Measures of Pseudorandomness for Finite Sequences: Minimal Values

Uniform distribution theory, 2017

In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic progressions, in particular, in cryptography one needs binary sequences whose short subsequences are also well-distributed relative to arithmetic progressions. Thus we introduce weighted measures of pseudorandomness of binary sequences to study this property. We study the typical and minimal values of this measure for binary sequences of a given length.

Cryptography and Communications, 2017

We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating functions or from their generating finite automatons, respectively.

Periodica Mathematica Hungarica, 2010

Binary and quaternary sequences are the most important sequences in view of many practical applications. Any quaternary sequence can be decomposed into two binary sequences and any two binary sequences can be combined into a quaternary sequence using the Gray mapping. We analyze the relation between the measures of pseudorandomness for the two binary sequences and the measures for the corresponding quaternary sequences, which were both introduced by Mauduit and Sárközy. Our results show that each 'pseudorandom' quaternary sequence corresponds to two 'pseudorandom' binary sequences which are 'uncorrelated'.

Acta Arithmetica, 2005

Finite fields and their applications, 2022

The correlation measure of order k is an important measure of pseudorandomness for binary sequences. This measure tries to look for dependence between several shifted versions of a sequence. We study the relation between the correlation measure of order k and two other pseudorandom measures: the N th linear complexity and the N th maximum order complexity. We simplify and improve several state-of-the-art lower bounds for these two measures using the Hamming bound as well as weaker bounds derived from it.

The Ramanujan Journal, 2014

We study the relationship between two measures of pseudorandomness for families of binary sequences: family complexity and cross-correlation measure introduced by Ahlswede et al. in 2003 and recently by Gyarmati et al., respectively. More precisely, we estimate the family complexity of a family

Problemy Peredachi Informatsii, 39:1, 2003

In the first part of this article, we answer Kolmogorov's question (stated in 1963 in [1]) about exact conditions for the existence of random generators. Kolmogorov theory of complexity permits of a precise definition of the notion of randomness for an individual sequence. For infinite sequences, the property of randomness is a binary property, a sequence can be random or not. For finite sequences, we can solely speak about a continuous property, a measure of randomness. Is it possible to measure randomness of a sequence t by the extent to which the law of large numbers is satisfied in all subsequences of t obtained in an “admissible way”? The case of infinite sequences was studied in [2]. As a measure of randomness (or, more exactly, of nonrandomness) of a finite sequence, we consider the specific deficiency of randomness δ (Definition 5). In the second part of this paper, we prove that the function δ/ln(1/δ) characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied.

Russian Mathematical Surveys. 45:1, 1990

How can one define that an individual binary sequence is random? The paper presents a comparative study of different approaches to this problem. The three properties of randomness to be analysed are: stochastic, chaotic, and typic (i.e., random in the senses given by von Mises, Kolmogorov, and Martin-Löf, respectively). The randomness of a (binary) sequence is to be considered with respect to some probability distribution over the set of all the sequences, e.g., the uniform Bernoulli distribution. The concept underlying the typicness is that of effectively zero sets of sequences: a sequence is said to be typic if it is not contained in any effectively zero set of sequences. The concept underlying the chaoticness is that of monotone entropy (complexity) KM(x) of any given finite sequence x: a sequence is said to be chaotic if KM(x)=−logP(Ωx)+O(1) for any initial segment x of this sequence. (Ωx is the set of all infinite sequences initiated by x, and P(Ωx) is the measure of this set in the given probability distribution.) The Levin-Schnorr theorem shows that a sequence is typic (w.r.t. the uniform Bernoulli distribution) iff it is chaotic. A sequence x is stochastic (i.e., random in the von Mises’ sense) w.r.t the uniform Bernoulli distribution if for any infinite subsequence of x chosen by some “admissible rule of choice”, the frequency of occurrence of the symbol 1 in initial segments of this subsequence tends to 1/2 as the lengths of these segments increase infinitely. Here the notion “admissible rule of choice” is requested to be defined. Two definitions of this notion introduced by Church and by Kolmogorov-Loveland are considered, and, according to them two definitions of the notion “stochastic sequence” are obtained. Unfortunately, these two definitions are not equivalent, and every of them is weaker than that of a typic, or chaotic, sequence.

2018

Generating random numbers and random sequences that are indistinguishable from truly random sequences is an important task for cryptography. To measure the randomness, statistical randomness tests are applied to the generated numbers and sequences. Knuth test suite is the one of the first statistical randomness suites. This suite, however, is mostly for real number sequences and the parameters of the tests are not given explicitly. In this work, we review the tests in Knuth Test Suite. We give test parameters in order for the tests to be applicable to integer and binary sequences and make suggestions on the choice of these parameters. We clarify how the probabilities used in the tests are calculated according to the parameters and provide formulas to calculate the probabilities. Also, some tests, like Permutation Test and Max-of-t-test, are modified so that the test can be used to test integer sequences. Finally, we apply the suite on some widely used cryptographic random number sou...

Informatica

Generating sequences of random numbers or bits is a necessity in many situations (cryptography, modeling, simulations, etc.. .). Those sequences must be random in the sense that their behavior should be unpredictable. For example, the security of many cryptographic systems depends on the generation of unpredictable values to be used as keys. Since randomness is related to the unpredictable property, it can be described in probabilistic terms, studying the randomness of a sequence by means of a hypothesis test. A new statistical test for randomness of bit sequences is proposed in the paper. The created test is focused on determining the number of different fixed length patterns that appear along the binary sequence. When 'few' distinct patterns appear in the sequence, the hypothesis of randomness is rejected. On the contrary, when 'many' different patterns appear in the sequence, the hypothesis of randomness is accepted. The proposed can be used as a complement of other statistical tests included in suites to study randomness. The exact distribution of the test statistic is derived and, therefore, it can be applied to short and long sequences of bits. Simulation results showed the efficiency of the test to detect deviation from randomness that other statistical tests are not able to detect. The test was also applied to binary sequences obtained from some pseudorandom number generators providing results in keeping with randomness. The proposed test distinguishes by fast computation when the critical values are previously calculated.

2017

We prove a relation between two measures of pseudorandomness, the arithmetic autocorrelation, and the correlation measure of order k. Roughly speaking, we show that any binary sequence with small correlation measure of order k up to a sufficiently large k cannot have a large arithmetic correlation. We apply our result to several classes of sequences including Legendre sequences defined with polynomials.

Indagationes Mathematicae, 2009

We estimate the linear complexity profile of m-ary sequences in terms of their correlation measure, which was introduced by Mauduit and Sárközy. For prime m this is a direct extension of a result of Brandstätter and the second author. For composite m, we define a new correlation measure for m-ary sequences, relate it to the linear complexity profile and estimate it in terms of the original correlation measure. We apply our results to sequences of discrete logarithms modulo m and to quaternary sequences derived from two Legendre sequences. s n+L = c L−1 s n+L−1 + · · · + c 0 s n , 0 n N − L − 1, MSC: 11K36, 94A55, 94A60

In this work, after a first analysis of random sequences and of the algorithmic impossibility to decide whether a certain arbitrary sequence may be defined random, we analyze some properties of pseudo-random sequences and in particular Blum-Blum-Shum (y = B(x) mod n) algorithm of which an implementation based upon some symmetry properties of quadratic congruencies is provided. In this way, a reduction of computational complexity is obtained, whose high value was responsible for a use of the same generator that was limited to the sole creation of short sequences.

Acta Arithmetica, 2008

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