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Linear complexity is an important pseudo-random measure of the key stream sequence in a stream cipher system. The 1-error linear complexity is used to measure the stability of the linear complexity, which means the minimal linear complexity of the new sequence by changing one bit of the original key stream sequence. This paper contributes to calculating the exact values of the linear complexity and 1-error linear complexity of the binary key stream sequence with two prime periods defined by Ding–Helleseth generalized cyclotomy. We provide a novel method to solve such problems by employing the discrete Fourier transform and the M–S polynomial of the sequence. Our results show that, by choosing appropriate parameters p and q, the linear complexity and 1-error linear complexity can be no less than half period, which shows that the linear complexity of this sequence not only meets the requirements of cryptography but also has good stability.

The linear complexity and the error linear complexity are two important security measures for stream ciphers. We construct periodic sequences from function fields and show that the error linear complexity of these periodic sequences is large. We also give a lower bound for the error linear complexity of a class of nonperiodic sequences.

The k-error linear complexity is an important cryptographic measure of pseudorandom sequences in stream ciphers. In this paper, we investigate the k-error linear complexity of p2-periodic binary sequences defined from the polynomial quotients modulo p, which are the generalizations of the well-studied Fermat quotients. Indeed, first we determine exact values of the k-error linear complexity over the finite field F2 for these binary sequences under the assumption of 2 being a primitive root modulo p2, and then we determine their k-error linear complexity over the finite field Fp. Theoretical results obtained indicate that such sequences possess ＇good＇ error linear complexity.

The Euler quotient modulo an odd-prime power pr （r 〉 1） can be uniquely decomposed as a p-adic number of the form （u（p- 1）Pr- 1 _ 1）/pr ≡- ao （u） ＋ a1 （u）p ＋... ＋at- 1 （u）Pr- 1 （mod pr）, gcd（u, p） = 1, where 0 ≤ aj（u） 〈 p for 0 ≤ j≤r - 1 and we set all aj（u） = 0 if gcd（u,p） 〉 1. We firstly study certain arithmetic properties of the level sequences （aj（u））u≥0 over Fp via introducing a new quotient. Then we determine the exact values of linear complexity of （aj（u））u≥0 and values of k-error linear complexity for binary sequences defined by （aj （U））u≥0.

We will show that the computation of the intercriteria counters can be done in O(n log n) time (quasi-linear complexity). Up to this point, all implementations have used O(n2) operations, which does not allow processing of data over hundreds of thousands.

In order to solve the problem of poor classification performance of traditional algorithms for basketball tactics, we propose a scientific training model for basketball tactics computer swarm intelligence algorithm. We designed a basketball tactical recognition model (TacViT) based on player trajectory data in NBA games. The TacViT model employs a Vision Transformer (ViT) as its backbone network. It utilizes a multi-head attention module to extract rich global trajectory features and integrates a trajectory filter to enhance the interaction of feature information between the court lines and player trajectories. The trajectory filter learns long-term spatial correlations in the frequency domain with logarithmic linear complexity, thereby improving the representation of player position features. We transformed the sequence data from the sports vision system (SportVU) into trajectory maps and constructed a basketball tactical dataset (PlayersTrack). The experimental results demonstrate that TacViT achieves an accuracy of 81.4%, which is 15.6% higher than the unmodified ViT-S model. Additionally, TacViT exhibits superior performance in precision, recall, and computational efficiency. The PlayersTrack dataset contains 10,000 trajectory images, each with a resolution of 256x256 pixels. The TacViT architecture introduces a novel trajectory filter module and a multi-head attention mechanism, which together enable efficient feature extraction. Key evaluation metrics include accuracy, precision, recall, and FLOPS. These results highlight the significant improvement in classification performance for basketball tactics recognition.

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math. 44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc. 21(2) (2019), 355–380].

Modern software oriented symmetric ciphers have become a key feature in utilizing word-oriented cryptographic primitives. Using the output sequence, in the order of its generation, of a word-oriented crypto- graphic primitive in the same way as traditional bit-oriented primitives, we can expose the intrinsic weakness of these primitives, especially for word-oriented linear feedback shift registers, T-functions, and so on. Two new methods for using word-oriented cryptographic primitives are presented in this paper, that is, the extracted state method and cascading extracted coordinate method. Using a T-function as an example, we research the different cryptographic properties of the output sequences of the original method and the two proposed methods, focusing mainly on period, linear complexity, and k-error linear complexity. Our conclusions show that the proposed methods could enhance at low cost the cryptographic properties of the output sequence. As a result, since the new methods are simple and easy to implement, they could be used to design new word-oriented cryptographic primitives.

Since the introduction of the Kolmogorov complexity of binary sequences in the 1960s, there have been significant advancements on the topic of complexity measures for randomness assessment, which are of fundamental importance in theoretical computer science and of practical interest in cryptography. This survey reviews notable research from the past four decades on the linear, quadratic and maximum-order complexities of pseudo-random sequences, and their relations with Lempel–Ziv complexity, expansion complexity, 2-adic complexity and correlation measures.

Quaternary sequences with high linear complexity and N -adic complexity have been widespread concerned in cryptography. By using the Gray mapping, we construct a new class generalized cyclotomic quaternary sequences with period pq over Z 4 . We determine the linear complexity over F 4 and 4-adic complexity of the sequences. The results show that these sequences have high linear complexity and 4-adic complexity.