Robert Brouzet

A Homomorphic Method for Sharing Secret Images

In this paper, we present a new method for sharing images between two parties exploiting homomorphic property of public key cryptosystem. With our method, we show that it is possible to multiply two encrypted images, to decrypt the resulted image and after to extract and reconstruct one of the two original images if the second original image is available. Indeed, extraction and reconstruction of original image at the receiving end is done with the help of carrier image. Experimental results and security analysis show the effectiveness of the proposed scheme.

On What Is the Almost-Near Principle

Abstract We introduce a class of functions satisfying what we call the Almost-Near property, or briefly the (AN) property. The motivation of this investigation is provided by the phenomenon “almost implies near” which appears in many various situations. We study some elementary properties of these functions and develop several examples (such as polynomial functions, complex exponential and covering maps).

Analysis of homomorphic properties of RSA-based cryptosystem for image sharing

Secret sharing between two or more parties, exploiting multiplicative homomorphic properties of RSA, may result in false data blocks during the extraction of message. This paper investigates factors that lead to such false blocks and suggest a controlled solution by analyzing their probability distribution. In this paper we prove that false blocks do exist and show that larger sizes of the prime product reduce the error probability of false blocks in extracted messages.

How to secretly share the treasure map of the captain

In this paper we present a new approach for sharing a secret image between l users exploiting additive homomorphic property of Paillier algorithm. With a traditional approach, when a dealer wants to share an image between l players, the secret image must be sequentially encrypted l + 1 times using l + 1 keys (secret or public keys). When the dealer and the l players want to extract the secret image, they must decrypt sequentially, keeping the same order of the encryption step, by using l + 1 keys (secret or private). With the proposed approach, during the encryption step, each player encrypts his own secret image using the same public key given by the dealer, the dealer encrypts the secret image to be shared with the same key and then the l secret encrypted images plus the encrypted image to be shared are multiplied between them to get a scrambled image. After this step, the dealer can securely use the private key to decrypt this scrambled image to get a new scrambled image which corresponds to the addition of the l + 1 original images because of the additive homomorphic property of Paillier algorithm. When the l players want to extract the secret image, they do not need the dealer and to use keys. Indeed, with our approach, to extract the secret image, the l players need only to subtract their own secret image from the scrambled image. In this paper we illustrate our approach with an example of a captain who wants to share a secret treasure map between l pirates. Experimental results and security analysis show the effectiveness of the proposed scheme.

To Be (a Circle) or Not to Be

Summary We begin with a high school exercise, determining whether a particular equation describes an arc of a circle. We give several proofs that it does not, exploring many properties of circles along the way. Yet the curve is the arc of a conic.We then explore which equations of a generalized family are arcs of conics and give some properties of the resulting Lamé curves.

Comparison of three solutions to correct erroneous blocks to extract an image of a multiplicative homomorphic cryptosystem

Multiplicative homomorphic properties of a cryptosystem can be used in various applications requiring security, protection and authentication e.g. digital fingerprinting, electronic voting, on line betting etc. Secret sharing between two or more parties exploiting multiplicative homomorphic property of RSA results into erroneous blocks while extracting the message. The generation of these erroneous blocks limits the capabilities of homomorphic properties of RSA to be used in its full extend. This paper provides three different approaches as solutions to the problem of erroneous blocks in image. These solutions are: mean value approach, shortest distance approach and image preprocessing approach. It has been observed that shortest distance approach results into good PSNR but it is computationally expensive. The best approach with high PSNR is image preprocessing approach before sharing process, which results into no erroneous blocks in the extracted image, thus no extra extraction techniques are required.

Analysis of the jump and approximate gradient for solutions of Mumford-Shah type energy functionals

The main objective of this paper is to introduce and illustrate a new tool stemming from Young measure theory in order to capture concentrations, jump sign and gradient oscillations of sequences of SBV-functions. We show how this notion of measure can be applied for the analysis of approximating solutions of Mumford-Shah type energy functionals in the one dimensional case.

Variational approximation of a constraint signal by a Mumford-Shah type energy functional

In this paper, the main objective is to establish an existence result of a variational model in image segmentation constrained by a given vector field. In the one dimensional case, we give a discrete version converging in a variational way to the continuous model. We finally describe the numerical analysis of this model with application in image segmentation.

Exemples de pseudo-métriques compatibles avec une forme symplectique; cas particulier des structures pseudo-kählériennes

Resume Nous donnons une methode explicite pour construire des varietes symplectiques munies d’une pseudo-metrique compatible. Elles apparaissent comme espace total d’une classe speciale de fibrations etudiee en [1]. Parmi elles, quelques exemples de varietes pseudo-kahleriennes compactes a courbure non nulle sont exhibes.