Detecting Distributed Denial-of-Service Attacks Using Kolmogorov Complexity Metrics

This paper describes an approach to detecting distributed denial of service (DDoS) attacks that is based on fundamentals of Information Theory, specifically Kolmogorov Complexity. A theorem derived using principles of Kolmogorov Complexity states that the joint complexity measure of random strings is lower than the sum of the complexities of the individual strings when the strings exhibit some correlation. Furthermore, the joint complexity measure varies inversely with the amount of correlation. We propose a distributed active network-based algorithm that exploits this property to correlate arbitrary traffic flows in the network to detect possible denial-of-service attacks. One of the strengths of this algorithm is that it does not require special filtering rules and hence it can be used to detect any type of DDoS attack. We implement and investigate the performance of the algorithm in an active network. Our results show that DDoS attacks can be detected in a manner that is not sensitive to legitimate background traffic.This research has been funded by the Defense Advanced Research Projects Agency (DARPA) contract F30602-01-C-0182 and managed by the Air Force Research Laboratory (AFRL) Information Directorate.General Electric Global Research Center, Niskayuna, New York.     

Axiomatizing Kolmogorov Complexity

We revisit the axiomatization of Kolmogorov complexity given by Alexander Shen, currently available only in Russian language. We derive an axiomatization for conditional plain Kolmogorov complexity. Next we show that the axiomatic system given by Shen cannot be weakened (at least in any natural way). In addition we prove that the analogue of Shen??s axiomatic system fails to characterize prefix-free Kolmogorov complexity.     

Kolmogorov Complexity for Possibly Infinite Computations

In this paper we study the Kolmogorov complexity for non-effective computations, that is, either halting or non-halting computations on Turing machines. This complexity function is defined as the length of the shortest input that produce a desired output via a possibly non-halting computation. Clearly this function gives a lower bound of the classical Kolmogorov complexity. In particular, if the machine is allowed to overwrite its output, this complexity coincides with the classical Kolmogorov complexity for halting computations relative to the first jump of the halting problem. However, on machines that cannot erase their output –called monotone machines–, we prove that our complexity for non effective computations and the classical Kolmogorov complexity separate as much as we want. We also consider the prefix-free complexity for possibly infinite computations. We study several properties of the graph of these complexity functions and specially their oscillations with respect to the complexities for effective computations.     

Nonreducible Descriptions for the Conditional Kolmogorov Complexity

Assume that a program p produces an output string b for an input string a: p(a) = b. We look for a “reduction” (simplification) of p, i.e., a program q such that q(a) = b but q has Kolmogorov complexity smaller than p and contains no additional information as compared to p (this means that the conditional complexity K(q|p) is negligible). We show that, for any two strings a and b (except for some degenerate cases), one can find a nonreducible program p of any arbitrarily large complexity (any value larger than K(a) + K(b|a) is possible).     

An Additivity Theorem for Plain Kolmogorov Complexity

We prove the formula C(a,b)=K(a|C(a,b))+C(b|a,C(a,b))+O(1) that expresses the plain complexity of a pair in terms of prefix-free and plain conditional complexities of its components.     

Kolmogorov-Loveland Stochasticity and Kolmogorov Complexity

Merkle et al. (Ann. Pure Appl. Logic 138(1–3):183–210, 2006) showed that all Kolmogorov-Loveland stochastic infinite binary sequences have constructive Hausdorff dimension 1. In this paper, we go even further, showing that from an infinite sequence of dimension less than H(\frac 12+d)\mathcal {H}(\frac {1}{2}+\delta) (ℋ being the Shannon entropy function) one can extract by an effective selection rule a biased subsequence with bias at least δ. We also prove an analogous result for finite strings.     

Modeling Time-Bounded Prefix Kolmogorov Complexity

In the literature, prefix Kolmogorov complexity is defined either in terms of self-delimiting Turing machines or in terms of partial recursive prefix functions. These notions of prefix Kolmogorov complexity are equivalent because, as Chaitin showed, every partial recursive prefix function can be simulated by a self-delimiting Turing machine. However, the simulation given by Chaitin's construction is not efficient, and so questions regarding the time-bounded equivalence of these notions remained unresolved. Here we closely examine these questions. As our main result, we show that every partial recursive prefix function can be simulated with polynomial efficiency by a self-delimiting Turing machine if and only if P = NP. Thus, it is unlikely that Chaitin's construction can be used to show the polynomial-time equivalence of these notions of prefix Kolmogorov complexity. Here we further examine the relationships between these notions of time-bounded prefix Kolmogorov complexity. Received March 25, 1997, and in final form October 8, 1999.     

A Strange Application of Kolmogorov Complexity

This paper deals with two similar inequalities: where K denotes simple Kolmogorov entropy (i.e., the very first version of Kolmogorov complexity having been introduced by Kolmogorov himself) and KP denotes prefix entropy (self-delimiting complexity by the terminology of Li and Vitanyi [1]). It turns out that from (1) the following well-known geometric fact can be inferred: where V is a set in three-dimensional space, S _xy , S _yz , S _xz are its three two-dimensional projections, and |W| is the volume (or the area) of W . Inequality (2), in its turn, is a corollary of the well-known Cauchy—Schwarz inequality. So the connection between geometry and Kolmogorov complexity works in both directions. Received April 20, 1993, and in final form December 6, 1993.     

