# Review of the Lineal Complexity Calculation through Binomial Decomposition-Based Algorithms

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## Abstract

**:**

## 1. Introduction

## 2. Shift Registers and the Concept of Linear Complexity

- L binary stages, which are interconnected and numbered $(0,1,2,\dots ,L-1)$ from left to right. Each stage stores a unique bit.
- The L-degree feedback or connection polynomial$$p(x)={x}^{L}+{c}_{1}{x}^{L-1}+{c}_{2}{x}^{L-2}+\dots +{c}_{L-1}x+{c}_{L}$$
- A non-zero initial state (stage contents) at the initial instant.

**Definition**

**1.**

#### An LFSR-Based Sequence Generator

- (a)
- A PN-sequences $\{{a}_{n}\}$ generated by an L-stage LFSR and a shifted version of such a sequence, notated $\{{b}_{n}\}$. Both sequences are related by the expression $\{{b}_{n}\}=\{{a}_{n+p}\}$, p being an integer. Thus, $\{{b}_{n}\}$ is nothing but the PN-sequence $\{{a}_{n}\}$ circularly rotated p positions with $(p=0,1,2\dots ,{2}^{L}-2)$.
- (b)
- A simple decimation rule defined as:$$\left\{\begin{array}{c}\mathrm{If}{a}_{n}=1\mathrm{then}{b}_{n}\mathrm{is}\mathrm{output},\hfill \\ \mathrm{If}{a}_{n}=0\mathrm{then}{b}_{n}\mathrm{is}\mathrm{discarded}\mathrm{and}\mathrm{no}\mathrm{bit}\mathrm{is}\mathrm{output}.\hfill \end{array}\right.$$

- All the generalized self-shrunken sequences are balanced apart from the identically 1 sequence [25] (Theorem 1).
- By construction, the family of generalized self-shrunken sequences consists of ${2}^{L}-1$ sequences of ${2}^{L-1}$ bits each of them. Thus, the length of any generalized sequence will be ${2}^{L-1}$ or divisors. At any rate, the length of these sequences will always be a power of 2.
- The family of generalized sequences plus the identically null sequence has structure of Abelian group where the group operation is the bit-wise sum mod 2. the neutral element is the identically null sequence and every sequence is its own inverse element [25] (Theorem 2).
- The sequence produced by the self-shrinking generator is a member of this family for $p={2}^{L-1}$, see [22].

**Example**

**1.**

## 3. Binomial Sequences

#### 3.1. Introduction to Binomial Sequences

**Definition**

**2.**

- Given the binomial sequence $\left\{\left(\genfrac{}{}{0pt}{}{n}{k}\right)\right\}$ with $k={2}^{m}+i$ where m is a non-negative integer and the index i takes values in the interval $0\le i<{2}^{m}$, then we have that [12] (Proposition 3):
- (a)
- The binomial sequence $\{\left(\genfrac{}{}{0pt}{}{n}{k}\right)\}$ has length $l={2}^{m+1}$.
- (b)
- The formation rule of this binomial sequence is:$${\left\{\left(\genfrac{}{}{0pt}{}{n}{{2}^{m}+i}\right)\right\}}_{0\le n<{2}^{m+1}}=\left\{\begin{array}{cc}0\hfill & \mathrm{if}0\le n{2}^{m}+i,\hfill \\ {\left(\genfrac{}{}{0pt}{}{n}{i}\right)}_{mod2}\hfill & \mathrm{if}{2}^{m}+i\le n{2}^{m+1}.\hfill \end{array}\right.$$

