A Degree Bound for the c-Boomerang Uniformity

2025/10/21 by Steiner, Matthias Johann
#11T06 #14G50 #14H50 #94A60 #Algebraic Geometry (math.AG) #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.18506

Abstract

Let \mathbbF_q be a finite field, and let F ∈ \mathbbF_q [X] be a polynomial with d = deg ( F ) such that gcd ( d, q ) = 1. In this paper we prove that the c-Boomerang uniformity, c ≠ 0, of F is bounded by - d² if c² ≠ 1, - d ⋅ (d - 1) if c = -1, - d ⋅ (d - 2) if c = 1. For all cases of c, we present tight examples for F ∈ \mathbbF_q [X]. Additionally, for the proof of c = 1 we establish that the bivariate polynomial F (x) - F (y) + a ∈ k [x, y], where k is a field of characteristic p and a ∈ k ∖ \ 0 \, is absolutely irreducible if p \nmid deg ( F ).

Citations

• The Differential and Boomerang Properties of a Class of Binomials
• A new class of S-boxes with optimal Feistel boomerang uniformity
• A connection between the boomerang uniformity and the extended differential in odd characteristic and applications
• In-depth analysis of S-boxes over binary finite fields concerning their differential and Feistel boomerang differential uniformities
• On the Niho type locally-APN power functions and their boomerang spectrum
• Generalized Triangular Dynamical System: An Algebraic System for Constructing Cryptographic Permutations over Finite Fields
• Low c-differential and c-boomerang uniformity of the swapped inverse function
• Investigations on c-Boomerang Uniformity and Perfect Nonlinearity
• C-differentials, multiplicative uniformity and (almost) perfect\n c-nonlinearity

Related

• Differential uniformity of polynomials of degree 10
• Two Classes of Power Mappings with Boomerang Uniformity 2
• Sharp Effective Finite-Field Nullstellensatz
• Trinomials with high differential uniformity
• A connection between the boomerang uniformity and the extended differential in odd characteristic and applications