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On Various Nonlinearity Measures for Boolean Functions.

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Cryptography and Communications : Discrete Structures, Boolean Functions and Sequences
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Summary

Six common cryptographic security measures, including nonlinearity and algebraic degree, are incomparable. This means a function strong in one area may be weak in another, impacting overall cryptographic function security.

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Area of Science:

  • Cryptography
  • Information Security
  • Theoretical Computer Science

Background:

  • Cryptographic function security relies on distance from linearity.
  • Several measures quantify this distance, including nonlinearity, algebraic degree, annihilator immunity, algebraic thickness, normality, and multiplicative complexity.

Purpose of the Study:

  • To investigate the relationships between six common measures of cryptographic function security.
  • To determine if these measures are comparable or incomparable.

Main Methods:

  • The study analyzes the theoretical properties of six distinct cryptographic measures.
  • It demonstrates incomparability by constructing specific functions to contrast their performance against different measures.

Main Results:

  • Six common measures of cryptographic distance (nonlinearity, algebraic degree, annihilator immunity, algebraic thickness, normality, multiplicative complexity) are shown to be incomparable.
  • New connections between two of these measures are established.
  • A lower bound for the multiplicative complexity of collision-free functions is provided.

Conclusions:

  • The incomparability of these security measures implies that a comprehensive security assessment requires considering multiple metrics.
  • Understanding these relationships is crucial for designing robust cryptographic functions.
  • Further research into the multiplicative complexity of specific function types is warranted.