- The linear complexity of the binomial sequence $\left\{\left(\genfrac{}{}{0pt}{}{n}{{2}^{m}+i}\right)\right\}$ with m and i defined as above is $LC={2}^{m}+i+1$, see [12] (Theorem 13).
- Every binary sequence ${\{{s}_{n}\}}_{n\ge 0}$ whose length is a power of 2 can be written as linear combination of binomial sequences [12] (Theorem 2). This combination is called the Binomial Decomposition of ${\{{s}_{n}\}}_{n\ge 0}$. Such a decomposition allows us to analyze fundamental properties of the sequence, e.g., length and linear complexity.
- Given a sequence ${\{{s}_{n}\}}_{n\ge 0}$ with binomial decomposition $\{{s}_{n}\}={\sum}_{i=1}^{r}\left\{\left(\genfrac{}{}{0pt}{}{n}{{k}_{i}}\right)\right\}$, where $0\le {k}_{1}<{k}_{2}<\cdots <{k}_{r}$ are integer indices, then its linear complexity is given by $LC={k}_{r}+1$, see [12] (Corollary 14).
- Given a sequence ${\{{s}_{n}\}}_{n\ge 0}$ with binomial decomposition $\{{s}_{n}\}={\sum}_{i=1}^{r}\left\{\left(\genfrac{}{}{0pt}{}{n}{{k}_{i}}\right)\right\}$, where $0\le {k}_{1}<{k}_{2}<\cdots <{k}_{r}$ are integer indices, then its length l is that of the binomial sequence $\left\{\left(\genfrac{}{}{0pt}{}{n}{{k}_{r}}\right)\right\}$, i.e., the length of the binomial sequence of maximum index in its binomial decomposition, see [32] (Theorem 1).

#### 3.2. Binomial Decomposition of GSS-Sequences

## 4. Different Algorithms to Compute the Linear Complexity of a Sequence

- For the sake of readability, In the sequel the binomial coefficient $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ just denotes the k-$th$ binomial sequence.
- The term ${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{i,j}$ represents the sub-sequence of $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ between the i-$th$ and j-$th$ bits.
- The term ${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{j}$ stands for the sub-sequence corresponding to the j first bits of $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$.

#### 4.1. Berlekamp-Massey Algorithm

#### 4.2. Binomial Decomposition Algorithm or BD-Algorithm

- According to Item 3 (in Section 3.1), the sequence $seq$ of length $l={2}^{m}$ can be decomposed into r binomial sequences of the form:$$seq=\left(\genfrac{}{}{0pt}{}{n}{{k}_{1}}\right)+\cdots +\left(\genfrac{}{}{0pt}{}{n}{{k}_{r}}\right).$$
- According to Item 4 (in Section 3.1), the lineal complexity of $seq$ is that of the binomial sequence of maximum index $\left(\genfrac{}{}{0pt}{}{n}{{k}_{r}}\right)$ in its binomial decomposition. Since the indices of the binomial sequences are written in increasing order, then $LC$ is computed by means of the following equation:$$LC={k}_{r}+1.$$

Algorithm 1 The BD-algorithm. |

Require:$seq$: the sequence to be analyzed |

$binom=[\varnothing ]$, ${k}_{r}=0$ |

for $i=0;ilength(seq);i$++ do |

if $se{q}_{i}==1$ then |

$seq+=\left(\genfrac{}{}{0pt}{}{n}{i}\right)$ |

$binom.add(i)$ |

${k}_{r}=i$ |

end if |

end for |

return $binom$ and $LC={k}_{r}+1$: binomial decomposition and $LC$ of $seq$. |

#### Improvement of the BD-Algorithm

#### 4.3. Half-Interval Search Algorithm

#### 4.3.1. Symmetry of the Binomial Sequences

**Theorem**

**1.**

- If k the index of the binomial sequence is $k<\frac{l}{2}$, then the two sub-sequences in Equation (2) are equal.
- If k the index of the binomial sequence is $k\ge \frac{l}{2}$, then the two sub-sequences in Equation (2) are written as:$${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{l}=\left(zero{s}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{i}\right)}_{\frac{l}{2}}\right),$$

**Proof.**

- Since $k<\frac{l}{2}$, then k can be written as $k={2}^{j}+i$, where j and i are non-negative integers such that $j<m-1$ and $0\le i<{2}^{j}$. According to Item 1(a) in Section 3.1, the binomial sequence $\left(\genfrac{}{}{0pt}{}{n}{k}\right)=\left(\genfrac{}{}{0pt}{}{n}{{2}^{j}+i}\right)$ has length $\tilde{l}={2}^{j+1}$ where the maximum length is ${\tilde{l}}_{max}={2}^{m-1}$ when $j=m-2$ and the minimum length ${\tilde{l}}_{min}={2}^{0}$ when $j=0$. At any rate, $\tilde{l}$ is a power of 2 as well as $\tilde{l}<{2}^{m}$ and, therefore, the first and second sub-sequences in Equation (2) are equal.
- Since $k\ge \frac{l}{2}={2}^{m-1}$, then k can be written as $k={2}^{m-1}+i$ with $0\le i<{2}^{m-1}$. According to Item 1(a) in Section 3.1, the binomial sequence $\left(\genfrac{}{}{0pt}{}{n}{k}\right)=\left(\genfrac{}{}{0pt}{}{n}{{2}^{m-1}+i}\right)$ has length $\tilde{l}=l={2}^{m}$. Moreover, according to Item 1(b) in Section 3.1$${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{\frac{l}{2},l-1}={\left(\genfrac{}{}{0pt}{}{n}{{2}^{m-1}+i}\right)}_{\frac{l}{2},l-1}={\left(\genfrac{}{}{0pt}{}{n}{i}\right)}_{\frac{l}{2}}.$$Thus, the sub-sequence ${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{l}$ satisfies the Equation (3) as well as the $\frac{l}{2}$ first terms are zeros.

