We survey some results in quantum cryptography. After a brief introduction to classical cryptography, we provide the quantum-mechanical background needed to present some fundamental protocols from quantum cryptography. In particular, we review quantum key distribution via the BB84 protocol and its security proof, as well as the related quantum bit commitment protocol and its proof of insecurity.

The three domatic number problem asks whether a given undirected graph can be partitioned into at least three dominating sets, i.e., sets whose closed neighborhood equals the vertex set of the graph. Since this problem is NP-complete, no polynomial-time algorithm is known for it. The naive deterministic algorithm for this problem runs in time 3n , up to polynomial factors. In this paper, we design an exact deterministic algorithm for this problem running in time 2.9416 n . Thus, our algorithm can handle problem instances of larger size than the naive algorithm in the same amount of time. We also present another deterministic and a randomized algorithm for this problem that both have an even better performance for graphs with small maximum degree.

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695 n (up to polynomial factors). This result improves the previous bound of 2.8805 n , which is due to Fomin, Grandoni, Pyatkin, and Stepanov. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree ∆(G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe [RR05] whenever ∆(G) ≥ 5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.

Abstract. We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of k given values is complete for BH 2k (NP), the 2kth level of the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP = BH 2 (NP). We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner's conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP.

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695 n (up to polynomial factors). This result improves the previous bound of 2.8805 n , which is due to Fomin, Grandoni, Pyatkin, and Stepanov. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree ∆(G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe [RR05] whenever ∆(G) ≥ 5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.