Explicit Constructions of Maximum-Cardinality Families of Plateaued Functions with Pairwise Disjoint Walsh Supports
Abstract
Families of plateaued Boolean functions with pairwise disjoint Walsh supports are useful in secondary constructions of cryptographic Boolean functions. Of particular interest are maximum-cardinality families whose members admit no nonzero linear structures. To the best of our knowledge, the previously known general construction attaining both properties is spectral (Hodžić et al., IEEE Trans. Inf. Theory 65(9): 5865--5879, 2019). In that work, explicit algebraic normal forms are not generally provided, and no general method is established for prescribing a common algebraic degree for all family members. In this paper, we present two new explicit algebraic constructions within a unified framework, one based on linear functions and the other on partially linear functions with bent components. Let p≥ 2 and q≥ 0 satisfy q<2p−p−1, and set m=p+q. Both constructions yield maximum-cardinality families of 2q+1 (q+1)-plateaued Boolean functions with pairwise disjoint Walsh supports. No member admits a nonzero linear structure, and every member has an explicit generalized Maiorana--McFarland representation. The first construction produces functions in m+p+1 variables and realizes any prescribed common algebraic degree 3≤ d≤ p+1, provided that q<∑i=2d−1binompi; its maximum attainable degree p+1 is optimal. The second construction produces functions in n+p+1 variables, where n>m and n−m is even, and realizes any prescribed common algebraic degree 3≤ d≤ p+(n−m)/2, provided that q<∑i=2min{d−1,p}binompi; its maximum attainable degree p+(n−m)/2 is next-to-optimal.
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BibTeX
@misc{wang2026explicit,
title = {{Explicit Constructions of Maximum-Cardinality Families of Plateaued Functions with Pairwise Disjoint Walsh Supports}},
author = {Chen Wang and Xiaoyan Zhang and Chunming Tang and Zhengchun Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21389},
eprint = {2609.21389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21389},
}
RIS
TY - UNPB
TI - Explicit Constructions of Maximum-Cardinality Families of Plateaued Functions with Pairwise Disjoint Walsh Supports
AU - Chen Wang
AU - Xiaoyan Zhang
AU - Chunming Tang
AU - Zhengchun Zhou
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.21389
UR - https://arxiv.org/abs/2609.21389
AB - Families of plateaued Boolean functions with pairwise disjoint Walsh supports are useful in secondary constructions of cryptographic Boolean functions. Of particular interest are maximum-cardinality families whose members admit no nonzero linear structures. To the best of our knowledge, the previously known general construction attaining both properties is spectral (Hodžić et al., IEEE Trans. Inf. Theory 65(9): 5865--5879, 2019). In that work, explicit algebraic normal forms are not generally provided, and no general method is established for prescribing a common algebraic degree for all family members. In this paper, we present two new explicit algebraic constructions within a unified framework, one based on linear functions and the other on partially linear functions with bent components. Let p≥ 2 and q≥ 0 satisfy q<2^p−p−1, and set m=p+q. Both constructions yield maximum-cardinality families of 2^q+1 (q+1)-plateaued Boolean functions with pairwise disjoint Walsh supports. No member admits a nonzero linear structure, and every member has an explicit generalized Maiorana--McFarland representation. The first construction produces functions in m+p+1 variables and realizes any prescribed common algebraic degree 3≤ d≤ p+1, provided that q<∑_i=2^d−1binompi; its maximum attainable degree p+1 is optimal. The second construction produces functions in n+p+1 variables, where n>m and n−m is even, and realizes any prescribed common algebraic degree 3≤ d≤ p+(n−m)/2, provided that q<∑_i=2^min{d−1,p}binompi; its maximum attainable degree p+(n−m)/2 is next-to-optimal.
ER -