Linear complexity of generalized NTU sequences

The linear complexity of sequences is defined and its main properties reviewed. The linear complexity of periodic sequences is examined in detail and an extensive list of its properties is formulated. The discrete Fourier transform (DFT) of a finite sequence is then connected to the linear complexity of a periodic sequence by Blahut's theorem. Cyclic codes are given a DFT formulation that relates their minimum distance to the least linear complexity among certain periodic sequences. To illustrate the power of this approach, a slight generalization of the Hartmann-Tzeng lower bound on minimum distance is proved, as well as the Bose-Chaudhuri-Hocquenghem lower bound. Other applications of linear complexity in the theory of cyclic codes are indicated.

The need for stream ciphers in communication and information cryptographic systems is rapidly grown in the last twenty years. The stream ciphers usually use a pseudo-random number generator with large period, large linear complexity and good statistical properties. At present a small number of software-oriented stream ciphers are known. Due to this reasons in our paper we suggest a new pseudo random number generator, named P-adic shrinking-multiplexing generator, and give its statistical analysis.

We study the linear complexity of sequences over the prime field \({\mathbb{F}_d}\) introduced by Sidel’nikov. For several classes of period length we can show that these sequences have a large linear complexity. For the ternary case we present exact results on the linear complexity using well known results on cyclotomic numbers. Moreover, we prove a general lower bound on the linear complexity profile for all of these sequences. The obtained results extend known results on the binary case. Finally we present an upper bound on the aperiodic autocorrelation.KeywordsSidel’nikov sequenceLinear complexityLinear complexity profileAperiodic autocorrelation

In this paper we study relationships between the linear complexities of a sequence when treated as a sequence over two distinct fields. We obtain bounds for one linear complexity in the form of a constant multiple of the other, where the constant depends only on the fields, not on the particular sequence.

Transmission of the information in any form requires security. Security protocols used for communication rely on the use of random numbers. Pseudo-random numbers are required with good statistical properties and efficiency. The use of a single chaotic map may not produce enough randomness. The turbulence is padded into the existing map to improve its chaotic behaviour and increase the periodicity. A Pseudo-random number generator (PRNG) with this architecture is devised to generate random bit sequences from secret keys. The statistical properties of newly constructed PRNG are tested with NIST SP 800–22 statistical test suite and were shown to have good randomness. To ensure its usability in cryptographic applications, we analysed the size of its key space, key sensitivity, and performance speed. The test results show that the newly designed PRNG has a 3.6% increase in key space and a 5% increase in its performance speed compared to existing chaotic PRNGs. The novel PRNG with faster performance is found suitable for lightweight cryptographic applications.

Pseudo-random number generators (PRNG) are a fundamental element of many security algorithms. We introduce a novel approach to their implementation, by proposing the use of generative adversarial networks (GAN) to train a neural network to behave as a PRNG. Furthermore, we showcase a number of interesting modifications to the standard GAN architecture. The most significant is partially concealing the output of the GAN’s generator, and training the adversary to discover a mapping from the overt part to the concealed part. The generator therefore learns to produce values the adversary cannot predict, rather than to approximate an explicit reference distribution. We demonstrate that a GAN can effectively train even a small feed-forward fully connected neural network to produce pseudo-random number sequences with good statistical properties. At best, subjected to the NIST test suite, the trained generator passed around 99% of test instances and 98% of overall tests, outperforming a number of standard non-cryptographic PRNGs.

In parallel Monte Carlo simulations, it is highly desirable to have a system of pseudo-random number generators that has good statistical properties and allows reproducibility of random sequences used by the individual processes. In this work, we discuss a distributed implementation of such a system on computer networks. The system employs linear congruential pseudorandom number generators obtained from the work “Random Number Generators for MIMD Parallel Processors” [1] and is organized as a virtual, complete binary tree. Nodes of the tree are the generators and the processes using them. The relationship between a node and its subtrees reflects that of a process and its children. Any binary tree up to a million nodes can be maintained. To achieve high utilization of the generators, we implement a dynamically expanding complete binary tree. A server is maintained on a host on our computer network and application programs obtain the generators from the interface provided by the system. The system guarantees that no generator is allocated more than once to an application process and no extraneous correlations are introduced. The pseudorandom number system uses the client/server model, communicating via the Remote Procedure Calls (RPC).

