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Generalized $p$-ary $\cPS$ Bent Functions
Alexander Kholosha, Mohit Pal
TL;DR
Complete classification of p-ary bent functions remains difficult, motivating explicit constructions. This paper uses partial m-spreads to generalize two p-ary PS classes; its stated bentness criteria yield regular bent functions, while the dual-class question for general m remains open.
Problem
Complete classification of bent functions is described as hopeless even in the Boolean case and more complicated for p-ary bent functions.
Source: §1 · p. 1Method
Results
Takeaways & Limitations
The construction extends the p-ary PS− and PS+ classes proposed by Lisoněk and Lu.
Source: §1 · p. 2, §3 · p. 4For general m, which class contains the dual of an m-PS− function remains open.
Source: §3 · p. 7
Abstract
from arXiv · showhide
We use $m$-dimensional partial spreads of $\fp^{2n}$, where $p$ is an odd prime, $n$ is a positive integer and $m$ divides $n$, to construct two classes of bent functions from $\fp^{2n}$ to $\fp$. Our construction generalizes the classes of $p$-ary $\cPS^{-}$ and $\cPS^{+}$ bent functions proposed by P. Lison\v ek and H. Y. Lu (Des. Codes Cryptogr. 73 (2014), 209--216).
1 Introduction
Bent functions are defined by maximal distance from affine functions, equivalently by constant-magnitude Walsh coefficients, and their complete classification is described as seemingly out of reach. The paper proposes two p-ary bent-function classes using partial m-spreads, with m dividing n, and also considers generalized Dillon exponents.
1 Introduction
- Bent functions have maximum Hamming distance from affine functions, equivalently constant absolute-square Walsh coefficients; their nonzero-direction derivatives are balanced.The paper notes that complete classification appears hopeless even in the Boolean case and is more complicated for p-ary bent functions.Source: §1 · p. 1
- The paper generalizes Lisoněk and Lu’s p-ary PS− and PS+ classes using partial m-spreads of F_p^(2n), where m divides n.The introduction places this work among primary constructions that build bent functions from scratch.Source: §1 · pp. 1–2
- The final section asks whether monomial bent functions with a generalized Dillon exponent can lie outside the PS− class.Source: §1 · p. 2
2 Preliminaries
The preliminaries establish algebraic-degree bounds for functions constant on a subspace and recall degree bounds for p-ary bent functions. In particular, a nonzero constant on an m-dimensional subspace forces degree at least (p − 1)m.
2 Preliminaries
- Weakly regular p-ary bent functions have degree at most n(p − 1)/2, while non-weakly-regular ones have degree at most n(p − 1)/2 + 1.The text notes that the regular p-ary PS bent functions attain the maximal algebraic degree.Source: §2 · p. 3
- A function vanishing at zero and equal to a nonzero constant on an m-dimensional subspace has algebraic degree at least (p − 1)m.The proof restricts the function to the subspace and uses Lagrange interpolation to show the restricted degree is (p − 1)m.Source: §2 · p. 3
- Any function of degree below (p − 1)m cannot be constant on the nonzero points of an m-dimensional subspace.Source: §2 · p. 3
3 p-ary m-PS−class
The section characterizes bent functions that are constant on the nonzero parts of a partial m-spread, defining the resulting regular functions as the p-ary m-PS− class. It also gives examples of functions in these generalized classes and notes a boundary on the theorem and an open question about duals.
Theorem 2
- For functions constant on each nonzero spread subspace, the theorem characterizes bentness through m | n and conditions on the function’s level sets.It also specifies Walsh values at nonzero inputs under the stated conditions and concludes that bent functions satisfying the theorem are regular.Source: §3 · p. 4
Definition 1
- The functions satisfying Theorem 2 are called the p-ary m-PS− class, and the theorem’s bent functions are regular.The Walsh value at zero is p^n; the two stated cases give p^n or p^nζ_p^t at nonzero inputs.Source: §3 · pp. 4, 7
Scope and duals
- The condition p^n > 3 is necessary: at p = 3 and n = 1, the theorem’s claim that all such bent functions are regular is false.The section also notes that the dual-class question for general m-PS− functions remains open.Source: §3 · p. 7
Example 1
- The function F is bent of algebraic degree four, and all its component functions are regular bent functions from the completed class M.Source: §3 · p. 7
Example 2
- Example 2 likewise gives a degree-four ternary vectorial bent function with components in the completed class M; for even n, its components are 2-PS− but not PS−.Source: §3 · p. 8
Example 3
- The section also supplies a degree-four ternary 2-PS− example over F_3^8 and a degree-four quinary 1-PS− example over F_5^4 that is not in PS−.Source: §3 · p. 8
4 p-ary m-PS+ class
The section extends the p-ary PS+ construction to partial m-spreads and characterizes when the resulting functions are bent. It also gives their Walsh values and identifies the resulting class as p-ary m-PS+.
4 p-ary m-PS+ class
- The result generalizes [7, Theorem 3.6] and suggests constructing bent functions from partial m-spreads.Source: §4 · p. 8
- Under the theorem's hypotheses, bentness is equivalent to m|n and the stated spread-subspace counts across the value fibers.The hypotheses include odd p, m<2n, p^n>3, constancy on each T*, and f^-1(0)=the union of S with f(0)=z≠0.Source: §4 · pp. 8–9
- The two alternatives allocate p^m−1 subspaces to f^-1(z), with other fibers receiving N or N+p^(n−m) subspaces under the stated cases.Source: §4 · p. 9
- The theorem gives W_f(a)=p^n or p^nζ_p^t under the respective count conditions and states that bent functions are regular.Source: §4 · p. 9
- For m=n, the corollary requires p^n−1 spread elements in each non-z fiber and p^n−1+1 in the z fiber.Source: §4 · p. 11
- The functions satisfying Theorem 3's conditions are named the p-ary m-PS+ class.Source: §4 · p. 11
5 Generalized Dillon Bent Functions
The section asks whether a monomial with a generalized Dillon exponent can be bent without belonging to the n-PS− class. It relates these exponents to m-PS− membership and studies a parameterized family.
5 Generalized Dillon Bent Functions
- The section considers functions formed as sums of traces of monomials α_j x^{t(p^m−1)}, with m|n and 0<t<(p^{2n}−1)/(p^m−1).The coefficients α_t lie in F_{p^{2n}}.Source: §5 · p. 11
- For the Desarguesian m-spread, each member is a multiplicative coset of F_{p^m}; a bent function constant on each T* belongs to the m-PS− class and is called generalized Dillon bent.The passage states that these cosets satisfy Theorem 2's conditions.Source: §5 · p. 12
- The generalized Dillon monomial classes extend the known Boolean Dillon class [3] and ternary bent functions from [4, Theorem 2].Source: §5 · p. 12
- Under m|2n and gcd(t, (p^{2n}−1)/(p^m−1))=1, Theorem 5 defines a p-ary function family and specifies its Walsh transform coefficients.The supplied passage does not state the coefficient values.Source: §5 · p. 12
- When p^n>3, bentness implies m|n and places f in the regular m-PS− class.This implication is stated for the monomial family under study.Source: §5 · p. 14