Mathematicians find optimal cryptographic functions with unique security features
On APN Functions with Boomerang Uniformity One over $\mathbb F_{3^n}$: Differential and Boomerang Spectra and CCZ-Inequivalence
Cryptography and Security
Summary
Some special mathematical functions help keep information safe and secret. The paper studies certain functions over numbers related to 3, showing they have the best possible resistance to a specific kind of security attack called the boomerang attack. The authors also classify these functions into different groups that are fundamentally different from each other, helping experts understand their unique properties. This work introduces an infinite family of functions with excellent security behavior that were not known before, expanding the toolbox for secure communication.
APN functionsboomerang uniformityperfect nonlinear functionsfinite fieldsDembowski–Ostrom polynomialdifferential uniformityCCZ equivalenceEA equivalencepresemifieldscryptographic functions
Authors
Namhun Koo, Soonhak Kwon, Minwoo Ko, Byunguk Kim
Abstract
Let $q=3^n$, where $n>1$ is odd, and let $g:\Fq\to\Fq$ be a perfect nonlinear (PN) function represented by a Dembowski--Ostrom (DO) polynomial. Put $τ=g(1)$, let $ε$ be the indicator of $\Fthree^*$, and, for $c\in\Fq$, define $\widetilde G_c(x):=g(x+c)+τε(x)$. We prove that every $\widetilde G_c$ is APN and has boomerang uniformity either one or two. More precisely, \[ β_{\widetilde G_c}=1 \quad\Longleftrightarrow\quad c\in\mathcal C_g :=\{c\in\Fq\setminus\Fthree:g(c)+τ\notin g(\Fq)\}, \qquad |\mathcal C_g|=\frac{q-3}{2}, \] whereas $β_{\widetilde G_c}=2$ for the remaining $(q+3)/2$ parameters. We determine the common differential spectrum and complete boomerang spectra of all the functions $\widetilde G_c$. Since boomerang uniformity one is the least possible for an APN function over a finite field of odd characteristic, this gives, to the best of our knowledge, the first general construction yielding infinite families of APN functions attaining this optimum. This common differential spectrum rules out CCZ equivalence with every power function and every Ness--Helleseth-type binomial. We also prove that CCZ equivalence between sign-switches of DO PN functions forces EA equivalence between the original PN functions. Using the orders of the nuclei of the associated presemifields, we exhibit, for infinitely many odd $n$, three pairwise CCZ-inequivalent PN functions over $\F_{3^n}$, one from each of the Gold $f_1$, Ding--Yuan $f_3$, and Bierbrauer $f_5$ families. Consequently, over each such field, our construction produces three pairwise CCZ-inequivalent APN functions with boomerang uniformity one. The smallest extension degree obtained in this way is $n=45$.