Explicit infinite family of Maiorana McFarland bent functions settles conjecture
A Novel Approach to Counterexamples of the Polujan-Pott Conjecture via Set-Partition Permutations
Information Theory
Summary
This paper solves a long-standing mathematical problem by constructing a special kind of complex function called a Maiorana–McFarland bent function for infinitely many variables. The authors use advanced algebra structures to create a transformation that simplifies these functions, showing they fit perfectly into known mathematical designs. Their work fully proves a conjecture made by Polujan and Pott, clarifying the properties of these functions.
What this means in practice
- •For cryptography engineers: Use explicit bent functions to design cryptographic systems with proven complexity and resistance properties.
- •For coding theory developers: Incorporate the construction of these bent functions to build error-correcting codes based on symplectic designs.
A theory result. No direct application yet.
Authors
Yansheng Wu, Jiaxin Wang, Jong Yoon Hyun
Abstract
In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana--McFarland bent functions $f_t$ in $2(2^t-1)$ variables with algebraic degree $°(f_t) = t + 1$ for any integer $t \ge 2$. Our construction builds upon a minimal commutative algebra $I_t$, which naturally induces a triangular set-partition polynomial permutation $P_t$. By identifying an elementary abelian subgroup within the direct sum $ I_t \oplus I_t^*$, we establish an explicit nonlinear coordinate transformation that pulls $f_t$ back to a canonical quadratic form. This linearizes the translation development $\operatorname{Dev}(D_{f_t})$ under an exotic group structure and proves that it is isomorphic to the classical symplectic design $S^\pm(2(2^t-1))$, thereby fully resolving the conjecture.