Burer’s key assumption for semidefinite and doubly nonnegative relaxations

Burer has shown that completely positive relaxations of nonconvex quadratic programs with nonnegative and binary variables are exact when the binary variables satisfy a so-called key assumption. Here we show that introducing binary slack variables to obtain an equivalent problem that satisfies the key assumption will not improve the semidefinite relaxation. In contrast, such slack variables will improve the doubly nonnegative relaxation, but the same improvement can be obtained in a simpler fashion by adding certain linear inequality constraints.     

On key applications of the Turán–Kubilius inequality

Lithuanian Mathematical Journal - After briefly describing the origin and the scope of the Turán–Kubilius inequality, we show how this important inequality leads to the law of large...     

Polynomial-time key recovery attack on the Faure–Loidreau scheme based on Gabidulin codes

Encryption schemes based on the rank metric lead to small public key sizes of order of few thousands bytes which represents a very attractive feature compared to Hamming metric-based encryption schemes where public key sizes are of order of hundreds of thousands bytes even with additional structures like the cyclicity. The main tool for building public key encryption schemes in rank metric is the McEliece encryption setting used with the family of Gabidulin codes. Since the original scheme proposed in 1991 by Gabidulin, Paramonov and Tretjakov, many systems have been proposed based on different masking techniques for Gabidulin codes. Nevertheless, over the years most of these systems were attacked essentially by the use of an attack proposed by Overbeck. In 2005 Faure and Loidreau designed a rank-metric encryption scheme which was not in the McEliece setting. The scheme is very efficient, with small public keys of size a few kiloBytes and with security closely related to the linearized polynomial reconstruction problem which corresponds to the decoding problem of Gabidulin codes. The structure of the scheme differs considerably from the classical McEliece setting and until our work, the scheme had never been attacked. We show in this article that for a range of parameters, this scheme is also vulnerable to a polynomial-time attack that recovers the private key by applying Overbeck’s attack on an appropriate public code. As an example we break in a few seconds parameters with 80-bit security claim. Our work also shows that some parameters are not affected by our attack but at the cost of a lost of efficiency for the underlying schemes.     

Revisiting key schedule’s diffusion in relation with round function’s diffusion

We study the weakness of key schedules from an observation: many existing attacks use the fact that the key schedules poorly distribute key bits in the diffusion path of round functions. This reminds us of the importance of the diffusion’s relation between key schedules and round functions. We present new cryptanalysis results by exploring such diffusion relation and propose a new criterion for necessary key schedule diffusion. We discuss potential attacks and summarize the causes for key schedules without satisfying this criterion. One major cause is that overlapping between the diffusion of key schedules and round functions leads to information leakage of key bits. Finally, a measure to estimate our criterion for recursive key schedules is presented. Today designing key schedule still lacks practical and necessary principles. For a practical key schedule with limited diffusion, our work adds more insight to its requirements and helps to maximize the security level.     

An efficient and secure Diffie–Hellman key agreement protocol based on Chebyshev chaotic map

This paper proposes a new efficient and secure Diffie–Hellman key agreement protocol based on Chebyshev chaotic map. The proposed key agreement protocol uses the semi-group property of Chebyshev polynomials to agree Diffie–Hellman based session key. The proposed protocol provides strong security compared with the previous related protocols. In addition, the proposed protocol does not require any timestamp information and greatly reduces computational costs between communication parties. As a result, the proposed protocol is more practical and provides computational/communicational efficiency compare with several previously proposed key agreement protocols based on Chebyshev chaotic map.     

A model for developing students’ example space: the key role of the teacher

Our research addresses the role of examples to foster the students’ development of the mathematical concepts, and of their mathematical ways of thinking. We consider the notion of example space introduced by Watson and Mason (Mathematics as a constructive activity: learners generating examples, 2005), particularly when it is not formed by a simple juxtaposition of examples, rather it is endowed by a certain structure. Such a structure is provided by the semiotic actions and by the theoretic and logical dimensions of the mathematical activities. However, the formation of structured example spaces is far from being an automatic process. In this paper, we focus on the genesis of examples and on the role of the teacher in helping the students to structure their examples spaces through the so-called cognitive apprenticeship method. We point out that the genesis of examples is often accomplished within a complex cyclic dynamics, the “cycle of examples production and modification”. We illustrate it by means of two emblematic episodes from a classroom discussion. We show that the teacher’s intervention can be crucial in helping the students to modify a wrong example, to generate the right one for the task and to start the long-term process of building up the structure of their own space of examples.     

The analysis of the number of fixed points in the key extending algorithm of RC4

The probabilities of the state transitions of the initial value So in the S table of RC4 are described by a kind of bistochastic matrices, and then a computational formula for such bistochastic matrices is given, by which the mathematical expectation of the number of fixed points in the key extending algorithm of RC4 is obtained. As a result, a statistical weakness of the key extending algorithm of RC4 is presented.