**Proposition**

**1.**

**Proof.**

Algorithm 2 Classification of the Binomial sequences. |

Given the sub-sequence ${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{l}$: |

if $k<\frac{l}{2}$ then |

${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{l}:=({\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{\frac{l}{2}})$ |

else |

${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{l}:=(zero{s}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{k-\frac{l}{2}}\right)}_{\frac{l}{2}})$ |

end if |

- ${M}_{0}$ and ${M}_{1}$ are $((i-1)\times \frac{l}{2})$ sub-matrices that, according to Theorem 1, satisfy the equality ${M}_{0}={M}_{1}$.
- ${M}_{2}$ is the $((r-i+1)\times \frac{l}{2})$ identically null sub-matrix.
- ${M}_{3}$ is the $((r-i+1)\times \frac{l}{2})$ sub-matrix representing the decomposition of a new sequence of length $\frac{l}{2}$ coming from the bit-wise sum of the two halves of $seq$. Therefore, from ${M}_{3}$ the matrix representation can be extended recursively.

**Example**

**2.**

#### 4.3.2. Description of the Half-Interval Search Algorithm

Algorithm 3 The half-interval search Algorithm |

Require:$seq$: sequence to be analyzed |

$k=0$ |

while $length(seq)>1$ do |

$l=length(seq)$ |

sum = $se{q}_{0,\frac{l}{2}-1}$ + $se{q}_{\frac{l}{2},l-1}$ |

if $sum\ne {0}_{\frac{l}{2}}$ then |

$seq=sum$ |

$k+=\frac{l}{2}$ |

else |

$seq=se{q}_{0,\frac{l}{2}-1}$ |

end if |

end while |

return k: maximum index k and $LC$ of $seq$. |

**Example**

**3.**

- Step 1: $\begin{array}{cc}\begin{array}{}\\ \\ sum=\end{array}& \begin{array}{ccc}& 0001& 1101\\ +& 1000& 1011\\ & 1001& 0110\end{array}\end{array}$

- Step 2: $\begin{array}{cc}\begin{array}{}\\ \\ sum=\end{array}& \begin{array}{ccc}& 10& 01\\ +& 01& 10\\ & 11& 11\end{array}\end{array}$

- Step 3: $\begin{array}{cc}\begin{array}{}\\ \\ sum=\end{array}& \begin{array}{ccc}& 1& 1\\ +& 1& 1\\ & 0& 0\end{array}\end{array}$

- Step 4: $\begin{array}{cc}\begin{array}{}\\ \\ sum=\end{array}& \begin{array}{cc}& 1\\ +& 1\\ & 0\end{array}\end{array}$

- Output: the maximum binomial sequence $\left(\genfrac{}{}{0pt}{}{n}{12}\right)\Rightarrow LC=k+1=12+1=13$.

#### 4.4. Matrix Binomial Decomposition or m-BD Algorithm

**Example**

**4.**

_{16}will havec

_{3}= c

_{4}= c

_{6}= c

_{8}= c

_{9}= c

_{10}= c

_{12}= 1 while the remaining components equal zero. The coefficients c

_{i}= 1 correspond to the binomial sequences $\left(\genfrac{}{}{0pt}{}{n}{i}\right)$ that appear in the binomial decomposition of seq

_{16}.

