48 research outputs found
DLP–based cryptosystems with Pell cubics
No full textThe classical Pell equation x
2−dy
2 = 1 can be extended to the cubic case considering
the points (x, y, z) ∈ F
3
such that, for fixed r ∈ F,
x
3 + ry
3 + r
2
z
3 − 3rxyz = 1.
The set of solutions over a finite field Fq equipped with a generalized Brahmagupta product
is a cyclic group for some choices of q and r. In these cases, novel cryptosystems can be built
exploiting the discrete logarithm problem over this group. This paper focuses on the study of
ElGamal-based cryptosystems as well as digital signature schemes with the Pell cubic. Finally,
a comparison in terms of security, data-size and performance among these cryptosystems and
the classical versions with finite fields, elliptic curves and also with Pell conics is provided
Pell Equation - Theory and applications to cryptography
No full textThe Pell equation is a classical topic in number theory. There are well known methods for solving this equation, but there are still several important issues. One of the most interesting from the point of view of cryptographic applications is the study of its solutions over a generic field, in which case new interesting open problems arise. This work focuses on studying the theoretical and practical potential of the Pell equation in this context. Firstly, the required theoretical results from the state of the art are collected using a new unique and simple notation. This allows to obtain easily and elegantly new properties also for the generalization of the Pell equation in the cubic case. Then, all the theoretical results are adopted to formulate new public–key encryption and digital signature schemes with security based on the integer factorization problem or on the discrete logarithm problem, namely new RSA–like and ElGamal cryptosystems, and new Digital Signature Algorithms. The obtained cryptosystems are compared in terms of security, data–size and performance with the classical alternatives, and the results are very interesting especially in the case of the quadratic Pell equation. Finally, the properties of the Pell equation are exploited for defining new powerful probabilistic primality tests, related to the Lucas test included in the widely used Baillie–PSW test. In particular, the new primality tests are equipped with adaptations of the Selfridge method for choosing the parameters, resulting in very powerful tests
Developments on primality tests based on linear recurrent sequences of degree two
No full textSome probabilistic primality tests, like the strong Lucas test that is part of the
widely used Baillie-PSW test, are defined through linear recurrent sequences. When adopting
linear recurrent sequences of degree two, the simple version of the Lucas test as well as tests
based on the Pell hyperbola can be generalized obtaining new powerful primality tests. This
paper describes a deeper analysis of these two generalized tests in order to find the best
parameters by number of pseudoprimes, i.e., the instances of the tests with less composite
integers that are declared primes. The Selfridge method for choosing the parameters of the
Lucas test can be adapted to the generalized tests and, when adopting the parameters among
those with best statistical results, the resulting tests have no pseudoprimes up to
On the cubic Pell equation over finite fields
No full textThe classical Pell equation can be extended to the cubic case considering
the elements of norm one in Z[
√3 r], which satisfy
x
3 + ry
3 + r
2
z
3 − 3rxyz = 1.
The solution of the cubic Pell equation is harder than the classical case, indeed a
method for solving it as Diophantine equation is still missing [3]. In this paper, we
study the cubic Pell equation over finite fields, extending the results that hold for
the classical one. In particular, we provide a novel method for counting the number
of solutions in all possible cases depending on the value of r. Moreover, we are also
able to provide a method for generating all the solutions
Exploring Deep Learning for In-Field Fault Detection in Microprocessors
No full textNowadays, due to technology enhancement, faults are increasingly compromising all kinds of computing machines, from servers to embedded systems. Recent advances in ma- chine learning are opening new opportunities to achieve fault detection exploiting hardware metrics inspection, thus avoiding the use of heavy software techniques or product-specific errors reporting mechanisms. This paper investigates the capability of different deep learning models trained on data collected through simulation-based fault injection to generalize over different software applications
A symbiosis between cellular automata and genetic algorithms
No full textCellular automata are systems which use a rule to describe the evolution of a population in a discrete lattice, while genetic algorithms are procedures designed to find solutions to optimization problems inspired by the process of natural selection. In this paper, we introduce an original implementation of a cellular automaton whose rules use a fitness function to select for each cell the best mate to reproduce and a crossover operator to determine the resulting offspring. This new system, with a proper definition, can be both a cellular automaton and a genetic algorithm. We show that in our system the Conway’s Game of Life can be easily implemented and, consequently, it is capable of universal computing. Moreover two generalizations of the Game of Life are created and also implemented with it. Finally, we use our system for studying and implementing the prisoner’s dilemma and rock-paper-scissors games, showing very interesting behaviors and configurations (e.g., gliders) inside these games
Supporting CPS Modeling Through a New Method for Solving Complex Non-holomorphic Equations
No full textPrimality tests, linear recurrent sequences and the Pell equation
No full textWe study new primality tests based on linear recurrent sequences of degree two exploiting a matrix approach. The classical Lucas test arises as a particular case and we see
how it can be easily improved. Moreover, this approach shows clearly how the Lucas
pseudoprimes are connected to the Pell equation and the Brahamagupta product. We
also introduce two new specific primality tests, which we will call generalized Lucas
test and generalized Pell test. We perform some numerical computations on the new
primality tests and we do not find any pseudoprime up to 238. Moreover, we combined
the generalized Lucas test with the Fermat test up to 264 and we did not find any
composite number that passes the test. We get the same result using the generalized
Pell test
