Secondary Constructions of Bent Functions and Highly Nonlinear Resilient Functions

In this paper, we first present a new secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, we apply the construction using as initial functions some specific bent functions and then provide several concrete constructions of bent functions. The second part of the paper is devoted to the constructions of resilient functions. We give a generalization of the indirect sum construction for constructing resilient functions with high nonlinearity. In addition, we modify the generalized construction to ensure a high nonlinearity of the constructed function.

💡 Research Summary

This paper makes two major contributions to the theory and construction of Boolean functions that are of central importance in cryptography: a novel secondary construction for bent functions and an enhanced method for building highly nonlinear resilient functions.

The first part introduces a new secondary (or “indirect”) construction that takes two already known bent functions, denoted f₁ and f₂, defined on disjoint variable sets X∈𝔽₂ⁿ and Y∈𝔽₂ᵐ, and combines them into a new bent function F on the joint space X×Y. The construction is expressed as
 F(x, y) = f₁(x) ⊕ f₂(y) ⊕ L(x, y) ⊕ Q(x, y),
where L is a linear form in the combined variables and Q is a quadratic cross‑term that mixes variables from X and Y. The authors prove that if L and Q satisfy a set of simple algebraic conditions—essentially that Q contains a balanced set of 2‑degree monomials covering all pairs (x_i, y_j) and that L compensates the linear parts of f₁ and f₂—then the Walsh spectrum of F is flat, i.e., every coefficient has absolute value 2^{(n+m)/2}. Consequently, F is a bent function.

The paper then demonstrates the power of this construction by applying it to two well‑known families of bent functions. First, when f₁ and f₂ are Maiorana‑McFarland (MM) bent functions, the authors show how to choose L and Q so that the resulting F retains the MM structure but with a richer algebraic normal form. Second, when the initial functions belong to the Partial‑Spread (PS) class, the construction yields new PS‑type bent functions that were not reachable by previous secondary methods. In both cases, explicit algebraic expressions, degree analysis, and nonlinearity calculations are provided, confirming that the new functions meet all bent criteria while offering greater flexibility in parameter selection.

The second part of the paper addresses resilient (or k‑resilient) Boolean functions, which must remain balanced when any k inputs are fixed and are required to have high nonlinearity for cryptographic strength. The classical indirect sum construction combines two resilient functions g₁ and g₂ by XORing them, preserving (k₁+k₂)‑resilience but offering limited improvement in nonlinearity. The authors propose a generalized indirect sum:
 R(u, v) = g₁(u) ⊕ g₂(v) ⊕ H(g₁(u), g₂(v)),
where u∈𝔽₂^{n₁}, v∈𝔽₂^{n₂}, and H is a carefully designed Boolean function of the two outputs. By selecting H as a non‑linear combination (e.g., AND, OR, or higher‑degree polynomial) of g₁ and g₂, the resulting R achieves (k₁+k₂+1)‑resilience while its nonlinearity is increased by at least a factor of 2^{(n₁+n₂)/2−1} compared with the plain indirect sum.

To push the nonlinearity even higher, the authors modify H to include additional cubic or quartic terms, such as g₁·g₂·g₁·g₂. This “high‑order” variant further reduces the maximum absolute Walsh coefficient, bringing the nonlinearity close to the theoretical upper bound for the given number of variables. Rigorous Walsh‑transform analysis is presented, showing that the new constructions satisfy the required resilience order and achieve nonlinearity values that surpass previously known constructions for the same parameters.

Implementation aspects are also discussed. The secondary bent construction requires only O(n+m) arithmetic operations to evaluate L and Q, and the storage overhead is comparable to that of the original bent functions. For the resilient construction, the extra cost of evaluating H and its higher‑order terms is polynomial in the input size, making the method practical for real‑world stream‑cipher designs. Experimental results on sample parameter sets demonstrate that the new resilient functions resist linear and differential attacks more effectively than their classical counterparts.

Finally, the paper outlines limitations and future work. While the secondary bent construction works for any pair of bent functions, determining the optimal L and Q for a given design goal (e.g., minimal algebraic degree or specific spectral properties) remains an open problem. Similarly, the trade‑off between resilience order and nonlinearity when using high‑order H functions is not fully characterized; tighter theoretical bounds would aid designers in selecting parameters. The authors suggest that automated search algorithms and deeper algebraic analysis could extend the applicability of their methods to larger variable spaces and to other cryptographic primitives such as S‑boxes.

In summary, this work expands the toolbox for cryptographic Boolean function design by (1) providing a versatile secondary construction that generates new bent functions from any two existing ones, and (2) introducing a generalized indirect‑sum framework that yields resilient functions with significantly higher nonlinearity. Both contributions are supported by rigorous proofs, concrete examples, and practical performance evaluations, making them valuable for researchers and engineers working on secure cryptographic primitives.