Elliptic Curve Systems Research Articles

Optimal Elliptic-Curve Subspaces for Applications in Double-Authenticated Requests in Mobile Distributed Data Mining

Mathematical models based on elliptic curves have been intensively studied since their applicability in data security systems was discovered. In this article, the authors describe the optimal way to select particular subspaces over which elliptic curves are defined, showing the applicability of these subspaces in secure data transfer. Access to large databases and analyses of the requests made to these databases are required daily by a variety of users, including legal entities. An attack on these communication systems causes violations in privacy and damage to/theft of data that can be worth EUR tens of billions annually. For requests made between computers, encryption methods can be used as these systems have adequate computing power and energy. For requests made from fixed and mobile systems, if the data are distributed heterogeneously, the computing power required to authenticate both the users and the answering entities determines the efficiency of the proposed solution. To address this limitation, our study proposes a double-authentication method based on particular elliptic-curve systems.

Security Improvement in Elliptic Curve Cryptography

This paper proposed different approaches to enhance the performance of the Elliptic Curve Cryptography (ECC) algorithm. ECC is vulnerable to attacks by exploiting the public parameters of ECC to solve Discrete Logarithm Problem (DLP). Therefore, these public parameters should be selected safely to obviate all recognized attacks. This paper presents a new generator function to produce the domain parameters for creating the elliptic curve; a secure mechanism is used in the proposed function to avoid all possible known attacks that attempts to solve the Elliptic Curve Discrete Logarithm Problem (ECDLP). Moreover, an efficient algorithm has been proposed for choosing two base points from the curve in order to generate two subgroups in a secure manner. The purpose of the aforementioned algorithm is to offer more confidence for the user since it is not built upon a hidden impairment that it could be subsequently utilized to retrieve user’s private key. The Elliptic Curve Diffie Hellman (ECDH) algorithm is implemented to exchange a session key between the communicating parties in a secure manner. Beside, a preprocessing operation is performed on the message to enhance the diffusion property and consequently leads to increase the strength against cryptanalysis attack. Finally, the dual encryption/decryption algorithm is implemented using different session keys in each stage of the encryption to boost immunity against any attack on the digital audio transmission. The gained results show the positive effect of the dual elliptic curve system in terms of speed and confidentiality without needing any extra time for encryption.

Toward Efficient Convertible Authenticated Encryption Schemes Using Self-Certified Public Key System

Convertible authenticated encryption (CAE) schemes enable the signer to send a confidential message and its corresponding signature to the designated recipient. The recipient can also convert the signature into a conventional one which can be verified by anyone. Integrating the properties of self-certified public key systems, this paper presents efficient and computationally indistinguishable self-certified CAE schemes for strengthening the security of E-Commerce applications. Additionally, we also adapt the proposed schemes to elliptic curve systems for facilitating the applications of limited computing power and insufficient storage space. The proposed schemes are secure against known existential active attacks, satisfy the semantic security requirement, and have the following advantages: (i) No extra certificate is required since the tasks of authenticating the public key and verifying the signature can be simultaneously carried out within one step, which helps reducing computation efforts and communication overheads. (ii) In case of a later dispute, the recipient can convert the signature into an ordinary one for the public arbitration. (iii) The signature conversion can be solely performed by the recipient without additional computation efforts or communication overheads. (iv) The recipient of the signature can prove himself, if needed, to anyone that he is actually the designated recipient.

A General Threshold Signature Scheme Based on Elliptic Curve

Based on Elliptic Curve cryptosystem, a threshold signature scheme characterized by (k,l) joint verification for (t,n) signature is put forward. After being signed by a signer company employing (t, n) threshold signature scheme, the informationmis transmitted to a particular verifier company, and then the signature is verified through the cooperation ofkones from the verifier company withlmembers, so as to realize a directional transmission between different companies. Finally, the application examples of the company encryption communication system, the generating polynomial of company private key and public key were given. The security of this scheme is based on Shamir threshold scheme and Elliptic Curve system, and due to the advantages of Elliptic Curve, the scheme enjoys wider application in practice.

