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Boolean Differential Equations – Solution and Applications
Last modified: 2014-01-28
Abstract
The \textit{Boolean Differential Calculus} (BDC) significantly extends the Boolean Algebra because not only Boolean values of the set $\mathbb{B}=\left\{ 0,1 \right\}$, but also changes of Boolean values or Boolean functions can be described.
A \textit{Boolean Differential Equation} (BDE) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solution,
algorithms to calculate the solution of a BDE, and selected applications.
In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithm to solve a BDE that is restricted to all vectorial derivatives of $f(\mathbf{x})$. We explain formulas for transformation of other derivative operations to vectorial derivatives in order to solve more general BDEs. Selected from the wide field of applications, we show BDEs which solve important tasks of circuit design and cryptography.
The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBOOLE problem program (PRP) of the freely available XBOOLE-Monitor quickly solves some BDEs.
A \textit{Boolean Differential Equation} (BDE) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solution,
algorithms to calculate the solution of a BDE, and selected applications.
In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithm to solve a BDE that is restricted to all vectorial derivatives of $f(\mathbf{x})$. We explain formulas for transformation of other derivative operations to vectorial derivatives in order to solve more general BDEs. Selected from the wide field of applications, we show BDEs which solve important tasks of circuit design and cryptography.
The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBOOLE problem program (PRP) of the freely available XBOOLE-Monitor quickly solves some BDEs.
Keywords
Boolean Differential Calculus; Boolean Differential Equation; circuit design; cryptography; XBOOLE.