Space curves, X-ranks and cuspidal projections

Let $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.

Application of Divisors on a Hyperelliptic Curve in Python

The paper studies hyperelliptic curves of the genus g > 1, divisors on them and their applications in Python programming language. The basic necessary definitions and known properties of hyperelliptic curves are demonstrated, as well as the notion of polynomial function, its representation in unique form, also the notion of rational function, norm, degree and conjugate to a polynomial are presented. These facts are needed to calculate the order of points of desirable functions, and thus to quickly and efficiently calculate divisors. The definition of a divisor on a hyperelliptic curve is shown, and the main known properties of a divisor are given. There are also an example of calculating a divisor of a polynomial function, reduced and semi-reduced divisors are described, theorem of the existence of such a not unique semi-reduced divisor, and theorem of the existence of a unique reduced divisor, which is equivalent to the initial one, are proved. In particular, a semi-reduced divisor can be represented as an GCD of divisors of two polynomial functions. It is also demonstrated that each reduced divisor can be represented in unique form by pair of polynomials [a(x), b(x)], which is called Mumford representation, and several examples of its representation calculation are given. There are shown Cantor’s algorithms for calculating the sum of two divisors: its compositional part, by means of which a not unique semi-reduced divisor is formed, and the reduction part, which gives us a unique reduced divisor. In particular, special case of the compositional part of Cantor’s algorithm, doubling of the divisor, is described: it significantly reduces algorithm time complexity. Also the correctness of the algorithms are proved, examples of applications are given. The main result of the work is the implementation of the divisor calculation of a polynomial function, its Mumford representation, and Cantor’s algorithm in Python programming language. Thus, the aim of the work is to demonstrate the possibility of e↵ective use of described algorithms for further work with divisors on the hyperelliptic curve, including the development of cryptosystem, digital signature based on hyperelliptic curves, attacks on such cryptosystems.

RFID Authentication Scheme based on Hyperelliptic Curve Signcryption

The implementation of efficient security mechanisms for Radio Frequency Identification (RFID) system has always been a continuous challenge due to its limited computing resources. Previously, hash-based, symmetric-key cryptography-based and elliptic curve cryptography based security protocols were proposed for RFID system. However, these protocols are not suitable because some of them failed to fulfill the RFID security requirements, and some of them produce high computational overhead. Recently researchers have focused on developing an efficient security mechanism based on hyperelliptic curve cryptography (HECC) which provides high security with 80 bits lower-key size. In this paper, we propose an efficient RFID authentication scheme based on hyperelliptic curve Signcryption. The proposed authentication scheme provides the required security features for the RFID system as well as security from potential attacks. We validated our proposed scheme’s security by utilizing a widely used simulation tool, Automated Validation of Internet Security Protocols and Applications (AVISPA). Furthermore, the results reveal that the computational, communication and storage overheads of the proposed scheme are much less than the other recently proposed schemes. Compared to the most recently published work based on ECC Signcryption, our scheme is 70% efficient in terms of computational overhead, 42.7% efficient in terms of communication overhead, and 50% efficient in terms of storage overhead. Therefore, the proposed scheme is more efficient as compared to the recently published work in this domain. Hence, it is an attractive solution for resource-limited devices like RFID tags.

RFID Authentication Scheme based on Hyperelliptic Curve Signcryption

The implementation of efficient security mechanisms for Radio Frequency Identification (RFID) system has always been a continuous challenge due to its limited computing resources. Previously, hash-based, symmetric-key cryptography-based and elliptic curve cryptography based security protocols were proposed for RFID system. However, these protocols are not suitable because some of them failed to fulfill the RFID security requirements, and some of them produce high computational overhead. Recently researchers have focused on developing an efficient security mechanism based on hyperelliptic curve cryptography (HECC) which provides high security with 80 bits lower-key size. In this paper, we propose an efficient RFID authentication scheme based on hyperelliptic curve Signcryption. The proposed authentication scheme provides the required security features for the RFID system as well as security from potential attacks. We validated our proposed scheme’s security by utilizing a widely used simulation tool, Automated Validation of Internet Security Protocols and Applications (AVISPA). Furthermore, the results reveal that the computational, communication and storage overheads of the proposed scheme are much less than the other recently proposed schemes. Compared to the most recently published work based on ECC Signcryption, our scheme is 70% efficient in terms of computational overhead, 42.7% efficient in terms of communication overhead, and 50% efficient in terms of storage overhead. Therefore, the proposed scheme is more efficient as compared to the recently published work in this domain. Hence, it is an attractive solution for resource-limited devices like RFID tags.