**Remark**

**1.**

#### 4.4.1. Description of the m-BD Algorithm

Algorithm 4 The m-BD Algorithm |

Require:$seq=[{s}_{0},{s}_{1},\dots ,{s}_{{2}^{m}-1}]$ and the binomial matrix ${H}_{m}=({\tilde{h}}_{0},{\tilde{h}}_{1},\dots ,{\tilde{h}}_{{2}^{m}-1})$ |

$i={2}^{m}-1$ |

${i}_{max}=0$ |

while $i>0$ do |

${c}_{i}=[{s}_{0},{s}_{1},\dots ,{s}_{{2}^{m}-1}]\xb7{\tilde{h}}_{i}$ |

if ${c}_{i}==0$ then |

$i--$ |

else |

${i}_{max}=i$ |

$i=0$ |

end if |

end while |

return $LC={i}_{max}+1$: Linear complexity of $seq$. |

#### 4.4.2. Sequences with Maximum $LC$:

**Theorem**

**2.**

**Proof.**(⇒)

**Corollary**

**1.**

**Corollary**

**2.**

#### 4.4.3. Sequences with Quasi-Maximum $LC$

**Theorem**

**3.**

- The sequence $\{{s}_{n}\}$ has an even number of ones.
- It satisfies the equality:$$\sum _{i=0}^{l/2-1}{s}_{2\xb7i}=1.$$

**Proof.**(⇒)

- $\{{s}_{n}\}$ must have an even number of ones, otherwise by Theorem 2 the sequence would have maximum linear complexity.
- Quasi-maximum linear complexity implies that ${c}_{{2}^{m}-2}=1$, but ${c}_{{2}^{m}-2}$ is the product mod 2 of the sequence $[{s}_{0},{s}_{1},\dots ,{s}_{{2}^{m}-1}]$ multiplied by the column ${\tilde{h}}_{{2}^{m}-2}$ in the binomial matrix (the $1010\dots 10$ column), thus$${c}_{{2}^{m}-2}=\sum _{i=0}^{l/2-1}{s}_{2\xb7i}.$$Hence, ${c}_{{2}^{m}-2}=1$ when the number of terms $({s}_{2\xb7i})$ (terms with even indices) equal to 1 is an odd number.

- (⇐)

- If the sequence $\{{s}_{n}\}$ has an even number of ones, then ${c}_{{2}^{m}-1}=0$.
- If $\{{s}_{n}\}$ satisfies the equality$$\sum _{i=0}^{l/2-1}{s}_{2\xb7i}=1,$$

