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Boolean Functions with Multiplicative Complexity 3 and 4.

Çağdaş Çalık1, Meltem Sönmez Turan1, René Peralta1

  • 1NIST Computer Security Division, 100 Bureau Dr, Gaithersburg, MD 20899.

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Summary
This summary is machine-generated.

This study characterizes affine equivalence classes for Boolean functions with multiplicative complexity (MC) 3 and 4. It introduces a new lower bound for MC, enabling the construction of optimal circuits for functions with MC 4 or less.

Keywords:
Affine equivalenceBoolean functionsMultiplicative complexity

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Area of Science:

  • Boolean function analysis
  • Circuit complexity theory
  • Cryptography and digital circuit design

Background:

  • Multiplicative complexity (MC) quantifies AND gates in Boolean circuits.
  • Previous work characterized functions with MC 1 and 2.
  • Understanding MC is crucial for efficient circuit design.

Purpose of the Study:

  • Identify affine equivalence classes for functions with MC 3 and 4.
  • Establish a new lower bound for multiplicative complexity.
  • Provide formulas for counting functions with MC 3 and 4.

Main Methods:

  • Utilizing the dimension (dim(f)) of Boolean functions.
  • Applying the concept of linearity dimension.
  • Leveraging techniques from prior research on MC.

Main Results:

  • Confirmed no new equivalence classes for MC 3 beyond existing ones.
  • Identified 1277 affine equivalence classes for functions with MC 4.
  • Established a lower bound MC >= [dim(f)/2].
  • Derived closed formulas for the number of n-variable functions with MC 3 and 4.

Conclusions:

  • The study successfully classifies functions with MC 3 and 4.
  • The new lower bound aids in complexity analysis.
  • Results facilitate the construction of AND-optimal circuits for functions with MC up to 4.