The cubic version of the Lucas cryptosystem is set up based on the cubic recurrence relation of the Lucas function by Said and Loxton [‘A cubic analogue of the RSA cryptosystem’, Bull. Aust. Math. Soc.68 (2003), 21–38]. To implement this type of cryptosystem in a limited environment, it is necessary to accelerate encryption and decryption procedures. Therefore, this paper concentrates on improving the computation time of encryption and decryption in cubic Lucas cryptosystems. The new algorithm is designed based on new properties of the cubic Lucas function and mathematical techniques. To illustrate the efficiency of our algorithm, an analysis was carried out with different size parameters and the performance of the proposed and previously existing algorithms was evaluated with experimental data and mathematical analysis.

A Message Authentication Code (MAC) is a function that takes a message and a key asparameters and outputs an authentication of the message. MAC are used to guarantee thelegitimacy of messages exchanged through a network, since generating a correctauthentication requires the knowledge of the key defined secretly by trusted parties.However, an attacker with access to a sufficiently large number of message/authenticationpairs may use a brute force algorithm to infer the secret key: from a set containinginitially all possible key candidates, subsequently remove those that yield an incorrectauthentication, proceeding this way for each intercepted message/authentication pair untila single key remains. In this paper, we determine an exact formula for the expected numberof message/authentication pairs that must be used before such form of attack issuccessful, along with an asymptotical bound that is both simple and tight. We conclude byillustrating a modern application where this bound comes in handy, namely the estimationof security levels in reflection-based verification of software integrity.

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences with given joint 2-adic complexity.

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau $ expansion of integers in the number fields $\mathbb{Q}\left( \sqrt{-3} \right)$ and $\mathbb{Q}\left( \sqrt{-7} \right)$. The (window) nonadjacent form of $\tau $ -expansion of integers in $\mathbb{Q}\left( \sqrt{-7} \right)$ was first investigated by Solinas. For integers in $\mathbb{Q}\left( \sqrt{-3} \right)$, the nonadjacent form and the window nonadjacent form of the $\tau $ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau $ expansions for integers in all Euclidean imaginary quadratic number fields.

In secret sharing, different access structures have different difficulty degrees for acceding to the secret. We give a numerical measure of how easy or how difficult is to recover the secret, depending only on the structure itself and not on the particular scheme used for realizing it. We derive some consequences.

The following article is based on the plenary address by Luiz Davidovich (Federal University of Rio de Janeiro), presented on April 14, 2004, at the 2004 MRS Spring Meeting in San Francisco. The field of quantum information is a discipline that aims to investigate methods for characterizing, transmitting, storing, compressing, and computationally utilizing the information carried by quantum states. It owes its rapid development over the last few years to several factors: the ability, developed in several laboratories, to control and measure simple microscopic systems; the discovery of fast quantum algorithms; and the recognition that Moore's law will soon lead to the single-atom limit of elementary computing gates.Cryptography and quantum computing are among the main applications in the field.They rely on the subtle and fundamental properties of the quantum world: the unavoidable disturbance associated with measurement, the superposition principle, and the nonlocal properties of entangled states. Progress in this area is intimately connected to a deep understanding of quantum physics: recent achievements include the experimental demonstration of teleportation and detailed investigations of the role of the environment in the quantum–classical transition. This article reviews basic concepts and recent developments in the field of quantum information, emphasizing the close ties between fundamental research and possible applications.