Paper 2026/1071

Event Algebras and Applications to Cryptography

Abstract

Discrete-step models are ubiquitous in many disciplines, in particular in Computer Science (e.g., computer systems, distributed and cryptographic protocols, etc.). The space of possible developments forms a tree (or forest) whose branches correspond to the possible discrete steps. Events are monotone predicates (or downsets) on the tree. Examples of events are input, output, forgery, consistency failure, or authentication failure events. Statements of interest about events are, for example, that a certain (``bad'') event can not occur. This paper introduces the concept of event algebras, a specific type of bounded distributive lattice $(E;\preceq,\wedge,\vee,∸,\top,\bot)$ with an additional operation $∸$, and shows that the event algebra axioms capture exactly and minimally the abstract mathematical structure of events in discrete-step models. An event inequality $e\preceq f$ can be read as ``event $e$ can not occur without event $f$ (having occurred).'' The most basic type of event algebra theorems, which are the scope of this paper, are inequalities between algebraic terms, for example, $a ∸ b \preceq (a ∸ c) \vee (c ∸ b)$, which hold universally, i.e., for any choice of the variables and for any event algebra. It is demonstrated that many fundamental statements in cryptography and other fields are direct implications of specific such universal event inequalities. For example, in a nutshell, the theorem stating the security of the well-known Hash-then-Sign paradigm is, in abstract form, the event inequality $e\preceq f\vee g$, where $e$ is the forgery event of the outer signature scheme, $f$ is the forgery event of the inner signature scheme, and $g$ is the (hash) collision event. The abstract algebraic treatment comes with the usual advantages: (1) generality, i.e., independence of modeling aspects such as computational and communication models or complexity and efficiency notions, (2) natural theorem composition, and (3) purely algebraic, minimal, and even formal proofs (here done in the Lean theorem prover).