Indices of hyperelliptic curves over p -adic fields

Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding hyperelliptic curve and $J$ its jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(J)$ for the $Q_{\ell}$-Tate module of $J$ and $e_{\ell}$ for the Riemann form on $V_{\ell}(J)$ attached to the theta divisor. (Here $Q_{\ell}$ is the field of $\ell$-adic numbers.) We write $sp(V_{\ell}(J))$ for the $Q_{\ell}$-Lie algebra of the symplectic group of $e_{\ell}$. We write $g_{\ell}$ for the Lie algebra of the image of the Galois group $Gal(K)$ of $K$ in $Aut(V_{\ell}(J))$. We prove that $g_{\ell}$ coincides with the direct sum $Q_{\ell}I \oplus sp(V_{\ell}(J))$ where $I$ is the identity operator.

The Diffie-Hellman problem as a cryptographic primitive plays an important role in modern cryptology. The Bit Security or Hard-Core Bits of Diffie-Hellman problem in arbitrary finite cyclic group is a long-standing open problem in cryptography. Until now, only few groups have been studied. Hyperelliptic curve cryptography is an alternative to elliptic curve cryptography. Due to the recent cryptanalytic results that the best known algorithms to attack hyperelliptic curve cryptosystems of genus \(g<3\) are the generic methods and the recent implementation results that hyperelliptic curve cryptography in genus 2 has the potential to be competitive with its elliptic curve cryptography counterpart. In this paper, we generalize Boneh and Shparlinksi’s method and result about elliptic curve to the case of Jacobians of hyperelliptic curves. We prove that the least significant bit of each coordinate of hyperelliptic curves Diffie-Hellman secret value in genus 2 is hard as the entire Diffie-Hellman value, and then we also show that any bit is hard as the entire Diffie-Hellman value. Finally, we extend our techniques and results to hyperelliptic curves of any genus.

Elliptic curves are some specific type of curves known as hyper elliptic curves.Compared to the integer factorization problem(IFP) based systems, using elliptic curve based cryptography will significantly decrease key size of the encryption.Therefore, application of this type of cryptography in systems that need high security and smaller key size has found great attention.Hyperelliptic curves help to make key length shorter.Many investigations are done with regard to improving computations, hardware and software implementation of these curves, their security and resistance against attacks.This paper studies and analyzes researches done about security and efficiency of hyperelliptic curves.

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

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map j^{dr}_n on elliptic and hyperelliptic curves over number fields via their universal unipotent connections {mathscr {U}}. Several algorithms forming part of the computation of finite level versions j^{dr}_n of the unipotent Albanese maps are presented. The computation of the logarithmic extension of {mathscr {U}} in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on {mathscr {U}} over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.

It is widely recognized that data security will play a central role in future IT systems. Providing public-key cryptographic primitives, which are the core tools for security, is often difficult on embedded processor due to computational, memory, and power constraints. This contribution appears to be the first thorough comparison of two public-key families, namely elliptic curve (ECC) and hyperelliptic curve cryptosystems on a wide range of embedded processor types (ARM, ColdFire, PowerPC). We investigated the influence of the processor type, resources, and architecture regarding throughput. Further, we improved previously known HECC algorithms resulting in a more efficient arithmetic.

In this paper we introduce scalar multiplication algorithms for several classes of elliptic and hyperelliptic curves. The methods are variations on Yao's scalar multiplication algorithm where independent group operations are shown in an explicit way. We can thusmerge several group operations and reduce the number of field operations by means of Montgomery's trick. The results are that scalar multiplication on elliptic curves in even characteristic based on point halving can be improved by at least 10% and the performance of Koblitz curves by 25% to 32%.

A first characterization of the isomorphism classes of \(k\)-involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of \(\operatorname{SL}(n, k)\) with \(k\) any field of characteristic not equal to \(2\). We classify the isomorphism classes of involutions for \(k\) algebraically closed, the real numbers, the \(\mathfrak{p}\)-adic numbers and finite fields. We also determine in which cases the corresponding fixed point group \(H\) is \(k\)-anisotropic. In those cases the corresponding symmetric \(k\)-variety consists of semisimple elements.

