The concept of timed-released encryption with pre-open capability (TRE-PC) was introduced by Hwang, Yum and Lee. In a TREPC scheme, a message is encrypted in such a way that it can only be decrypted at a certain point in time or if the sender releases a piece of trapdoor information known as a pre-open key. This paper examines the security model for a TRE-PC scheme, demonstrates that a TRE-PC s...

The distinguishing number D(G) of a graph G is the least cardinal number א such that G has a labeling with א labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes...

We begin the study of distinguishing geometric graphs. Let G be a geometric graph. An automorphism of the underlying graph that preserves both crossings and noncrossings is called a geometric automorphism. A labelling, f : V (G) → {1, 2, . . . , r}, is said to be rdistinguishing if no nontrivial geometric automorphism preserves the labels. The distinguishing number of G is the minimum r such th...

The distinguishing number ∆(X) of a graph X is the least positive integer n for which there exists a function f : V (X) → {0, 1, 2, · · · , n−1} such that no nonidentity element of Aut(X) fixes (setwise) every inverse image f−1(k), k ∈ {0, 1, 2, · · · , n − 1}. All infinite, locally finite trees without pendant vertices are shown to be 2distinguishable. A proof is indicated that extends 2-disti...

Definition 1.2 (Distinguishing number of a graph). Let Γ = (V,E) be a graph and let f : V → [r] be a coloring of the set of vertices by r colors. The map f need not be surjective, in fact when r > |V | it cannot be surjective. We say that f is r-distinguishing if the only automorphism of Γ that fixes the coloring f is the trivial automorphism. The distinguishing number of Γ is denoted by D(Γ) a...

A coloring of the vertices of a graph G is said to be distinguishing provided that no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, denoted D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing number, first introduced by Albertson and Collins in 1996, has been widely studied and a number of interesting res...