43 research outputs found

    Geometry of the del Pezzo surface y^2=x^3+Am^6+Bn^6

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    In this paper, we give an effective and efficient algorithm which on input takes non-zero integers AA and BB and on output produces the generators of the Mordell-Weil group of the elliptic curve over Q(t)\mathbb{Q}(t) given by an equation of the form y2=x3+At6+By^2=x^3+At^6+B. Our method uses the correspondence between the 240 lines of a del Pezzo surface of degree 1 and the sections of minimal Shioda height on the corresponding elliptic surface over Q‾\overline{\mathbb{Q}}. For most rational elliptic surfaces, the density of the rational points is proven by various authors, but the results are partial in case when the surface has a minimal model that is a del Pezzo surface of degree 1. In particular, the ones given by the Weierstrass equation y2=x3+At6+By^2=x^3+At^6+B, are among the few for which the question is unsolved, because the root number of the fibres can be constant. Our result proves the density of the rational points in many of these cases where it was previously unknown.Comment: 28 pages, 7 figure

    A database of genus 2 curves over the rational numbers

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    We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. This data has been incorporated into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe

    Dynamic S-BOX using Chaotic Map for VPN Data Security

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    A dynamic SBox using a chaotic map is a cryptography technique that changes the SBox during encryption based on iterations of a chaotic map, adding an extra layer of confusion and security to symmetric encryption algorithms like AES. The chaotic map introduces unpredictability, non-linearity, and key dependency, enhancing the overall security of the encryption process. The existing work on dynamic SBox using chaotic maps lacks standardized guidelines and extensive security analysis, leaving potential vulnerabilities and performance concerns unaddressed. Key management and the sensitivity of chaotic maps to initial conditions are challenges that need careful consideration. The main objective of using a dynamic SBox with a chaotic map in cryptography systems is to enhance the security and robustness of symmetric encryption algorithms. The method of dynamic SBox using a chaotic map involves initializing the SBox, selecting a chaotic map, iterating the map to generate chaotic values, and updating the SBox based on these values during the encryption process to enhance security and resist cryptanalytic attacks. This article proposes a novel chaotic map that can be utilized to create a fresh, lively SBox. The performance assessment of the suggested S resilience Box against various attacks involves metrics such as nonlinearity (NL), strict avalanche criterion (SAC), bit independence criterion (BIC), linear approximation probability (LP), and differential approximation probability (DP). These metrics help gauge the Box ability to handle and respond to different attack scenarios. Assess the cryptography strength of the proposed S-Box for usage in practical security applications, it is compared to other recently developed SBoxes. The comparative research shows that the suggested SBox has the potential to be an important advancement in the field of data security.Comment: 11 Page

    Computational tools for quadratic Chabauty

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    http://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfhttp://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfFirst author draf

    The Algorithmic Solution of Simultaneous Diophantine Equations

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    A new method is presented for solving pairs of simultaneous Diophantine equations, such as those which result from the 2-descent process on elliptic curves. The method works by determining a set of solutions modulo a prime P, raising each of these solutions to a set of P solutions modulo p2, and then determining a solution modulo p6 for each of the solutions modulo p2. These solutions modulo p6 lie on a lattice which is then reduced using a suitable lattice reduction algorithm. The required solution can then be written as a linear combination of the basis vectors for the lattice, and the coefficients in this combination are determined. The running time of this algorithm is O(N213) where N is a bound on the size of the solution required. Variations on the method are also presented. Following a 2-descent on elliptic curves of the form y2 = X3 +pX, where p =- 5 (mod 8) originally described by Bremner and Cassels [8], the methods are applied to various pairs of equations. Generators for the free abelian part of the group of rational points on each of these curves are presented, including the case p= 16421 which has a canonical height of 137.2290. By combining the method with existing techniques, we also find a generator for the set of points of infinite order on the curve y2 = X3 + 17477X. This point has canonical height h(P) = 406.4797. We also find a generator on the Mordell curve y2 = X3 + 7823, which is the only case missing from the tables of Gebel, Peth6 and Zimmer for the curves y2=x3+k with IkI :ý! 10000 [20]

    Chaos and Cellular Automata-Based Substitution Box and Its Application in Cryptography

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    Substitution boxes are the key factor in symmetric-key cryptosystems that determines their ability to resist various cryptanalytic attacks. Creating strong substitution boxes that have multiple strong cryptographic properties at the same time is a challenging task for cryptographers. A significant amount of research has been conducted on S-boxes in the past few decades, but the resulting S-boxes have been found to be vulnerable to various cyberattacks. This paper proposes a new method for creating robust S-boxes that exhibit superior performance and possess high scores in multiple cryptographic properties. The hybrid S-box method presented in this paper is based on Chua’s circuit chaotic map, two-dimensional cellular automata, and an algebraic permutation group structure. The proposed 16×16 S-box has an excellent performance in terms of security parameters, including a minimum nonlinearity of 102, the absence of fixed points, the satisfaction of bit independence and strict avalanche criteria, a low differential uniformity of 5, a low linear approximation probability of 0.0603, and an auto-correlation function of 28. The analysis of the performance comparison indicates that the proposed S-box outperforms other state-of-the-art S-box techniques in several aspects. It possesses better attributes, such as a higher degree of inherent security and resilience, which make it more secure and less vulnerable to potential attacks

    A novel symmetric image cryptosystem resistant to noise perturbation based on S8 elliptic curve S-boxes and chaotic maps

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    The recent decade has seen a tremendous escalation of multimedia and its applications. These modern applications demand diverse security requirements and innovative security platforms. In this manuscript, we proposed an algorithm for image encryption applications. The core structure of this algorithm relies on confusion and diffusion operations. The confusion is mainly done through the application of the elliptic curve and S8 symmetric group. The proposed work incorporates three distinct chaotic maps. A detailed investigation is presented to analyze the behavior of chaos for secure communication. The chaotic sequences are then accordingly applied to the proposed algorithm. The modular approach followed in the design framework and integration of chaotic maps into the system makes the algorithm viable for a variety of image encryption applications. The resiliency of the algorithm can further be enhanced by increasing the number of rounds and S-boxes deployed. The statistical findings and simulation results imply that the algorithm is resistant to various attacks. Moreover, the algorithm satisfies all major performance and quality metrics. The encryption scheme can also resist channel noise as well as noise-induced by a malicious user. The decryption is successfully done for noisy data with minor distortions. The overall results determine that the proposed algorithm contains good cryptographic properties and low computational complexity makes it viable to low profile applications
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