scholarly journals Constructing Picard curves with complex multiplication using the Chinese remainder theorem

2019 ◽  
Vol 2 (1) ◽  
pp. 21-36
Author(s):  
Sonny Arora ◽  
Kirsten Eisenträger
Keyword(s):  
Chinese Remainder Theorem ◽  
Complex Multiplication ◽  
Picard Curves

scholarly journals An inverse Jacobian algorithm for Picard curves

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Joan-C. Lario ◽  
Anna Somoza ◽  
Christelle Vincent
Keyword(s):  
Class Number ◽  
Complex Multiplication ◽  
Hyperelliptic Curves ◽  
Period Matrix ◽  
Isomorphism Classes ◽  
Jacobian Problem ◽  
Picard Curves ◽  
The Given ◽  
Small Period ◽  
Jacobian Algorithm

AbstractWe study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

scholarly journals Primes Dividing Invariants of CM Picard Curves

2019 ◽  
Vol 72 (2) ◽  
pp. 480-504 ◽  
Author(s):  
Pınar Kılıçer ◽  
Elisa Lorenzo García ◽  
Marco Streng
Keyword(s):  
Complex Multiplication ◽  
Good Reduction ◽  
Explicit Constructions ◽  
Picard Curves ◽  
Genus 2 ◽  
Reduction Of Curves

AbstractWe give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

scholarly journals Vertical Distribution Relations for Special Cycles on Unitary Shimura Varieties

2018 ◽  
Vol 2020 (13) ◽  
pp. 3902-3926
Author(s):  
Réda Boumasmoud ◽  
Ernest Hunter Brooks ◽  
Dimitar P Jetchev
Keyword(s):  
Vertical Distribution ◽  
Three Dimensional ◽  
Complex Multiplication ◽  
Horizontal Distribution ◽  
Unitary Groups ◽  
Shimura Varieties ◽  
Heegner Points ◽  
Universal Norm

Abstract We consider cycles on three-dimensional Shimura varieties attached to unitary groups, defined over extensions of a complex multiplication (CM) field $E$, which appear in the context of the conjectures of Gan et al. [6]. We establish a vertical distribution relation for these cycles over an anticyclotomic extension of $E$, complementing the horizontal distribution relation of [8], and use this to define a family of norm-compatible cycles over these fields, thus obtaining a universal norm construction similar to the Heegner $\Lambda $-module constructed from Heegner points.

scholarly journals Secure Video Surveillance Framework in Smart City

Sensors ◽  
2021 ◽  
Vol 21 (13) ◽  
pp. 4419
Author(s):  
Hao Li ◽  
Tianhao Xiezhang ◽  
Cheng Yang ◽  
Lianbing Deng ◽  
Peng Yi
Keyword(s):  
Video Surveillance ◽  
Smart City ◽  
Key Management ◽  
Surveillance System ◽  
Chinese Remainder Theorem ◽  
Smart Cities ◽  
Group Key Management ◽  
Video Surveillance System ◽  
Surveillance Systems ◽  
Key Update

In the construction process of smart cities, more and more video surveillance systems have been deployed for traffic, office buildings, shopping malls, and families. Thus, the security of video surveillance systems has attracted more attention. At present, many researchers focus on how to select the region of interest (RoI) accurately and then realize privacy protection in videos by selective encryption. However, relatively few researchers focus on building a security framework by analyzing the security of a video surveillance system from the system and data life cycle. By analyzing the surveillance video protection and the attack surface of a video surveillance system in a smart city, we constructed a secure surveillance framework in this manuscript. In the secure framework, a secure video surveillance model is proposed, and a secure authentication protocol that can resist man-in-the-middle attacks (MITM) and replay attacks is implemented. For the management of the video encryption key, we introduced the Chinese remainder theorem (CRT) on the basis of group key management to provide an efficient and secure key update. In addition, we built a decryption suite based on transparent encryption to ensure the security of the decryption environment. The security analysis proved that our system can guarantee the forward and backward security of the key update. In the experiment environment, the average decryption speed of our system can reach 91.47 Mb/s, which can meet the real-time requirement of practical applications.

scholarly journals Computationally Efficient Approach to Implementation of the Chinese Remainder Theorem Algorithm in Minimally Redundant Residue Number System

2021 ◽  
Author(s):  
Mikhail Selianinau
Keyword(s):  
Chinese Remainder Theorem ◽  
Number System ◽  
Residue Number System ◽  
Computer Applications ◽  
Efficient Approach ◽  
Residue Modulo ◽  
Modern Computer ◽  
New Variant ◽  
Residue Number ◽  
Residue Code

AbstractIn this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. Throughout the article, an efficient approach for implementing the CRT algorithm is described. The structure of the rank of an RNS number, a principal positional characteristic of the residue code, is investigated. It is shown that the rank of a number can be represented by a sum of an inexact rank and a two-valued correction to it. We propose a new variant of minimally redundant RNS, which provides low computational complexity for the rank calculation, and its effectiveness analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the excess residue modulo 2, the complexity of the rank calculation goes down from $O\left (k^{2}\right )$ O k 2 to $O\left (k\right )$ O k with respect to required modular addition operations and lookup tables, where k equals the number of non-redundant RNS moduli.

scholarly journals Multiparty weighted threshold quantum secret sharing based on the Chinese remainder theorem to share quantum information

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yao-Hsin Chou ◽  
Guo-Jyun Zeng ◽  
Xing-Yu Chen ◽  
Shu-Yu Kuo
Keyword(s):  
Quantum Information ◽  
Phase Shift ◽  
Quantum State ◽  
Secret Sharing ◽  
Chinese Remainder Theorem ◽  
Security Analysis ◽  
Quantum Secret Sharing ◽  
Security Protocol ◽  
Multiple Quantum ◽  
Cryptographic Primitive

AbstractSecret sharing is a widely-used security protocol and cryptographic primitive in which all people cooperate to restore encrypted information. The characteristics of a quantum field guarantee the security of information; therefore, many researchers are interested in quantum cryptography and quantum secret sharing (QSS) is an important research topic. However, most traditional QSS methods are complex and difficult to implement. In addition, most traditional QSS schemes share classical information, not quantum information which makes them inefficient to transfer and share information. In a weighted threshold QSS method, each participant has each own weight, but assigning weights usually costs multiple quantum states. Quantum state consumption will therefore increase with the weight. It is inefficient and difficult, and therefore not able to successfully build a suitable agreement. The proposed method is the first attempt to build multiparty weighted threshold QSS method using single quantum particles combine with the Chinese remainder theorem (CRT) and phase shift operation. The proposed scheme allows each participant has its own weight and the dealer can encode a quantum state with the phase shift operation. The dividing and recovery characteristics of CRT offer a simple approach to distribute partial keys. The reversibility of phase shift operation can encode and decode the secret. The proposed weighted threshold QSS scheme presents the security analysis of external attacks and internal attacks. Furthermore, the efficiency analysis shows that our method is more efficient, flexible, and simpler to implement than traditional methods.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

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