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A general secondary construction of Boolean functions including the indirect sum and its generalizations

Claude Carlet, Deng Tang

TL;DR

This work introduces a unified secondary-construction framework for Boolean functions, defined by $f(x,y)=g(x)\oplus h_{F(x)}(y)$ with a vectorial map $F:{\mathbb F}_2^s\to{\mathbb F}_2^k$, which subsumes the direct sum and indirect sum as special cases and extends to larger image sets ${\rm Im}(F)$. It derives explicit Walsh-transform expressions linking $W_f$ to $W_g$, $W_{g\oplus v\cdot F}$, and $W_{h_u}$, enabling precise analysis of when the resulting functions are bent, plateaued, or resilient. The authors then systematically study small-image scenarios ($|{\rm Im}(F)|=1,2,3,4$ and up to $8$), obtaining concrete formulas and identifying particular instantiations (e.g., gen1, gen2) that reproduce known constructions and yield new function families. This unification broadens the construction toolbox for cryptographic Boolean functions and provides a rigorous spectral framework for exploring their nonlinear properties.

Abstract

We study a secondary construction of Boolean functions, which generalizes the direct sum and the indirect sum. We detail how these two classic secondary constructions are particular cases of this more general one, as well as two known generalizations of the indirect sum. This unifies the known secondary constructions of Boolean functions. We study very precisely the Walsh transform of the constructed functions. This leads us to an interesting observation on the Walsh transforms $W_g,W_{g'},W_{g''}$, and $W_{g\oplus g'\oplus g''}$ when $g,g',g''$ are Boolean functions such that $(g\oplus g')(g\oplus g'')$ equals the zero function.

A general secondary construction of Boolean functions including the indirect sum and its generalizations

TL;DR

This work introduces a unified secondary-construction framework for Boolean functions, defined by with a vectorial map , which subsumes the direct sum and indirect sum as special cases and extends to larger image sets . It derives explicit Walsh-transform expressions linking to , , and , enabling precise analysis of when the resulting functions are bent, plateaued, or resilient. The authors then systematically study small-image scenarios ( and up to ), obtaining concrete formulas and identifying particular instantiations (e.g., gen1, gen2) that reproduce known constructions and yield new function families. This unification broadens the construction toolbox for cryptographic Boolean functions and provides a rigorous spectral framework for exploring their nonlinear properties.

Abstract

We study a secondary construction of Boolean functions, which generalizes the direct sum and the indirect sum. We detail how these two classic secondary constructions are particular cases of this more general one, as well as two known generalizations of the indirect sum. This unifies the known secondary constructions of Boolean functions. We study very precisely the Walsh transform of the constructed functions. This leads us to an interesting observation on the Walsh transforms , and when are Boolean functions such that equals the zero function.
Paper Structure (10 sections, 2 theorems, 51 equations)

This paper contains 10 sections, 2 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A generalized indirect sum construction
  4. The Walsh transform of the constructed functions
  5. The expression of $f$ and Relations (\ref{['E:WalshPreimage']}) and (\ref{['E:Walshfi']}) when $F$ has a small image set
  6. $Im (F)$ of sizes 1, 2 and 3
  7. $Im (F)$ of size 4
  8. Function (\ref{['gen1']}) as a particular case:
  9. $Im (F)$ of size at most 8
  10. Function \ref{['gen2']} as a particular case:

Key Result

Theorem 1

Let $k,s,t$ be three positive integers and $F=(f_1,f_2,\cdots, f_k)$ be a $(s,k)$-vectorial Boolean function. Let ${g}$ be a Boolean function in $s$ variables and for every $u\in{\mathbb F}_2^k$, let $h_u$ be a Boolean functions in $t$ variables, and let $f$ be the $n=s+t$-variable Boolean function For any $a=(a', a^{"})\in{\mathbb F}_2^s\times {\mathbb F}_2^t$ with $a'=(a_1,\cdots,a_s)$ and $a^{

Theorems & Definitions (11)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Remark 5
  • Lemma 1
  • proof
  • Remark 6
  • ...and 1 more