5,215 research outputs found
Security of almost ALL discrete log bits
Let G be a finite cyclic group with generator \alpha and with an encoding so that multiplication is computable in polynomial time. We study the security of bits of the discrete log x when given \exp_{\alpha}(x), assuming that the exponentiation function \exp_{\alpha}(x) = \alpha^x is one-way. We reduce he general problem to the case that G has odd order q. If G has odd order q the security of the least-significant bits of x and of the most significant bits of the rational number \frac{x}{q} \in [0,1) follows from the work of Peralta [P85] and Long and Wigderson [LW88]. We generalize these bits and study the security of consecutive shift bits lsb(2^{-i}x mod q) for i=k+1,...,k+j. When we restrict \exp_{\alpha} to arguments x such that some sequence of j consecutive shift bits of x is constant (i.e., not depending on x) we call it a 2^{-j}-fraction of \exp_{\alpha}. For groups of odd group order q we show that every two 2^{-j}-fractions of \exp_{\alpha} are equally one-way by a polynomial time transformation: Either they are all one-way or none of them. Our key theorem shows that arbitrary j consecutive shift bits of x are simultaneously secure when given \exp_{\alpha}(x) iff the 2^{-j}-fractions of \exp_{\alpha} are one-way. In particular this applies to the j least-significant bits of x and to the j most-significant bits of \frac{x}{q} \in [0,1). For one-way \exp_{\alpha} the individual bits of x are secure when given \exp_{\alpha}(x) by the method of Hastad, N\"aslund [HN98]. For groups of even order 2^{s}q we show that the j least-significant bits of \lfloor x/2^s\rfloor, as well as the j most-significant bits of \frac{x}{q} \in [0,1), are simultaneously secure iff the 2^{-j}-fractions of \exp_{\alpha'} are one-way for \alpha' := \alpha^{2^s}. We use and extend the models of generic algorithms of Nechaev (1994) and Shoup (1997). We determine the generic complexity of inverting fractions of \exp_{\alpha} for the case that \alpha has prime order q. As a consequence, arbitrary segments of (1-\varepsilon)\lg q consecutive shift bits of random x are for constant \varepsilon >0 simultaneously secure against generic attacks. Every generic algorithm using generic steps (group operations) for distinguishing bit strings of j consecutive shift bits of x from random bit strings has at most advantage O((\lg q) j\sqrt{t} (2^j/q)^{\frac14})
On the Design of Cryptographic Primitives
The main objective of this work is twofold. On the one hand, it gives a brief
overview of the area of two-party cryptographic protocols. On the other hand,
it proposes new schemes and guidelines for improving the practice of robust
protocol design. In order to achieve such a double goal, a tour through the
descriptions of the two main cryptographic primitives is carried out. Within
this survey, some of the most representative algorithms based on the Theory of
Finite Fields are provided and new general schemes and specific algorithms
based on Graph Theory are proposed
Insecurity of Quantum Secure Computations
It had been widely claimed that quantum mechanics can protect private
information during public decision in for example the so-called two-party
secure computation. If this were the case, quantum smart-cards could prevent
fake teller machines from learning the PIN (Personal Identification Number)
from the customers' input. Although such optimism has been challenged by the
recent surprising discovery of the insecurity of the so-called quantum bit
commitment, the security of quantum two-party computation itself remains
unaddressed. Here I answer this question directly by showing that all
``one-sided'' two-party computations (which allow only one of the two parties
to learn the result) are necessarily insecure. As corollaries to my results,
quantum one-way oblivious password identification and the so-called quantum
one-out-of-two oblivious transfer are impossible. I also construct a class of
functions that cannot be computed securely in any ``two-sided'' two-party
computation. Nevertheless, quantum cryptography remains useful in key
distribution and can still provide partial security in ``quantum money''
proposed by Wiesner.Comment: The discussion on the insecurity of even non-ideal protocols has been
greatly extended. Other technical points are also clarified. Version accepted
for publication in Phys. Rev.
A fast Bayesian approach to discrete object detection in astronomical datasets - PowellSnakes I
A new fast Bayesian approach is introduced for the detection of discrete
objects immersed in a diffuse background. This new method, called PowellSnakes,
speeds up traditional Bayesian techniques by: i) replacing the standard form of
the likelihood for the parameters characterizing the discrete objects by an
alternative exact form that is much quicker to evaluate; ii) using a
simultaneous multiple minimization code based on Powell's direction set
algorithm to locate rapidly the local maxima in the posterior; and iii)
deciding whether each located posterior peak corresponds to a real object by
performing a Bayesian model selection using an approximate evidence value based
on a local Gaussian approximation to the peak. The construction of this
Gaussian approximation also provides the covariance matrix of the uncertainties
in the derived parameter values for the object in question. This new approach
provides a speed up in performance by a factor of `hundreds' as compared to
existing Bayesian source extraction methods that use MCMC to explore the
parameter space, such as that presented by Hobson & McLachlan. We illustrate
the capabilities of the method by applying to some simplified toy models.
Furthermore PowellSnakes has the advantage of consistently defining the
threshold for acceptance/rejection based on priors which cannot be said of the
frequentist methods. We present here the first implementation of this technique
(Version-I). Further improvements to this implementation are currently under
investigation and will be published shortly. The application of the method to
realistic simulated Planck observations will be presented in a forthcoming
publication.Comment: 30 pages, 15 figures, revised version with minor changes, accepted
for publication in MNRA
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