Niederreiter showed that there is a class of periodic sequences which possess large linear complexity and large k-error linear complexity simultaneously. This result disproved the conjecture that there exists a trade-off between the linear complexity and the k-error linear complexity of a periodic sequence by Ding et al.. Using the entropy function in coding theory, we obtain three main results which hold for much larger k than those of Niederreiter et al.: a) sequences with maximal linear complexity and almost maximal k-error linear complexity with general periods; b) sequences with maximal linear complexity and maximal k-error linear complexity with special periods; c) sequences with maximal linear complexity and almost maximal k-error linear complexity in the asymptotic case with composite periods.

A family of novel pseudorandom number generators is proposed based on an evolutionary algorithm. These pseudorandom number generators are expressed as EAPRNG. EAPRNGs are driven by some existed simple and unsafe generators which are denoted by mother generators (MG), while outputs of the MGs are hidden by EAPRNGs. The analysis indicates that EAPRNGs are resistant to many known attacks, and that they have good statistical properties, such as uniform probability distribution over GF (2/sup l/) and large periods.

Most commonly-used pseudo-random number generators (PNGs) in computer systems are based on linear recurrence. These deterministic PNGs have fast and compact implementations, andean ensure very long periods. However, the points generated by linear PNGs in fact have a regular lattice structure and are thus not suit able for applications that rely on the assumption of uniformly distributed pseudo-random numbers (PNs). In this paper we propose and evaluate several fast and compact linear, non-linear, and hybrid PNGs for a field- programmable gate array (FPGA). The PNGs have excellent equidistribution properties and very small autocorrelations, and have very long repetition periods. The distribution and long-range correlation properties of the new generators are efficiently, and much more rapidly, estimated at hardware speeds using designed modules within the FPGA. The results of these statistical tests confirm that the combination of several linear PNGs or the combination of even one small non-linear PNG with a linear PNG significantly improves the statistical properties of the generated PNs.

The combination of nonlinear functions with LFSRs provides a good means for generating running keys (see chapter 5). The major problem is the difficulty of implementing complex and truly random nonlinear functions. What we would like to have is a simple and fast implementation of a large set of highly nonlinear functions which is readily indexed by the key. Moreover, any function randomly chosen from the large available set should, with virtual certainty, produce a sequence with good distribution properties. As we have seen in the preceding chapter, the knapsack provides an ideal instrument to meet these requirements, except perhaps for a slightly reduced operational speed when compared to direct or table implementations. The maximum order of most sum-bit functions fj,k (x) assures that large linear complexity may be attained. The knapsack also distributes good statistics in the sense that equally distributed selector variables xl,...,xN will, with virtual certainty, result in equally distributed bits sj of the binary representation of the partial sum S when the modulus is a power of 2 (or close to a power of 2). In the original knapsack problem the weights are public. When the knapsack is used to realize a nonlinear mapping, there is no reason to give this information away; giving it away can only make the attack easier.

A new Pseudo random number generator based on generalized Newton complex map with dynamic key

A new statistical test for uniform pseudo-random number generators (PRNGs) is presented. The idea is that a sequence of pseudo-random numbers should have numbers reappear with a certain probability. The expectation time that a repetition occurs provides the metric for the test. For linear congruential generators (LCGs) failure can be shown theoretically. Empirical test results for a number of commonly used PRNGs are reported, showing that some PRNGs considered to have good statistical properties fail. A sample implementation of the test is provided over the Internet.

Following the idea of Xing et al., we investigate a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence. With the help of the tool of exponential sums on elliptic curves, we study their periods, linear complexities, linear complexity profiles, distributions of r-patterns, periodic correlation, partial period distributions, and aperiodic correlation in detail. The results show that they have nice randomness.

In this paper pseudorandom number generator based on elliptic curve bilinear pairing is developed. Residue number system and approximate method are used for effictive realization of modular operations over finite field that allows to increase the speed of pseudorandom number generator for \(-\)256 by 2,15 times compared to similar PRNG that uses positional notation. The developed pseudorandom number generator based on neural network has as good statistical properties as random sequences from site random.org and passes Diehard tests.