A Note on the Construction of Differentially Uniform Permutations Using Extension Fields

Qichun WANG  Haibin KAN  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E95-A   No.11   pp.2080-2083
Publication Date: 2012/11/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E95.A.2080
Print ISSN: 0916-8508
Type of Manuscript: LETTER
Category: Cryptography and Information Security
Keyword: 
block ciphers,  vectorial Boolean functions,  extension fields,  

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Summary: 
Constructing APN or 4-differentially uniform permutations achieving all the necessary criteria is an open problem, and the research on it progresses slowly. In ACISP 2011, Carlet put forth an idea for constructing differentially uniform permutations using extension fields, which was illustrated with a construction of a 4-differentially uniform (n,n)-permutation. The permutation has optimum algebraic degree and very good nonlinearity. However, it was proved to be a permutation only for n odd. In this note, we investigate further the construction of differentially uniform permutations using extension fields, and construct a 4-differentially uniform (n,n)-permutation for any n. These permutations also have optimum algebraic degree and very good nonlinearity. Moreover, we consider a more general type of construction, and illustrate it with an example of a 4-differentially uniform (n,n)-permutation with good cryptographic properties.