6,874 research outputs found
On lattice profile of the elliptic curve linear congruential generators
Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG
Finite Fields: Theory and Applications
Finite fields are the focal point of many interesting geometric, algorithmic and combinatorial problems. The workshop was devoted to progress on these questions, with an eye also on the important applications of finite field techniques in cryptography, error correcting codes, and random number generation
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
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