TY - JOUR
T1 - A new construction of differentially 4-uniform (n, n-1)-functions
AU - Carlet, Claude
AU - Alsalami, Yousuf
N1 - Publisher Copyright:
© 2015 AIMS.
PY - 2015/11
Y1 - 2015/11
N2 - In this paper, a new way to construct differentially 4-uniform (n; n-1)-functions is presented. As APN (n; n)-functions, these functions offer the best resistance against differential cryptanalysis and they can be used as substitution boxes in block ciphers with a Feistel structure. Constructing such functions is assumed to be as difficult as constructing APN (n; n)-functions. A function in our family of functions can be viewed as the concatenation of two APN (n - 1; n - 1)-functions satisfying some necessary conditions. Then, we study the special case of this construction in which the two APN functions differ by an affne function. Within this construction, we propose a family in which one of the APN functions is a Gold function which gives the quadratic differentially 4-uniform (n, n-1)-function (formula presented) where (formula presented) and xnF2 with gcd(i,n-1)=1. We study the nonlinearity of this function in the case i = 1 because in this case we can use results from Carlitz which are unknown in the general case. We also give the Walsh spec- trum of this function and prove that it is CCZ-inequivalent to functions of the form L ₒ F where L is an affne surjective (n, n-1)-function and F is a known APN (n; n)-function for n ≤ 8, or the Inverse APN (n; n)-function for every n ≥ 5 odd, or any AB (n, n)-function for every n > 3 odd, or any Gold APN (n; n)-function for every n > 4 even.
AB - In this paper, a new way to construct differentially 4-uniform (n; n-1)-functions is presented. As APN (n; n)-functions, these functions offer the best resistance against differential cryptanalysis and they can be used as substitution boxes in block ciphers with a Feistel structure. Constructing such functions is assumed to be as difficult as constructing APN (n; n)-functions. A function in our family of functions can be viewed as the concatenation of two APN (n - 1; n - 1)-functions satisfying some necessary conditions. Then, we study the special case of this construction in which the two APN functions differ by an affne function. Within this construction, we propose a family in which one of the APN functions is a Gold function which gives the quadratic differentially 4-uniform (n, n-1)-function (formula presented) where (formula presented) and xnF2 with gcd(i,n-1)=1. We study the nonlinearity of this function in the case i = 1 because in this case we can use results from Carlitz which are unknown in the general case. We also give the Walsh spec- trum of this function and prove that it is CCZ-inequivalent to functions of the form L ₒ F where L is an affne surjective (n, n-1)-function and F is a known APN (n; n)-function for n ≤ 8, or the Inverse APN (n; n)-function for every n ≥ 5 odd, or any AB (n, n)-function for every n > 3 odd, or any Gold APN (n; n)-function for every n > 4 even.
KW - APN functions
KW - Block ciphers
KW - Differentially 4-uniform functions
KW - S-boxes
KW - Vectorial boolean functions
UR - http://www.scopus.com/inward/record.url?scp=84947803622&partnerID=8YFLogxK
U2 - 10.3934/amc.2015.9.541
DO - 10.3934/amc.2015.9.541
M3 - Article
AN - SCOPUS:84947803622
SN - 1930-5346
VL - 9
SP - 541
EP - 565
JO - Advances in Mathematics of Communications
JF - Advances in Mathematics of Communications
IS - 4
ER -