IDEM足够吗？构建高非线性幂等布尔函数
IDEM Enough? Evolving Highly Nonlinear Idempotent Boolean Functions
AI总结：
本文研究了使用多项式基表示法构建高非线性幂等布尔函数的方法，通过轨道编码强制幂等性，发现交叉和突变操作对问题解决有破坏性影响。
摘要：
幂等布尔函数是一个高度结构化的布尔函数子类，与在正常基表示下的旋转对称性和在多项式基中固定线性映射下的不变性密切相关。这些函数因其在密码学设计中的吸引力而备受关注，但其额外的代数约束使寻找高非线性性比无约束情况要困难得多。在本工作中，我们研究了使用多项式基表示法和标准原多项式构建高非线性幂等布尔函数的方法，适用于n=5到n=12的维度。我们的结果表明，由于交叉和突变操作的破坏性，构建幂等函数的问题较为困难。接下来，我们展示通过在轨道上编码真值表可以强制幂等性，从而得到大小等于不同平方轨道数量的紧凑基因组。
英文摘要：
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.
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