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Second-order differential uniformity for cryptographic Boolean functions

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posted on 2025-03-28, 15:10 authored by Connor O'Reilly

We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results.

History

School

  • Science

Department

  • Computer Science

Publisher

Loughborough University

Rights holder

© Connor O'Reilly

Publication date

2025

Notes

A Master's Thesis. Submitted in partial fulfilment of the requirements for the award of the degree of Master of Philosophy of Loughborough University.

Language

  • en

Supervisor(s)

Ana Sălăgean ; Robert Mercas

Qualification name

  • MPhil

Qualification level

  • Masters

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  • I have submitted a signed certificate

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