The Kolmogorov Expression Complexity of Logics

The purpose of the paper is to propose a completely new notion of complexity of logics in finite-model theory. It is the Kolmogorov variant of the Vardi'sexpression complexity. We define it by considering the value of the Kolmogorov complexityC(L[]) of the infinite stringL[] of all truth values of sentences ofLin . The higher is this value, the more expressive is the logicLin . If is a class of finite models, then the value ofC(L[]) over all ∈ is a measure of expressive power ofLin . Unboundedness ofC(L[])−C(L′[]) for ∈ implies nonexistence of a recursive interpretation ofLinL′. A version of this statement with complexities modulo oracles implies the nonexistence of any interpretation ofLinL′. Thus the valuesC(L[]) modulo oracles constitute an invariant of the expressive power of logics over finite models, depending on their real (absolute) expressive power, and not on the syntax. We investigate our notion for fragments of the infinitary logic ^ω_∞ω: least fixed point logic (LFP) and partial fixed point logic (PFP). We prove a precise characterization of 0–1 laws for these logics in terms of a certain boundedness condition placed onC(L[]). We get an extension of the notion of a 0–1 law by imposing an upper bound on the value ofC(L[]) growing not too fast with cardinality of , which still implies inexpressibility results similar to those implied by 0–1 laws. We also discuss classes in whichC(PFP^k[]) is very high. It appears that then PFP or its simple extension can define all the PSPACE subsets of .     

Time-Bounded Kolmogorov Complexity and Solovay Functions

A Solovay function is an upper bound g for prefix-free Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n. We obtain natural examples of computable Solovay functions by showing that for some constant c ₀ and all computable functions t such that c ₀ n??t(n), the time-bounded version K ^t of K is a Solovay function. By unifying results of Bienvenu and Downey and of Miller, we show that a right-computable upper bound g of K is a Solovay function if and only if ?? _g =??2^?g(n) is Martin-Löf random. We obtain as a corollary that the Martin-Löf randomness of the various variants of Chaitin??s ?? extends to the time-bounded case in so far as $\Omega _{ \textnormal{K}^{t}}$ is Martin-Löf random for any t as above. As a step in the direction of a characterization of K-triviality in terms of jump-traceability, we demonstrate that a set A is K-trivial if and only if A is O(g(n)?K(n))-jump traceable for all computable Solovay functions g. Furthermore, this equivalence remains true when the universal quantification over all computable Solovay functions in the second statement is restricted either to all functions of the form K ^t for some function t as above or to a single function K ^t of this form. Finally, we investigate into the plain Kolmogorov complexity C and its time-bounded variant C ^t of initial segments of computably enumerable sets. Our main theorem here asserts that every high c.e. Turing degree contains a c.e. set B such that for any computable function t there is a constant c _t >0 such that for all m it holds that C ^t (B?m)??c _t ?m, whereas for any nonhigh c.e. set A there is a computable time bound t and a constant c such that for infinitely many m it holds that C ^t (A?m)??logm+c. By similar methods it can be shown that any high degree contains a set B such that C ^t (B?m)??⁺ m/4. The constructed sets B have low unbounded but high time-bounded Kolmogorov complexity, and accordingly we obtain an alternative proof of the result due to Juedes et al. (Theor. Comput. Sci. 132(1?C2):37?C70, 1994) that every high degree contains a strongly deep set.     

On Strings with Trivial Kolmogorov Complexity

The Kolmogorov complexity of a string is the length of the shortest program that generates it. A binary string is said to have trivial Kolmogorov complexity if its complexity is at most the complexity of its length. Intuitively, such strings carry no more information than the information that is inevitably coded into their length (which is the same as the information coded into a sequence of 0s of the same length). We study the set of these trivial sequences from a computational perspective, and with respect to plain and prefix-free Kolmogorov complexity. This work parallels the well known study of the set of nonrandom strings (which was initiated by Kolmogorov and developed by Kummer, Muchnik, Stephan, Allender and others) and points to several directions for further research.     

Relating and Contrasting Plain and Prefix Kolmogorov Complexity

In (Bauwens and Shen, J. Symb. Log. 79(2), 620–632, 2013) a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. We argue that the proof technique is useful to simplify existing proofs and to solve open questions. We present a short proof of a result due to Robert Solovay that relates plain and prefix complexity:     

基于Kolmogorov复杂性的文本聚类算法改进

基于Kolmogorov复杂性的聚类算法虽然具有普适性、参数无关性的优点,但是应用到文本内容语义信息聚类时往往准确率较低。针对这一问题,提出了一种基于特征扩展的文本聚类改进算法——DEF-KC算法。该算法通过引用百度百科中特定词条的信息,对预处理过的文本中的关键词进行特征扩展,从而提高特征词的主题贡献度,增强文本的结构辨识度,并通过选取特定压缩算法近似计算Kolmogorov复杂性得到文本相似度,最后使用谱聚类算法进行聚类。实验结果表明,与传统的基于Kolmogorov复杂性的文本聚类算法相比,使用该算法时聚类准确率和召回率均得到了较大提升。     

Scaled Dimension and the Kolmogorov Complexity of Turing-Hard Sets

We study constructive and resource-bounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (Inf. Comput. 125(1), 13–31, 1996) established a small span theorem for P/poly-Turing reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (?3)rd-order scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turing-hard sets for ESPACE.     

采用整数变换降低率失真运动估计失真度计算量的研究

基于率失真的运动估计是提高视频编码器运动补偿精度的一个重要手段,但率失真失真度的计算太复杂,长期以来应用一直受限制.本文试用整型DCT变换代替DCT变换来优化计算失真度的纹理因子,从而提高计算速度.文章详细推导了优化计算公式,并用几组相关的实验数据来验证方法的有效性.