60,853 research outputs found
Number and length of attractors in a critical Kauffman model with connectivity one
The Kauffman model describes a system of randomly connected nodes with
dynamics based on Boolean update functions. Though it is a simple model, it
exhibits very complex behavior for "critical" parameter values at the boundary
between a frozen and a disordered phase, and is therefore used for studies of
real network problems. We prove here that the mean number and mean length of
attractors in critical random Boolean networks with connectivity one both
increase faster than any power law with network size. We derive these results
by generating the networks through a growth process and by calculating lower
bounds.Comment: 4 pages, no figure, no table; published in PR
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
Boolean feedback functions for full-length nonlinear shift registers, Journal of Telecommunications and Information Technology, 2004, nr 4
In the paper a heuristic algorithm for a random generation of feedback functions for Boolean full-length shift register sequences is presented. With the help of the algorithm one can generate n-stage Boolean full-length shift register sequences for (potentially) arbitrary n≥6. Some properties of the generated feedback functions are presented
Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for
decoding of a single message bit using a small number of queries to a corrupted
encoding. Despite decades of study, the optimal trade-off between query
complexity and codeword length is far from understood. In this work, we give a
new characterization of LDCs using distributions over Boolean functions whose
expectation is hard to approximate (in~~norm) with a small number of
samples. We coin the term `outlaw distributions' for such distributions since
they `defy' the Law of Large Numbers. We show that the existence of outlaw
distributions over sufficiently `smooth' functions implies the existence of
constant query LDCs and vice versa. We give several candidates for outlaw
distributions over smooth functions coming from finite field incidence
geometry, additive combinatorics and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only
required to work on average over a random message and a random message index
can be turned into true LDCs at the cost of only constant factors in the
parameters.Comment: A preliminary version of this paper appeared in the proceedings of
ITCS 201
Neutrosophic Boolean Function and Rejection Sampling in Post Quantum Cryptography
The use of random seeds to a deterministic random bit generator to generate uniform random sampling has been applied multiple times in post-quantum algorithms. The finalists Dilithium and Kyber use SHAKE and AES to generate the random sequence at multiple stages of the algorithm. Here we characterize one of the sampleing techniques available in Dilithium for a random sequence of length 256 with the help of the neutrosophic Boolean function. The idea of the neutrosophic Boolean function came from the theory of neutrosophy and it is useful to study any ternary distributions. We present the non-existence of neutrobalanced bent functions specifically with respect to the sampling named SampleInBall in Dilithium
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Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term “outlaw distributions” for such distributions since they “defy” the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently “smooth” functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters
Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read
A Boolean function of n bits is balanced if it takes the value 1 with
probability 1/2. We exhibit a balanced Boolean function with a randomized
evaluation procedure (with probability 0 of making a mistake) so that on
uniformly random inputs, no input bit is read with probability more than
Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for
which the corresponding probability is Theta(n^{-1/3} log n). We then show that
for any randomized algorithm for evaluating a balanced Boolean function, when
the input bits are uniformly random, there is some input bit that is read with
probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions,
there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page
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