TY - JOUR

T1 - On the differential equivalence of APN functions

AU - Gorodilova, Anastasiya

N1 - Publisher Copyright: © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2019/7/15

Y1 - 2019/7/15

N2 - Carlet et al. (Des. Codes Cryptogr. 15, 125–156, 1998) defined the associated Boolean function γF(a,b) in 2n variables for a given vectorial Boolean function F from F2n to itself. It takes value 1 if a≠0 and equation F(x) + F(x + a) = b has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function G in n variables, n ≤ 6, that is differentially equivalent to a given quadratic APN function F, can be represented as G = F + A, where A is affine. For the APN Gold function F, we completely described all affine functions A such that F and F + A are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial.

AB - Carlet et al. (Des. Codes Cryptogr. 15, 125–156, 1998) defined the associated Boolean function γF(a,b) in 2n variables for a given vectorial Boolean function F from F2n to itself. It takes value 1 if a≠0 and equation F(x) + F(x + a) = b has solutions. This article defines the differentially equivalent functions as vectorial functions having equal associated Boolean functions. It is an open problem of great interest to describe the differential equivalence class for a given Almost Perfect Nonlinear (APN) function. We determined that each quadratic APN function G in n variables, n ≤ 6, that is differentially equivalent to a given quadratic APN function F, can be represented as G = F + A, where A is affine. For the APN Gold function F, we completely described all affine functions A such that F and F + A are differentially equivalent. This result implies that the class of APN Gold functions up to EA-equivalence contains the first infinite family of functions, whose differential equivalence class is non-trivial.

KW - Almost perfect nonlinear function

KW - Boolean function

KW - Crooked function

KW - Differential equivalence

KW - Linear spectrum

KW - PERFECT

UR - http://www.scopus.com/inward/record.url?scp=85068152651&partnerID=8YFLogxK

U2 - 10.1007/s12095-018-0329-y

DO - 10.1007/s12095-018-0329-y

M3 - Article

AN - SCOPUS:85068152651

VL - 11

SP - 793

EP - 813

JO - Cryptography and Communications

JF - Cryptography and Communications

SN - 1936-2447

IS - 4

ER -