作为等距的连分式展开：线性、跳跃和2-Adic复杂性的迭代对数律
Continued Fraction Expansion as Isometry: The Law of the Iterated Logarithm for Linear, Jump, and 2--Adic Complexity
摘要：
在流密码和伪随机序列的密码分析中，线性复杂度、跳跃复杂度和2-adic复杂度的概念自然出现，以衡量给定字符串的（非）随机性。我们在F_q^ \infty上定义了一个等距K，它精确等价于实数上的Euclid算法来计算形式幂级数的连分式展开式。连分数扩展允许推导出输入序列的线性和跳跃复杂性曲线。因为K是等距的，所以得到的F_q^ \infty -序列是i.i.d.。对于I.I.D.输入.因此，线性和跳跃复杂性曲线可以通过伯努利实验（对于F_2：掷硬币）来建模，并且我们可以应用Revesz收集的非常精确的界限，其中包括迭代对数定律。第二个主题是2-adic跨度和复杂性，由Goresky和Klapper定义。我们再次导出一个等距，这次是在并元整数Z_2上，它在F_2}^ \infty上导出一个等距A。相应的跳跃复杂度的平均行为完全像抛硬币。索引术语：形式幂级数，等距，线性复杂度，跳跃复杂度，2-adic复杂度，2-adic跨度，重对数律，Levy类，流密码，伪随机序列
英文摘要：
In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2-adic complexity arise naturally to measure the (non)randomness of a given string. We define an isometry K on F_q^\infty that is the precise equivalent to Euclid's algorithm over the reals to calculate the continued fraction expansion of a formal power series. The continued fraction expansion allows to deduce the linear and jump complexity profiles of the input sequence. Since K is an isometry, the resulting F_q^\infty-sequence is i.i.d. for i.i.d. input. Hence the linear and jump complexity profiles may be modelled via Bernoulli experiments (for F_2: coin tossing), and we can apply the very precise bounds as collected by Revesz, among others the Law of the Iterated Logarithm. The second topic is the 2-adic span and complexity, as defined by Goresky and Klapper. We derive again an isometry, this time on the dyadic integers Z_2 which induces an isometry A on F_2}^\infty. The corresponding jump complexity behaves on average exactly like coin tossing. Index terms: Formal power series, isometry, linear complexity, jump complexity, 2-adic complexity, 2-adic span, law of the iterated logarithm, Levy classes, stream ciphers, pseudorandom sequences
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