Forward (Δ) and Backward (∇) Difference Operators Basic Sets of Polynomials in and Their Effectiveness in Reinhardt and Hyperelliptic Domains

We generate, from a given basic set of polynomials in several complex variables , new basic sets of polynomials and generated by the application of the Δ and &#8711 operators to the set . All relevant properties relating to the effectiveness in Reinhardt and hyperelliptic domains of these new sets are properly deduced. The case of classical orthogonal polynomials is investigated in details and the results are given in a table. Notations are also provided at the end of a table.

Rational Points on Genus 3 Real Hyperelliptic Curves

We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C (). We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

Effective generalized equations of secure hyperelliptic curve digital signature algorithms

A hyperelliptic curve digital signature algorithm （HECDSA） can be viewed as the hyperelliptic curve analogue of the standard digital signature algorithm （DSA）. This article discusses divisor evaluations, the basic HECDSA, variants, two HECDSA equations and a 4-tuple HECDSA scheme, and puts forward a generalized equation for HECDSA. From this generalized equation, seven general HECDSA types are derived based on the efficiency requirements. Meanwhile, the securities of these general HECDSA types are analyzed in detail.

Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2

In this paper we study the computation of the number of isomorphism classes of hyperelliptic curves of genus 2 over finite fields Fq with q even. We show the formula of the number of isomorphism classes, that is, for q = 2m, if 4 (|) m, then the formula is 2q3 + q2 - q;if 4 | m, then the formula is 2q3 + q2 - q + 8. These results can be used in the classification problems and the hyperelliptic curve cryptosystems.

The Main Conjecture on Geometric MDS Codes From Hyperelliptic Curves

In Ref.[2] the author dealt with the Main conjecture on geometric codes andproved the correctness of the conjecture for codes arising from curves with genus 1 or2 when the cardinal of the ground field is large enough. In this note, the conjecturefor codes from hyperelliptic curves is attacked.

Counting Isomorphism Classes of Pointed Hyperelliptic Curves of Genus 4 over Finite Fields with Even Characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.

An Efficient Proxy Blind Signcryption Scheme for IoT

Recent years have witnessed growing scientific research interest in the Internet of Things(IoT)technologies,which supports the development of a variety of applications such as health care,Industry 4.0,agriculture,ecological data management,and other various domains.IoT utilizes the Internet as a prime medium of communication for both single documents as well as multi-digital messages.However,due to the wide-open nature of the Internet,it is important to ensure the anonymity,untraceably,confidentiality,and unforgeability of communication with efficient computational complexity and low bandwidth.We designed a light weight and secure proxy blind signcryption for multi-digital messages based on a hyperelliptic curve(HEC).Our results outperform the available schemes in terms of computational cost and communication bandwidth.The designed scheme also has the desired authentication,unforgeability of warrants and/or plaintext,confidentiality,integrity,and blindness,respectively.Further,our scheme is more suitable for devices with low computation power such as mobiles and tablets.

HAMILTONIANS WITH TWO DEGREES OF FREEDOM ADMITTING A SINGLEVALUED GENERAL SOLUTION

Following the basic principles stated by Painlevé, we first revisit the process of selecting the admissible time-independent Hamiltonians H = （p1^2 ＋ p2^2）/2 ＋ V（q1, q2） whose some integer power qj^nj （t） of the general solution is a singlevalued function of the complez time t. In addition to the well known rational potentials V of Hénon-Heiles, this selects possible cases with a trigonometric dependence of V on qj. Then, by establishing the relevant confluences, we restrict the question of the explicit integration of the seven （three “cubic” plus four “quartic”） rational Hénon-Heiles cases to the quartic cases. Finally, we perform the explicit integration of the quartic cases, thus proving that the seven rational cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.

Hyper Elliptic Curve Based Certificateless Signcryption Scheme for Secure IIoT Communications

Industrial internet of things (IIoT) is the usage of internet of things(IoT) devices and applications for the purpose of sensing, processing andcommunicating real-time events in the industrial system to reduce the unnecessary operational cost and enhance manufacturing and other industrial-relatedprocesses to attain more profits. However, such IoT based smart industriesneed internet connectivity and interoperability which makes them susceptibleto numerous cyber-attacks due to the scarcity of computational resourcesof IoT devices and communication over insecure wireless channels. Therefore, this necessitates the design of an efficient security mechanism for IIoTenvironment. In this paper, we propose a hyperelliptic curve cryptography(HECC) based IIoT Certificateless Signcryption (IIoT-CS) scheme, with theaim of improving security while lowering computational and communicationoverhead in IIoT environment. HECC with 80-bit smaller key and parameterssizes offers similar security as elliptic curve cryptography (ECC) with 160-bitlong key and parameters sizes. We assessed the IIoT-CS scheme security byapplying formal and informal security evaluation techniques. We used Realor Random (RoR) model and the widely used automated validation of internet security protocols and applications (AVISPA) simulation tool for formalsecurity analysis and proved that the IIoT-CS scheme provides resistance tovarious attacks. Our proposed IIoT-CS scheme is relatively less expensivecompared to the current state-of-the-art in terms of computational cost andcommunication overhead. Furthermore, the IIoT-CS scheme is 31.25% and 51.31% more efficient in computational cost and communication overhead,respectively, compared to the most recent protocol.

One-way permutation on a family of surfaces

In the’recent decade, one-way functions have very. important applications in cryptograpy, especially in public key systems and generators of pseudorandom number. In fact, modem cryptograph is developed based on the assumption of the existence of oneway function. A one-way function which is also a permutation is called a one-way permutation.