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The Multiplicative Complexity of 6-variable Boolean Functions.

Çağdaş Çalık1, Meltem Sönmez Turan2, René Peralta1

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Summary
This summary is machine-generated.

This study determines the multiplicative complexity of Boolean functions, finding all 6-variable functions need at most 6 AND gates. This research advances understanding of Boolean function implementation with minimal AND gates.

Keywords:
06E3094A60Affine equivalenceBoolean functionsCircuit complexityCryptographyMultiplicative complexity

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Area of Science:

  • Computer Science
  • Digital Logic Design
  • Computational Complexity Theory

Background:

  • Multiplicative complexity quantifies the minimum AND gates for Boolean functions (AND, XOR, NOT).
  • Determining this complexity is computationally intractable for general functions.
  • Prior work established bounds for n ≤ 5 and existence proofs for n ≥ 7.

Purpose of the Study:

  • To develop a method for calculating the multiplicative complexity of Boolean functions.
  • To analyze the multiplicative complexity of all 6-variable Boolean functions.
  • To establish the maximum number of AND gates required for 6-variable functions.

Main Methods:

  • Proposed a novel method analyzing circuits with a specific number of AND gates.
  • Utilized the concept of affine equivalence of Boolean functions.
  • Calculated complexities for all 150,357 affine equivalence classes of 6-variable functions.

Main Results:

  • Established that any 6-variable Boolean function can be implemented using at most 6 AND gates.
  • Identified specific 6-variable Boolean functions that require exactly 6 AND gates.
  • Provided a comprehensive complexity map for 6-variable functions.

Conclusions:

  • The maximum multiplicative complexity for 6-variable Boolean functions is 6.
  • The proposed method effectively determines Boolean function multiplicative complexity.
  • This work contributes to efficient circuit design and complexity theory.