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.

翻译：幂等布尔函数构成布尔函数中一个高度结构化的子类，其与正规基表示下的旋转对称性以及多项式基下固定线性映射的不变性密切相关。这类函数作为密码学设计的候选对象具有吸引力，然而其额外的代数约束使得寻找高非线性度比无约束情形下更为困难。本文研究利用规范本原多项式下的多项式基表示，为维度$n=5$至$n=12$构造高度非线性幂等布尔函数的演化方法。结果表明，由于交叉与变异算子的破坏性，演化幂等函数的问题具有挑战性。进一步，我们证明可通过在轨道上编码真值表来强制实现幂等性，从而得到基因组大小等于不同平方轨道数量的紧凑编码方案。