Emparelhamentos hiperelípticos em criptografia

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.

Currently, SM9 algorithm has received more and more attention as a new cryptographic product. The SM9 algorithm encryption and decryption principle relies on the mapping relationship on the elliptic curve. Although this mapping relationship improves the security, it will slightly reduce the efficiency. The goal of this article is to improve the efficiency of the SM9 algorithm. Different from the traditional assembly line acceleration method, we decided to start with the basic operation of the algorithm itself. There is a bilinear pairing operation on the elliptic curve, which completes the point to point on the elliptic curve. The calculation complexity directly determines the SM9 algorithm. For this reason, we propose two new bilinear pair processing methods. The former uses the properties of isomorphic mapping to transfer the operations involved in the calculation of bilinear pairs from a large feature domain to a small feature domain, reducing the number of operations on the feature domain. The latter is for special operations in the bilinear pairing process, adding intermediate variables to convert them into low-time-consuming multiplication operations. According to the traditional Miller algorithm, the calculation of bilinear pairs requires 900 multiplication time units. Our solution can reduce this value to 700 and 800 multiplication time units respectively. In addition, the two algorithms have not changed the mapping relationship of the bilinear pair. On the premise of ensuring the correct mapping relationship, the efficiency of the SM9 algorithm is improved.

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.

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.

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 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.

The concept of policy-based cryptography is a promising paradigm for trust establishment and authorization in largescale open environments like the Internet and Mobile Networks. It aims at providing a framework for performing cryptographic operations with respect to policies formalized as monotone Boolean expressions written in standard normal forms. A policy involves conjunctions and disjunctions of conditions where each condition is fulfilled by a digital credential representing the signature of a specific credential issuer on a set of statements about a certain entity. Therefore, an entity fulfills a policy if any only if it has been issued a set of credentials fulfilling the combination of conditions specified by the policy.In this work, we focus on policy-based encryption schemes which allow to encrypt a message according to a policy so that only entities fulfilling the policy are able to decrypt the message. More generally, policy-based encryption belongs to an emerging family of encryption schemes sharing the ability to integrate encryption with access control structures. This ability is mainly enabled by bilinear pairings over elliptic curves and allows for several interesting applications in different contexts.A policy-based encryption scheme has to fulfill two primary requirements: on one hand, provable security under well defined attack models. On the other hand, efficiency, especially when dealing with the conjunctions and disjunctions of credential-based conditions.The contributions of our research work are twofold:1. The standard acceptable notion of security for public-key encryption schemes is indistinguishability against chosen ciphertext attacks. Hence, it is natural to require that a policy-based encryption scheme also satisfies this strong notion of security. However, the definition of this security notion must be adapted to the policy-based setting. Our first contribution is the definition of policy-oriented security model for policy-based encryption schemes as well as the development of an efficient policy-based encryption scheme that is provably secure under our security model in the random oracle model.2. Policy-based encryption schemes may suffer from the key-escrow property i.e. in addition to the legitimate holder of the credentials fulfilling the encryption policy, any collusion of credential issuers who are able to issue a set of credentials fulfilling the policy can decrypt the message. Our second contribution is to address this issue through the notion of policy-based public-key encryption. The latter allows encrypting a message not only with respect to a policy but also according to a public-key so that only an entity fulfilling the policy and having access to the corresponding private-key is able to decrypt the message. We developed a policy-based public-key encryption scheme from bilinear pairings and proved its security under the corresponding security model. Our proposal improves related work in terms of both security and efficiency.

A practical publicly verifiable secret sharing (PVSS) is constructed based on the bilinear pairing on elliptic curves, which has all advantages of B. Schoenmakerspsila PVSS and its secret is not the form of discrete logarithm, thus this PVSS is extremely practical. Moreover, in the schemepsilas distribution of shares phase, only using bilinearity of bilinear pairing, anybody can verify whether the participants received correct shares without implementing zero-knowledge proofs, without implementing the non-interactive protocol and without construction so called witness of shares applying Fiat-Shamirpsilas technique. Subsequently, in the schemepsilas reconstruction of secret phase, the released shares may be verified by anybody with the same method. Since the PVSS need not to implement non-interactive protocol to prevent malicious players. Therefore this scheme is simpler, more efficient and practical suitable for some especially case.

Secret sharing is a tool that we use in numerous cryptographic protocols. In this paper, we have proposed a multi-secret sharing scheme using elliptic curves and bilinear pairings. Accordingly, we have also proposed a compartmented multi-secret sharing scheme. The schemes are verifiable and efficient. The fundamental reason for utilizing bilinear pairing with elliptic curves is to provide similar security (as in existing schemes) with smaller key sizes. The computational complexities of the proposed schemes have been determined. We have included explicit examples of the schemes using SageMath.

Blind signature is an interesting cryptographic primitive which allows user to get signature on his document from signatory authority, without leaking any information. Blind signature is useful in many e-commerce applications where user’s anonymity is the main concern. Since the Zhang et al., was the first to propose the identity based blind signature, many schemes based on bilinear pairing have been proposed. But the computational cost of pairing operation on elliptic curve is around 20 times the point multiplication on an elliptic curve. In order to save the running time, we present a new Identity-Based Blind Signature (ID-BS) scheme whose security is based on elliptic curve discrete logarithm problem (ECDLP). Performance comparison shows that proposed scheme reduces the cost of computation. Security analysis shows that proposed scheme is secure against the adversary and achieves the property of blindness and Non-forgeabillity. At the end; we propose an e-cash payment system based on our ID-based blind signature scheme.

Attribute-based blind signature is used to realize the blind signature of messages by multiple people, and it is suitable for electronic payment services in the cloud. Most existing attribute-based blind signature schemes have complex bilinear pairing operations in the process of signature and verification, which leads to low algorithm efficiency. To solve this problem, in this study, an efficient pairing-free attribute-based blind signature scheme based on ordered binary decision diagram is proposed, and the security analysis of the new scheme is given under the difficulty of the elliptic curve discrete logarithm problem (ECDLP). The proposed scheme is based on elliptic curve cryptography (ECC). In this scheme, scalar multiplication on the elliptic curve, which is relatively lightweight, is used to replace bilinear pairing with a large computational cost in traditional attribute-based blind signature schemes. The computational overhead in the signature and verification processes is reduced, making the solution more suitable for resource-constrained cloud terminal user scenarios. On the other hand, in order to improve the computational efficiency of the attribute-based blind signature scheme, a more expressive and computationally efficient ordered binary decision diagram (OBDD) structure is used to construct the access structure of the signer. The OBDD access structure can simultaneously support the positive and negative values of attributes in the access structure at the same time. Compared with existing attribute-based blind signature schemes, the proposed scheme has better performance and storage, and meets the needs of resource-constrained cloud end users.

Chapter 91 - Advanced Data Encryption

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb{Q}(\zeta_n+\zeta^{-1}_n)$.