1,358 research outputs found
Normal Elliptic Bases and Torus-Based Cryptography
We consider representations of algebraic tori over finite fields.
We make use of normal elliptic bases to show that, for infinitely many
squarefree integers and infinitely many values of , we can encode
torus elements, to a small fixed overhead and to -tuples of
elements, in quasi-linear time in .
This improves upon previously known algorithms, which all have a
quasi-quadratic complexity. As a result, the cost of the encoding phase is now
negligible in Diffie-Hellman cryptographic schemes
Encryption, Elliptic Curves, and the Symmetries of Differential Equations
In cryptography, encryption is the process of encoding messages in such a way that only authorized parties can access them. The intended information, referred to as plaintext, is encrypted using an encryption algorithm, generating ciphertext that can only be read if decrypted. Public key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner. In a public key encryption system, any person can encrypt a message using the public key, but such a message can be decrypted only with the private key. Elliptic curve cryptography (ECC) is a particularly powerful approach to public-key cryptography based on tori or more precisely elliptic curves.
The purpose of this talk is to discuss the mathematics employed in elliptic curve encryption which is based on the algebraic structure of elliptic curves, in particular on the ability to add points. Such group structure on a torus is evident if we represent it as a fundamental domain in the complex plane with its edges identified. Once the group structure has been defined in the complex plane, the group structure on a torus is evident. In turn, an elliptic curve is parameterized over the complex plane by the Weierstrass elliptic function. Moreover, the Weierstrass elliptic function allows to identify the defining quantities of a torus with those of an elliptic curve using modular forms
On Modular Inverses of Cyclotomic Polynomials and the Magnitude of their Coefficients
Let p and r be two primes and n, m be two distinct divisors of pr. Consider
the n-th and m-th cyclotomic polynomials. In this paper, we present lower and
upper bounds for the coefficients of the inverse of one of them modulo the
other one. We mention an application to torus-based cryptography.Comment: 21 page
Discrete Logarithms in Generalized Jacobians
D\'ech\`ene has proposed generalized Jacobians as a source of groups for
public-key cryptosystems based on the hardness of the Discrete Logarithm
Problem (DLP). Her specific proposal gives rise to a group isomorphic to the
semidirect product of an elliptic curve and a multiplicative group of a finite
field. We explain why her proposal has no advantages over simply taking the
direct product of groups. We then argue that generalized Jacobians offer poorer
security and efficiency than standard Jacobians
Parameterizable Byzantine Broadcast in Loosely Connected Networks
We consider the problem of reliably broadcasting information in a multihop
asynchronous network, despite the presence of Byzantine failures: some nodes
are malicious and behave arbitrarly. We focus on non-cryptographic solutions.
Most existing approaches give conditions for perfect reliable broadcast (all
correct nodes deliver the good information), but require a highly connected
network. A probabilistic approach was recently proposed for loosely connected
networks: the Byzantine failures are randomly distributed, and the correct
nodes deliver the good information with high probability. A first solution
require the nodes to initially know their position on the network, which may be
difficult or impossible in self-organizing or dynamic networks. A second
solution relaxed this hypothesis but has much weaker Byzantine tolerance
guarantees. In this paper, we propose a parameterizable broadcast protocol that
does not require nodes to have any knowledge about the network. We give a
deterministic technique to compute a set of nodes that always deliver authentic
information, for a given set of Byzantine failures. Then, we use this technique
to experimentally evaluate our protocol, and show that it significantely
outperforms previous solutions with the same hypotheses. Important disclaimer:
these results have NOT yet been published in an international conference or
journal. This is just a technical report presenting intermediary and incomplete
results. A generalized version of these results may be under submission
Chaotic dynamical systems associated with tilings of
In this chapter, we consider a class of discrete dynamical systems defined on
the homogeneous space associated with a regular tiling of , whose most
familiar example is provided by the dimensional torus \T ^N. It is proved
that any dynamical system in this class is chaotic in the sense of Devaney, and
that it admits at least one positive Lyapunov exponent. Next, a
chaos-synchronization mechanism is introduced and used for masking information
in a communication setup
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