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Testing a cellular automata construction method to obtain 9-variable cryptographic Boolean functions

Thomas Prévost, Bruno Martin

TL;DR

This work proposes a method for constructing 9-variable cryptographic Boolean functions from the iterates of 5-variable cellular automata rules and analyzes how these functions are preserved after extension to 9-variable Boolean functions.

Abstract

We propose a method for constructing 9-variable cryptographic Boolean functions from the iterates of 5-variable cellular automata rules. We then analyze, for important cryptographic properties of 5-variable cellular automata rules, how they are preserved after extension to 9-variable Boolean functions. For each cryptographic property, we analyze the proportion of 5-variable cellular automata rules that preserve it for each of the 48 affine equivalence classes.

Testing a cellular automata construction method to obtain 9-variable cryptographic Boolean functions

TL;DR

This work proposes a method for constructing 9-variable cryptographic Boolean functions from the iterates of 5-variable cellular automata rules and analyzes how these functions are preserved after extension to 9-variable Boolean functions.

Abstract

We propose a method for constructing 9-variable cryptographic Boolean functions from the iterates of 5-variable cellular automata rules. We then analyze, for important cryptographic properties of 5-variable cellular automata rules, how they are preserved after extension to 9-variable Boolean functions. For each cryptographic property, we analyze the proportion of 5-variable cellular automata rules that preserve it for each of the 48 affine equivalence classes.
Paper Structure (17 sections, 1 theorem, 2 equations, 2 figures)

This paper contains 17 sections, 1 theorem, 2 equations, 2 figures.

Table of Contents

  1. UNIFORM CELLULAR AUTOMATA
  2. CRYPTOGRAPHIC PROPERTIES OF BOOLEAN FUNCTIONS
  3. Representation of Boolean functions
  4. Affine equivalence classes
  5. Balancedness
  6. Correlation immunity
  7. Strict avalanche criterion
  8. Propagation criterion
  9. Algebraic degree
  10. ANALYSIS OF THE CRYPTOGRAPHIC PROPERTIES OF 5-VARIABLE BOOLEAN FUNCTIONS
  11. OUR METHOD TO EXTEND BOOLEAN FUNCTIONS FROM 5 TO 9 VARIABLES
  12. Iterates of a cellular automaton
  13. Testing the different properties of 9-variable Boolean functions
  14. RESULTS
  15. List of affine equivalence classes of 5-variable Boolean functions
  16. ...and 2 more sections

Key Result

theorem 1

A $n$-variable Boolean function $f$ is k-order correlation immune, $1 \le k \le n$ if and only if for every $\omega \in \mathbb{F}_2^n$ such that $1 \le w_h(\omega) \le k$, $S_f(\omega) = 0$.

Figures (2)

  • Figure 1: A 1-dimensional uniform cellular automaton with the single 3-bit local rule 224. 224 is the binary representation of the rule's truth table.
  • Figure 2: Extension of the 5-variable cellular automata rule $f$ to the 9-variable Boolean function $g$. The 9 input bits of $g$ are placed in the cells of the uniform cellular automaton. We apply the local rule $f$ twice, then the value of the central cell of the cellular automaton is the output of the function $g$. Hatched cells are no longer necessary at the corresponding time step to calculate the output of $g$.

Theorems & Definitions (7)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • theorem 1
  • definition 5
  • definition 6