Andre Souto

Assessment of Disagreement: A New Information-Based Approach

PURPOSE Disagreement on the interpretation of diagnostic tests and clinical decisions remains an important problem in medicine. As no strategy to assess agreement seems to be fail-safe to compare the degree of agreement, or disagreement, we propose a new information-based approach to assess disagreement. METHODS The sum over all logarithms of possible outcomes of a variable is a valid measure of the amount of information contained in the variable. The proposed measure is based on the amount of information contained in the differences between two observations. This measure is normalized and satisfies the flowing properties: it is a metric, scaled invariant and it has differential weighting. RESULTS Two real examples and a simulation were used to illustrate the usefulness of the proposed measure to compare the degree of disagreement. CONCLUSIONS Used as complement to the existing methods, we believe our approach can be useful to compare the degree of disagreement among different populations.

Entropy Measures vs. Kolmogorov Complexity

Kolmogorov complexity and Shannon entropy are conceptually different measures. However, for any recursive probability distribution, the expected value of Kolmogorov complexity equals its Shannon entropy, up to a constant. We study if a similar relationship holds for R´enyi and Tsallis entropies of order α, showing that it only holds for α = 1. Regarding a time-bounded analogue relationship, we show that, for some distributions we have a similar result. We prove that, for universal time-bounded distribution mt(x), Tsallis and Renyi entropies converge if and only if α is greater than 1. We also establish the uniform continuity of these entropies.

Sophistication as Randomness Deficiency

The sophistication of a string measures how much structural information it contains. We introduce naive sophistication, a variant of sophistication based on randomness deficiency. Naive sophistication measures the minimum number of bits needed to specify a set in which the string is a typical element. Thanks to Vereshchagin and Vitanyi, we know that sophistication and naive sophistication are equivalent up to low order terms. We use this to relate sophistication to lossy compression, and to derive an alternative formulation for busy beaver computational depth.

Universality of quantum Turing machines with deterministic control

A simple notion of quantum Turing machine with deterministic, classical control is proposed and shown to be powerful enough to compute any unitary transformation which is computable by a finitely generated quantum circuit. An efficient universal machine with the s-m-n property is presented. The BQP class is recovered. A robust notion of plain Kolmogorov complexity of quantum states is proposed and compared with those previously reported in the literature.

Sophistication vs Logical Depth

Sophistication and logical depth are two measures that express how complicated the structure in a string is. Sophistication is defined as the minimal complexity of a computable function that defines a two-part description for the string that is shortest within some precision; the second can be defined as the minimal computation time of a program that is shortest within some precision. We show that the Busy Beaver function of the sophistication of a string exceeds its logical depth with logarithmically bigger precision, and that logical depth exceeds the Busy Beaver function of sophistication with logarithmically bigger precision. We also show that sophistication is unstable in its precision: constant variations can change its value by a linear term in the length of the string.

Quantum blind signature with an offline repository

We propose a quantum blind signature scheme that achieves perfect security under the assumption of an honest offline repository. The security of the protocol also relies on perfect private quantum channels, which are achievable using quantum one-time pads with keys shared via a quantum key distribution (QKD) protocol. The proposed approach ensures that signatures cannot be copied and that the sender must compromise to a single message, which are important advantages over classical protocols for certain applications.

Enhancing Privacy with Quantum Networks

Using quantum networks to distribute symmetric keys has become a usable and commercial technology available under limitations that are acceptable in many application scenarios. The fact that the security is implemented directly at the hardware level, and moreover, relies on the laws of physics instead of conjectured hardness assumptions, justifies the use of quantum security in many cases. Limitations include 100 km communication range and installation of quantum channels between each pair of users of the network. Presently, with the current lack of trust in commercial security solutions, mostly due to the Snowden crisis, there is the need to improve such solutions. In this paper we discuss how quantum networks can be used to setup secure multiparty computation (SMC), allowing for instance for private data mining, electronic elections among other security functionalities. SMC relies mostly on establishing an efficient oblivious transfer protocol. We present a bit-string quantum oblivious transfer protocol based on single-qubit rotations that can be implemented with current technology based on optics and whose security relies only on the laws of physics.

Entropy measures vs. algorithmic information

Algorithmic entropy and Shannon entropy are two conceptually different information measures, as the former is based on size of programs and the later in probability distributions. However, it is known that, for any recursive probability distribution, the expected value of algorithmic entropy equals its Shannon entropy, up to a constant that depends only on the distribution. We study if a similar relationship holds for Rényi and Tsallis entropies of order α, showing that it only holds for Rényi and Tsallis entropies of order 1 (i.e., for Shannon entropy). Regarding a time bounded analogue relationship, we show that, for distributions such that the cumulative probability distribution is computable in time t(n), the expected value of time-bounded algorithmic entropy (where the alloted time is nt(n) log(nt(n))) is in the same range as the unbounded version. So, for these distributions, Shannon entropy captures the notion of computationally accessible information. We prove that, for universal time-bounded distribution mt(x), Tsallis and Rényi entropies converge if and only if a is greater than 1.

Kolmogorov complexity cores

We study the relationship between complexity cores of a language and the descriptional complexity of the characteristic sequence of the language based on Kolmogorov complexity. We prove that a recursive set A has a complexity core if for all constants c, the computational depth (the difference between time-bounded and unbounded Kolmogorov complexities) of the characteristic sequence of A up to length n is larger than c infinitely often. We also show that if a language has a complexity core of exponential density, then it cannot be accepted in average polynomial time, when the strings are distributed according to a time bounded version of the universal distribution.

Witness Hiding Without Extractors or Simulators

In a witness hiding protocol the prover tries to convince the verifier that he knows a witness to an instance of an \(\mathbf{NP}\) problem without revealing the witness. We propose a new look at witness hiding based on the information conveyed in each particular instance of the protocol.