## 5. Algorithm Comparison

#### 5.1. Algorithm Analysis

#### 5.2. Experimental Results

#### 5.3. Different Use-Cases

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chin, W.L.; Li, W.; Chen, H.H. Energy big data security threats in IoT-based smart grid communications. IEEE Commun. Mag.
**2017**, 55, 70–75. [Google Scholar] [CrossRef] - Meyer, D.; Haase, J.; Eckert, M.; Klauer, B. New attack vectors for building automation and IoT. In Proceedings of the IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society, Beijing, China, 29 October–1 November 2017; pp. 8126–8131. [Google Scholar]
- Gallegos-Segovia, P.L.; Bravo-Torres, J.F.; Argudo-Parra, J.J.; Sacoto-Cabrera, E.J.; Larios-Rosillo, V.M. Internet of things as an attack vector to critical infrastructures of cities. In Proceedings of the 2017 International Caribbean Conference on Devices, Circuits and Systems (ICCDCS), Cozumel, Mexico, 5–7 June 2017; pp. 117–120. [Google Scholar]
- Rouf, I. Security and Privacy Vulnerabilities of In-Car Wireless Networks: A Tire Pressure Monitoring System Case Study. In Proceedings of the USENIX Security Symposium, Washington, DC, USA, 11–13 August 2010; Volume 10. [Google Scholar]
- Cynthia, J.; Sultana, H.P.; Saroja, M.; Senthil, J. Security protocols for IoT. In Ubiquitous Computing and Computing Security of IoT; Springer: Berlin/Heidelberg, Germany, 2019; pp. 1–28. [Google Scholar]
- Mavromoustakis, C.X.; Mastorakis, G.; Batalla, J.M. Internet of Things (IoT) in 5G mobile technologies; Springer: Berlin/Heidelberg, Germany, 2016; Volume 8. [Google Scholar]
- NIST Lightweight Cryptography Project. Available online: https://csrc.nist.gov/Projects/Lightweight-Cryptography (accessed on 15 February 2021).
- McGinthy, J.M. Solutions for Internet of Things Security Challenges: Trust and Authentication. Ph.D. Thesis, Virginia Tech, Blacksburg, VI, USA, 2019. [Google Scholar]
- NIST Lightweight Cryptography Project Round 2 Candidates. Available online: https://csrc.nist.gov/Projects/lightweight-cryptography/round-2-candidates. (accessed on 15 February 2021).
- Dubrova, E.; Hell, M. Espresso: A stream cipher for 5G wireless communication systems. Cryptogr. Commun.
**2017**, 9, 273–289. [Google Scholar] [CrossRef][Green Version] - Massey, J. Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory
**1969**, 15, 122–127. [Google Scholar] [CrossRef][Green Version] - Cardell, S.D.; Fúster-Sabater, A. Binomial Representation of Cryptographic Binary Sequences and Its Relation to Cellular Automata. Complexity
**2019**. [Google Scholar] [CrossRef][Green Version] - Chang, K.Y.; Lee, M.K.; Lee, H.R.; Hong, D.W.; Kang, J.S.; Cho, H.S.; Chung, K.I. Method and Apparatus for Generating Keystream. US Patent 7,587,046, 8 September 2009. [Google Scholar]
- Kang, Y.S.; Kim, H.W.; Chung, K.I. Apparatus and Method for Protecting RFID Data. US Patent 8,386,794, 17 February 2013. [Google Scholar]
- Falk, R.; Merli, D. Programmable Logic Device, Key Generation Circuit and Method for Providing Security Information. EP Patent 3146520, 11 May 2016. [Google Scholar]
- Martin-Navarro, J.L.; Fúster-Sabater, A. Folding-BSD Algorithm for Binary Sequence Decomposition. Computers
**2020**, 9, 100. [Google Scholar] [CrossRef] - Cardell, S.D.; Climent, J.J.; Fúster-Sabater, A.; Requena, V. Representations of Generalized Self-Shrunken Sequences. Mathematics
**2020**, 8, 1006. [Google Scholar] [CrossRef] - Golomb, S.W. Shift Register Sequences; Aegean Park Press: Walnut Creek, CA, USA, 1967. [Google Scholar]
- Menezes, A.J.; van Oorschot, P.C.; Vanstone, S.A. Handbook of Applied Cryptography; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Paar, C.; Pelzl, J. Understanding Cryptography; Springer: Berlin, Germany, 2010. [Google Scholar]
- Cardell, S.D.; Fúster-Sabater, A. The t-Modified self-shrinking generator. In International Conference on Computational Science (ICCS 2018); Springer: Berlin/Heidelberg, Germany, 2018; pp. 653–663. [Google Scholar]
- Cardell, S.D.; Fúster-Sabater, A. Cryptography with Shrinking Generators: Fundamentals and Applications of Keystream Sequence Generators Based on Irregular Decimation; Series: Briefs in Mathematics; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Coppersmith, D.; Krawczyk, H.; Mansour, Y. The shrinking generator. In Annual International Cryptology Conference; Springer: Berlin/Heidelberg, Germany, 1993; pp. 22–39. [Google Scholar]
- Meier, W.; Staffelbach, O. The self-shrinking generator. In Communications and Cryptography; Springer: Berlin/Heidelberg, Germany, 1994; pp. 287–295. [Google Scholar]
- Hu, Y.; Xiao, G. Generalized self-shrinking generator. IEEE Trans. Inf. Theory
**2004**, 50, 714–719. [Google Scholar] [CrossRef] - Mihaljevic, M.J. A faster cryptanalysis of the self-shrinking generator. In Proceedings of the Information Security and Privacy, First Australasian Conference, ACISP’96, Wollongong, Australia, 24–26 June 1996; Lecture Notes in Computer Science. Springer: Berlin/Heidelberg, Germany, 1996; Volume 1172, pp. 182–189. [Google Scholar] [CrossRef]
- Simpson, L.; Golic, J.D.; Dawson, E. A Probabilistic Correlation Attack on the Shrinking Generator. In Proceedings of the Information Security and Privacy, Third Australasian Conference, ACISP’98, Brisbane, Australia, 13–15 July 1998; Lecture Notes in Computer Science. Springer: Berlin/Heidelberg, Germany, 1998; Volume 1438, pp. 147–158. [Google Scholar] [CrossRef]
- Caballero, P.; Fúster-Sabater, A.; Pazo, M.E. New Attack Strategy for the Shrinking Generator. J. Res. Pract. Inf. Technol.
**2009**, 23, 171–180. [Google Scholar] - Fúster-Sabater, A.; Pazo, M.E.; Caballero, P. A Simple Linearization of the Self-Shrinking Generator by Means of Linear Cellular Automata. Neural Netw.
**2010**, 23, 461–464. [Google Scholar] [CrossRef] - Cardell, S.D.; Fúster-Sabater, A.; Ranea, A.H. Linearity in decimation-based generators: An improved cryptanalysis on the shrinking generator. Open Math.
**2018**, 16, 646–655. [Google Scholar] [CrossRef] - Fúster-Sabater, A.; Cardell, S.D. Linear complexity of generalized sequences by comparison of PN-sequences. Rev. De La Real Acad. De Cienc. Exactas, Físicas Y Naturales. Ser. A. Matemáticas
**2020**, 114, 79. [Google Scholar] [CrossRef] - Fúster-Sabater, A. Generation of cryptographic sequences by means of difference equations. Appl. Math. Inform. Sci.
**2014**, 8, 475–484. [Google Scholar] [CrossRef] - Cusick, T.W.; Stanica, P. Cryptographic Boolean Functions and Applications; Academic Press: Cambridge, MA, USA, 2017. [Google Scholar]
- Martin-Navarro, J.L.; Fúster-Sabater, A. Folding-BSD Algorithm for Binary Sequence Decomposition. In International Conference on Computational Science and Its Applications (ICCSA 2020); Springer: Berlin/Heidelberg, Germany, 2020; pp. 345–359. [Google Scholar]

