New results on the state cycles of Trivium

Zhang, Shiyong; Chen, Gongliang

doi:10.1007/s10623-018-0493-2

Cited by 8 publications

(2 citation statements)

References 6 publications

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“…Simpson and Boztas proposed an alternate representation of Trivium using 3-bit word-based shift registers to determine the cycles of some internal state values [14]. Zhang et al presented some small cycles for the Galois NFSR in Trivium [15], but the Key/IV initialization guarantees that Trivium will never fall into these small cycles. So far neither the minimum period of keystreams generated by Trivium nor a reasonable lower bound of it has been determined since Trivium was proposed a decade ago.…”

Section: Introductionmentioning

confidence: 99%

GL-S-NFSR: A new NFSR structure and its period properties

Wang,

Tian,

2024

Cryptogr. Commun.

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Recently nonlinear feedback shift registers (NFSRs) have frequently been used as basic building blocks for stream ciphers. A major problem concerning NFSRs is to construct NFSRs with large periods. In this paper, a new NFSR structure whose period could be theoretically analyzed is proposed and studied, called GL-S-NFSR. A GL-S-NFSR is a selective cascade connection of a primitive Galois LFSR into a standard Galois NFSR with a linear simplified feedback function, where standard Galois NFSRs with linear simplified feedback functions are very useful in stream ciphers, e.g., Trivium. It is proved that the periods of the output sequences of a GL-S-NFSR are lower bounded by the product of all the Zsigmondy primes of 2 n − 1 with a probability close to 1 under a weak assumption, and particularly, if n is a prime, then 2 n − 1 divides the periods of the output sequences with a high probability, where n is the stage of the Galois LFSR. Besides, it is also proved that there are several registers satisfying that the periods are multiples of Zsigmondy primes without any assumption. Note that the main building block of Kreyvium consists of a standard Galois NFSR with a linear simplified feedback function and two pure cycling registers (PCRs). Periodic results on GL-S-NFSR are applied to Kreyvium by modifying one PCR to a primitive LFSR and the modified building block of Kreyvium is called M-Kreyvium. It is shown that the sequences involved in M-Kreyvium could have large periods with high probabilities.

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Section: Introductionmentioning

confidence: 99%

GL-S-NFSR: A new NFSR structure and its period properties

Wang,

Tian,

2024

Cryptogr. Commun.

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“…Over the years, NFSRs have received extensive attention due to their wide range of application scenarios. According to the structure, NFSRs can be divided into Fibonacci NFSRs [10,11], Galois NFSRs [11,12], Grain-like cascade NFSRs [13,14], Trivium-like cascade NFSRs [15,16] and so on. According to whether there is input, NFSRs can be divided into autonomous NFSRs and non-autonomous NFSRs.…”

Section: Introductionmentioning

confidence: 99%

Research Status of Nonlinear Feedback Shift Register Based on Semi-Tensor Product

Gao

Feng

2022

Mathematics

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Nonlinear feedback shift registers (NFSRs) are the main components of stream ciphers and convolutional decoders. Recent years have seen an increase in the requirement for information security, which has sparked NFSR research. However, the NFSR study is very imperfect as a result of the lack of appropriate mathematical tools. Many scholars have discovered in recent years that the introduction of semi-tensor products (STP) of matrices can overcome this issue because STP can convert the NFSR into a quasi-linear form. As a result of STP, new NFSR research has emerged from a different angle. In view of this, in order to generalize the latest achievements of NFSRs based on STP and provide some directions for future development, the research results are summarized and sorted out, broadly including the modeling of NFSRs, the analysis of the structure of NFSRs, and the study of the properties of NFSRs.

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Further results on the equivalence between Galois NFSRs and Fibonacci NFSRs

Zhao

Zhang

2019

Des. Codes Cryptogr.

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New results on the state cycles of Trivium

Cited by 8 publications

References 6 publications

GL-S-NFSR: A new NFSR structure and its period properties

GL-S-NFSR: A new NFSR structure and its period properties

Research Status of Nonlinear Feedback Shift Register Based on Semi-Tensor Product

Further results on the equivalence between Galois NFSRs and Fibonacci NFSRs

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