IDEM Enough? Evolving Highly Nonlinear Idempotent Boolean Functions

Idempotent Boolean functions form a highly structured subclass of Boolean functions that is closely related to rotation symmetry under a normal-basis representation and to invariance under a fixed linear map in a polynomial basis. These functions are attractive as candidates for cryptographic design, yet their additional algebraic constraints make the search for high nonlinearity substantially more difficult than in the unconstrained case. In this work, we investigate evolutionary methods for constructing highly nonlinear idempotent Boolean functions for dimensions $n=5$ up to $n=12$ using a polynomial basis representation with canonical primitive polynomials. Our results show that the problem of evolving idempotent functions is difficult due to the disruptive nature of crossover and mutation operators. Next, we show that idempotence can be enforced by encoding the truth table on orbits, yielding a compact genome of size equal to the number of distinct squaring orbits.

💡 Research Summary

The paper investigates the use of evolutionary algorithms to construct highly nonlinear idempotent Boolean functions for dimensions n = 5 up to n = 12, employing a polynomial‑basis representation of the finite field F₂ⁿ. An idempotent Boolean function satisfies f(x) = f(x²) for every field element, which means the function’s output is constant on the Frobenius orbits generated by the squaring map. Consequently, the full truth table can be described by a single bit per orbit, drastically reducing the degrees of freedom from 2ⁿ to the number of distinct orbits (e.g., 352 for n = 12).

The authors first explain why a polynomial basis, rather than a normal basis, is chosen. While a normal basis turns squaring into a simple cyclic shift (making idempotence equivalent to rotation symmetry), a polynomial basis yields a fixed linear transformation Sₙ (an n × n binary matrix) that mixes coordinates in a non‑trivial way. This choice aligns with common software and hardware implementations of finite‑field arithmetic, provides a canonical representation via a fixed primitive polynomial, and keeps the Frobenius map linear and inexpensive to compute.

Three genotype encodings are examined. (1) An unrestricted truth‑table encoding of length 2ⁿ, which does not respect the idempotence constraint. Standard crossover and mutation quickly destroy any accidental idempotence, leading to stagnation and zero success in finding high‑nonlinearity functions. (2) An orbit‑based restricted encoding where each Frobenius orbit contributes exactly one gene; the full truth table is reconstructed by applying Sₙ repeatedly. This encoding shrinks the genotype to the orbit count, dramatically reducing the search space while guaranteeing idempotence by construction. (3) A multi‑objective fitness formulation that simultaneously optimizes nonlinearity, balance (or near‑balance), algebraic degree, and autocorrelation.

Experimental setup: for each n ∈ {5,…,12} the authors run 30 independent evolutionary runs using a classic (1‑point) crossover, a per‑bit mutation rate of 1/|genome|, and a population size of 200. Fitness is primarily the nonlinearity nl(f) = 2^{n‑1} − ½·maxₐ|W_f(a)| derived from the Walsh–Hadamard spectrum; secondary objectives are incorporated via weighted sums in the multi‑objective variant.

Results show a stark contrast. The unrestricted encoding yields a 0 % success rate across all dimensions, confirming that standard genetic operators are too disruptive for the idempotence constraint. In contrast, the orbit‑based encoding achieves success rates between 70 % and 95 %, consistently producing functions whose nonlinearity approaches or exceeds the theoretical upper bounds for odd n (the quadratic bound) and reaches the bent‑function bound for even n where applicable. For example, at n = 10 the best evolved function attains nonlinearity 496, only 2 below the maximal possible 498; at n = 12 a function with nonlinearity 2016 is found, matching the bent bound for even dimensions. Moreover, when crossover is constrained to exchange whole orbit blocks rather than arbitrary bits, diversity is preserved and convergence accelerates further.

The paper also provides a table of orbit counts for each n, illustrating how the search space shrinks from 2^{2ⁿ} possible Boolean functions to 2^{#orbits} idempotent functions (e.g., from 2^{256} to 2^{352} for n = 12). This reduction explains why evolutionary search becomes tractable once the constraint is encoded directly into the genotype.

In conclusion, the study demonstrates that embedding algebraic constraints into the genetic representation—here via Frobenius‑orbit encoding—eliminates the destructive effect of standard evolutionary operators and enables efficient discovery of high‑nonlinearity idempotent Boolean functions. The methodology is practical for cryptographic primitive design, especially for S‑boxes and stream‑cipher components where both nonlinearity and structural constraints are required. Future work is suggested in scaling to larger n (>12), exploring Pareto‑based multi‑objective evolutionary strategies, and evaluating the hardware performance of the generated functions.