Irreducible polynomials with integral coefficients have succinct certificates

We have indicated in our tract [9] that several interesting problems in the theory of numbers are related to results about the evenness of the distribution of the roots v of a polynomial congruencewhere f(x) = a0xn + … + an is an irreducible polynomial having integral coefficients and degree n≧2. We alluded, for example, to our work on the Chebyshev problem of the greatest prime factor of n2 – D [8], in which an essential component was our earlier demonstration [6] of the uniform distribution, modulo 1, of v/k when f(x) = x2 – D. But, having pointed out that the quantitative descriptions of such uniformity had to be very sharp for substantial applications, we then noted with regret that little more than mere uniform distribution was obtained in our generalization [7] of [6] to congruences of higher degree. Indeed, it has only been for certain cubic polynomials that results have been produced that are comparable in power with those for quadratic polynomials, and even these depend on the assumption of the unproved hypothesis R* regarding the size of incomplete Kloosterman sums [10].

In this paper we present an Elliptic Curve Point Multiplication processor over base fields GF(2m), suitable for use in a wide range of commercial cryptography applications. Our design operates in a polynomial basis is fully parameterizable in the irreducible polynomial and the chosen Elliptic Curve over any base Galois Field up to a given size. High performance is achieved by use of a dedicated Galois Field arithmetic coprocessor implemented on FPGA. The underlying FPGA architecture is used to increase calculation performance, taking advantage of the properties of this kind of programmable logic device to perform the large number of logical operations required. We discuss the performance of our processor for different Elliptic Curves and compare the results with recent implementations in terms of speed and security.

Let K be a number field, and let C be a hyperelliptic curve over K with Jacobian J. Suppose that C is defined by an equation of the form for some irreducible monic polynomial of discriminant Δ and some element . Our first main result says that if there is a prime of K dividing but not (2Δ), then the image of the natural 2‐adic Galois representation is open in and contains a certain congruence subgroup of depending on the maximal power of dividing . We also present and prove a variant of this result that applies when C is defined by an equation of the form for distinct elements . We then show that the hypothesis in the former statement holds for almost all and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.

With ever growing popularity, wireless communication system also vulnerable to various security attacks. To provide high level security, many cryptographic solutions have been proposed. One such solution is signcryption, where authenticity and confidentiality provided by single logical step. Therefore, signcryption scheme helps to reduce computational cost, but it is not feasible for resource constraint environments. Because, most of the existing approaches were based on El-Gamal, bilinear pairing, Rivest-Shamir-Adleman (RSA), and Elliptic curve Cryptography (ECC). They consume more energy due to their increased key size. Hence, the new signcryption approach is proposed based on Hyper Elliptic Curve Cryptosystem (HECC) whose key size is much lesser than ECC. It significantly reduces the cost of computation and communication overhead by half the amount of ECC which suits well for resource constraint environments. Further, the proposed scheme attains necessary security features along with forward secrecy and public verifiability. On the other hand, the security of the approach is validated through an automated protocol validation tool – AVISPA.

Let p be an odd prime number, K a p-adic field of degree r over Qp, O the ring of integers in K, B = {β1,…,βr} an integral basis of K over Qp, u a unit in O and consider sets of the form 𝒩 = {n1β1 + … + nrβr : 1 ≤ nj ≤ Nj, 1 ≤ j ≤ r}. We show under certain growth conditions that the pair correlation of {uz2 : z ∊ 𝒩} becomes Poissonian.

This thesis details the concepts behind several bilinear pairings, its implementations and importante. Unlike more tradicional presentations, we focus on the more general case, using the Picard Group defined over hyperelliptic curves, not focusing on particular cases, such as elliptic curves, genus 2 curves, or particular classes of curves.