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On Degree-d Zero-Sum Sets of Full Rank
Beierle, Christof; Biryukov, Alex; Udovenko, Aleksei
2018
 

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Keywords :
annihilator; orthogonal matrix; nonlinear invariant; symmetric cryptography; trapdoor cipher
Abstract :
[en] A set S⊆Fn2 is called degree-d zero-sum if the sum ∑s∈Sf(s) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n−d−1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independent elements, and show relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices. We are particularly interested in the smallest of such sets and prove bounds on the minimum number of elements in a degree-d zero-sum set of rank n. The motivation for studying those objects comes from the fact that degree-d zero-sum sets of full rank can be used to build linear mappings that preserve special kinds of nonlinear invariants, similar to those obtained from orthogonal matrices and exploited by Todo, Leander and Sasaki for breaking the block ciphers Midori, Scream and iScream.
Research center :
Interdisciplinary Centre for Security, Reliability and Trust (SnT) > CryptoLUX
Disciplines :
Computer science
Author, co-author :
Beierle, Christof ;  University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust (SNT)
Biryukov, Alex ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC)
Udovenko, Aleksei  ;  University of Luxembourg > Interdisciplinary Centre for Security, Reliability and Trust (SNT)
Language :
English
Title :
On Degree-d Zero-Sum Sets of Full Rank
Publication date :
10 December 2018
Number of pages :
26
Focus Area :
Computational Sciences
Funders :
FNR - Fonds National de la Recherche [LU]
Available on ORBilu :
since 18 October 2019

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