arXiv subjects

John Proos

Publications and source records attributed to John Proos.

3 recordsLinked to original sources

arXiv2017

Subhypergraphs in non-uniform random hypergraphs

In this paper we focus on the problem of finding (small) subhypergraphs in a (large) hypergraph. We use this problem to illustrate that reducing hypergraph problems to graph problems by working with the 2-section is not always a reasonable approach. We begin by defining a generalization of the binomial random graph model to hypergraphs and formalizing several definitions of subhypergraph. The bulk of the paper focusses on determining the expected existence of these types of subhypergraph in random hypergraphs. We also touch on the problem of determining whether a given subgraph appearing in the 2-section is likely to have been induced by a certain subhypergraph in the hypergraph. To evaluate the model in relation to real-world data, we compare model prediction to two datasets with respect to (1) the existence of certain small subhypergraphs, and (2) a clustering coefficient.

math.CO↗

arXiv2016

Connectivity in Hypergraphs

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard while the other can be solved in polynomial time.

math.CO↗

arXiv2004

Shor's discrete logarithm quantum algorithm for elliptic curves

We show in some detail how to implement Shor's efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around 1000 qubits while factoring the security-wise equivalent 1024 bit RSA modulus would require about 2000 qubits. In this paper we only consider elliptic curves over GF($p$) and not yet the equally important ones over GF($2^n$) or other finite fields. The main technical difficulty is to implement Euclid's gcd algorithm to compute multiplicative inverses modulo $p$. As the runtime of Euclid's algorithm depends on the input, one difficulty encountered is the ``quantum halting problem''.

quant-ph↗