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Use of Simple Arithmetic Operations to Construct Efficiently Implementable Boolean functions Possessing High Nonlinearity and Good Resistance to Algebraic Attacks

Claude Carlet, Palash Sarkar

Abstract

We describe a new class of Boolean functions which provide the presently best known trade-off between low computational complexity, nonlinearity and (fast) algebraic immunity. In particular, for $n\leq 20$, we show that there are functions in the family achieving a combination of nonlinearity and (fast) algebraic immunity which is superior to what is achieved by any other efficiently implementable function. The main novelty of our approach is to apply a judicious combination of simple integer and binary field arithmetic to Boolean function construction.

Use of Simple Arithmetic Operations to Construct Efficiently Implementable Boolean functions Possessing High Nonlinearity and Good Resistance to Algebraic Attacks

Abstract

We describe a new class of Boolean functions which provide the presently best known trade-off between low computational complexity, nonlinearity and (fast) algebraic immunity. In particular, for , we show that there are functions in the family achieving a combination of nonlinearity and (fast) algebraic immunity which is superior to what is achieved by any other efficiently implementable function. The main novelty of our approach is to apply a judicious combination of simple integer and binary field arithmetic to Boolean function construction.
Paper Structure (24 sections, 13 theorems, 12 equations, 2 tables, 2 algorithms)

This paper contains 24 sections, 13 theorems, 12 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic normal form.
  4. Nonlinearity and Walsh transform.
  5. Algebraic resistance.
  6. Implementation efficiency.
  7. Vectorial functions.
  8. Relevant Previous Constructions
  9. Carlet-Feng (CF) functions.
  10. Hidden weight bit (HWB) functions.
  11. Generalised HWB (GHWB) functions.
  12. Cyclic weightwise functions.
  13. Inverse map.
  14. Construction of Interval $\lambda$-HWB Functions
  15. Post-processing
  16. ...and 9 more sections

Key Result

Proposition 1

Let $n$ and $m$ be positive integers with $n\geq m$, and let $F$ be a balanced $(n,m)$-vectorial function. Let $f$ be an $m$-variable Boolean function. Then $f\circ F$ is balanced if and only if $f$ is balanced.

Theorems & Definitions (17)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • ...and 7 more