism can be utilized in different hierarchy levels as shown in many publications. However, a comprehensive analysis on the effects of parallelization has not been previously presented. This paper provides tools for evaluating the use of parallelism and shows where it should be used in order to maximize efficiency. Special attention is given for a family of curves called Koblitz curves because they offer very efficient point multiplication. A new method where the latency of point multiplication is reduced with parallel field arithmetic processors is introduced. It is shown to outperform the previously presented multiple field multiplier techniques in the cases of Koblitz curves and generic curves with fixed base points. A highly efficient general elliptic curve cryptography processor architecture is presented and analyzed. Based on this architecture and analysis on the effects of parallelization, a few designs are implemented on an Altera Stratix II field-programmable gate array (FPGA).

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

Introduction. Let k be an algebraically closed ground field of characteristic zero. The following result is well-known (see Severi [1], Anhang F, p. 317). If En d is a complete (i.e., maximal) irreducible algebraic system, defined over k, of plane algebraic (not necessarily irreducible) curves of a given order n, such that the general curve C* of En d /k has d nodes (and no other singularities), then the dimension of En d is equal to 3n + p 1, where p = ((n 1)(n 2)/2) d is the genus of C*. We note that 3n + p 1 is equal to (n (n + 3)/2) d. Here N = (n(n + 3)/2) is the dimension of the (linear) system of all plane curves of order n. This is in agreement with the intuitive expectation that the requirement that a curve of order n possess d nodes (in non-assigned position) imposes d independent algebraic (non-linear) conditions on the curve. The above result is easily proved and will be included by us in a more general theorem, the proof of which is the main object of this paper (Theorem 2, Section 3) and which deals with irreducible algebraic systems E of curves of order n in which the general curve C* of E/k has arbitrary singularities (but is free from multiple components; equivalently: C* is a reduced curve). This theorem asserts that if p is the genus of C* (the definition of the effective genus of a reducible-but reduced-curve is given in Section 3, formula (2)), then dim E c 3n + p 1, with equality if and only if C* has only nodes as singularities. It is easily seen (see Severi, loc. cit.) that if 0 c d ' (n(n 1)/2) then there always exists curves of order n having d nodes (the maximum

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p2) of a particular type of supersingular elliptic curve is at least as hard as solving the Diffie–Hellman problem in the XTR subgroup. This provides strong evidence for a negative answer to the question posed by Vanstone and Menezes at the Crypto 2000 Rump Session on the possibility of efficiently inverting the MOV embedding into the XTR subgroup. As a side result we show that the Decision Diffie–Hellman problem in the group of points on this type of supersingular elliptic curves is efficiently computable, which provides an example of a group where the Decision Diffie–Hellman problem is simple, while the Diffie–Hellman and discrete logarithm problems are presumably not. So-called distortion maps on groups of points on elliptic curves that play an important role in our cryptanalysis also lead to cryptographic applications of independent interest. These applications are an improvement of Joux’s one round protocol for tripartite Diffie–Hellman key exchange and a non-refutable digital signature scheme that supports escrowable encryption. We also discuss the applicability of our methods to general elliptic curves defined over finite fields which includes a classification of elliptic curve groups where distortion maps exist.

This paper discusses a solution to the problem of how to approximate an algebraic curve of any degree with piecewise parametric curves. The approximation can be performed using any differentiable parametric curve form. An implementation using rational cubic Bezier curves results in a G 2 piecewise approximation which experiences better than O(h 7) convergence. If the algebraic curve happens to be a cubic curve of genus zero, a parametric curve can be found which exactly fits the given algebraic curve to within floating point precision. An error bound is developed which is much tighter than previously published bounds.

Efficiently computable homomorphisms allow elliptic curve point multiplication to be accelerated using the Gallant–Lambert–Vanstone (GLV) method. Iijima, Matsuo, Chao and Tsujii gave such homomorphisms for a large class of elliptic curves by working over ${\mathbb{F}}_{p^{2}}$. We extend their results and demonstrate that they can be applied to the GLV method. In general we expect our method to require about 0.75 the time of previous best methods (except for subfield curves, for which Frobenius expansions can be used). We give detailed implementation results which show that the method runs in between 0.70 and 0.83 the time of the previous best methods for elliptic curve point multiplication on general curves.

The best algorithms for discrete logarithms in Jacobians of algebraic curves of small genus are based on index calculus methods coupled with large prime variations. For hyperelliptic curves, relations are obtained by looking for reduced divisors with smooth Mumford representation (Gaudry); for non-hyperelliptic curves it is faster to obtain relations using special linear systems of divisors (Diem, Kochinke). Recently, Sarkar and Singh have proposed a sieving technique, inspired by an earlier work of Joux and Vitse, to speed up the relation search in the hyperelliptic case. We give a new description of this technique, and show that this new formulation applies naturally to the non-hyperelliptic case with or without large prime variations. In particular, we obtain a speed-up by a factor approximately 3 for the relation search in Diem and Kochinke’s methods.