Non-linear error detection for elliptic curve cryptosystems

The authors propose applying systematic non-linear error-detection codes to protect elliptic curve point addition and doubling operations against active fault attacks. These codes provide nearly perfect error-detection capability (except with exponentially small probability) at reasonable overhead. The proposed technique is applied to secure point addition and doubling operations for both Weierstrass and Edwards curves using different coordinate systems (i.e. affine and projective). The authors observe that the Weierstrass-based elliptic curve systems can be protected with reasonable area overhead. However, due to its balanced normal form, Edwards formulation is more appropriate for the non-linear error-detection technique proposed here. In addition, the proposed technique is compared with the method discussed by Gaubatz et al. (2006), where an error-detection technique is proposed for robust public key arithmetic. When compared with their method, the proposed technique provides approximately the same level of security with much less overhead. For Edwards curves, the overhead of the proposed scheme is less than half (42–46%) of the overhead of scheme proposed by Gaubatz et al. (2006). In addition, the overhead of the proposed scheme is 52–81% of the overhead of scheme proposed by Gaubatz et al. (2006) for different versions of the Weierstrass curves.

A new process and framework for direct anonymous attestation based on symmetric bilinear maps

For the problem of the original direct anonymous attestation (DAA) scheme’s complexity and great time consumption, a new DAA scheme based on symmetric bilinear pairings is presented, which gives a practical solution to ECC-based TPM in protecting the privacy of the TPM. The scheme still includes five procedures or algorithms: Setup, Join, Sign, Verify and Rogue tagging, but gets rid of zero-knowledge proof and takes on a new process and framework, of which the main operations are addition, scalar multiplication and bilinear maps on supersingular elliptic curve systems. Moreover, the scheme adequately utilizes the properties of bilinear maps as well as the signature and verification of the ecliptic curve system itself. Compared with other schemes, the new DAA scheme not only satisfies the same properties, and shows better simplicity and high efficiency. This paper gives not only a detailed security proof of the proposed scheme, but also a careful performance analysis by comparing with the existing DAA schemes.

An RNS Implementation of an $F_{p}$ Elliptic Curve Point Multiplier

Elliptic curve point multiplication is considered to be the most significant operation in all elliptic curve cryptography systems, as it forms the basis of the elliptic curve discrete logarithm problem. Designs for elliptic curve cryptography point multiplication are area demanding and time consuming. Thus, the efficient realization of point multiplication is of fundamental importance for the performance of an elliptic curve system. In this paper, a hardware architecture of an elliptic curve point multiplier is proposed that exploits the intrinsic parallelism of the residue number system (RNS), in order to speed up the elliptic curve point calculations and minimize the area complexity of the elliptic curve point multiplier. The architecture proves to be the fastest among all known design approaches, while complexity is less than half of that of previous efforts. This architecture also supports the required input (binary-to-RNS) and output (RNS-to-binary) conversions. Through a graph-oriented approach, the area of the elliptic curve point multiplier is minimized, by optimizing the point addition and doubling algorithms. Also, through this approach, the number of execution steps for point addition is matched to the number of execution steps for point doubling. Additionally, the impact of various RNS bases, in terms of number of moduli and their bit lengths, on the area and speed of the proposed implementation is analyzed, in an effort to define the potential for using RNS in elliptic curve cryptography.

Software multiplication using Gaussian normal bases

Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over binary fields F2m. In this paper, we present new software algorithms for efficient multiplication over F2m that use a Gaussian normal basis representation. Two approaches are presented, direct normal basis multiplication and a method that exploits a mapping to a ring where fast polynomial-based techniques can be employed. Our analysis, including experimental results on an Intel Pentium family processor, shows that the new algorithms are faster and can use memory more efficiently than previous methods. Despite significant improvements, we conclude that the penalty in multiplication is still sufficiently large to discourage the use of normal bases in software implementations of elliptic curve systems

Constructive and destructive facets of Weil descent on elliptic curves

In this paper we look in detail at the curves which arise in the method of Galbraith and Smart for producing curves in the Weil restriction of an elliptic curve over a finite field of characteristic 2 of composite degree. We explain how this method can be used to construct hyperelliptic cryptosystems which could be as secure as cryptosystems based on the original elliptic curve. On the other hand, we show that the same technique may provide a way of attacking the original elliptic curve cryptosystem using recent advances in the study of the discrete logarithm problem on hyperelliptic curves. We examine the resulting higher genus curves in some detail and propose an additional check on elliptic curve systems defined over fields of characteristic 2 so as to make them immune from the methods in this paper.

An implementation of elliptic curve cryptosystems over F/sub 2/155

The authors describe a VLSI Galois field processor and how it can be applied to the implementation of elliptic curve groups. They demonstrate the feasibility of constructing very fast, and very secure, public key systems with a relatively simple device, and the possibility of putting such a system on a smart card. The registers necessary to implement the elliptic curve system will require less than 1 mm/sup 2/ (or less than 4%) of the area available on the card.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>