**Figure 3.**Comparison between the algorithms in the $LC$ calculation for all the possible GSS-sequences of a given length (Half-Interval scale).

**Figure 5.**Comparison between the algorithms in the $LC$ calculation for all the possible GSS-sequences of a given length (m-BD scale).

p-Rotation | $\{{\mathit{b}}_{\mathit{n}}\}$ Sequences | GSS-Sequences |
---|---|---|

0 | $\mathbf{1}$ 0 $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ 0 0 | 1111 |

1 | $\mathbf{0}$ 1 $\mathbf{1}$ $\mathbf{1}$ $\mathbf{0}$ 0 1 | 0110 |

2 | $\mathbf{1}$ 1 $\mathbf{1}$ $\mathbf{0}$ $\mathbf{0}$ 1 0 | 1100 |

3 | $\mathbf{1}$ 1 $\mathbf{0}$ $\mathbf{0}$ $\mathbf{1}$ 0 1 | 1001 |

4 | $\mathbf{1}$ 0 $\mathbf{0}$ $\mathbf{1}$ $\mathbf{0}$ 1 1 | 1010 |

5 | $\mathbf{0}$ 0 $\mathbf{1}$ $\mathbf{0}$ $\mathbf{1}$ 1 1 | 0101 |

6 | $\mathbf{0}$ 1 $\mathbf{0}$ $\mathbf{1}$ $\mathbf{1}$ 1 0 | 0011 |

PN-sequence | $\mathbf{1}$ 0 $\mathbf{1}$ $\mathbf{1}$ $\mathbf{1}$ 0 0 |

Binom. Coeff. | Binomial Sequences $\{\left(\genfrac{}{}{0pt}{}{\mathit{n}}{\mathit{k}}\right)\}$ | Length | Linear Complexity |
---|---|---|---|

$\left(\genfrac{}{}{0pt}{}{n}{0}\right)$ | $\left\{1,1,1,1,1,1,1,1,\dots \right\}$ | ${l}_{0}=1$ | $L{C}_{0}=1$ |

$\left(\genfrac{}{}{0pt}{}{n}{1}\right)$ | $\left\{0,1,0,1,0,1,0,1,\dots \right\}$ | ${l}_{1}=2$ | $L{C}_{1}=2$ |

$\left(\genfrac{}{}{0pt}{}{n}{2}\right)$ | $\left\{0,0,1,1,0,0,1,1,\dots \right\}$ | ${l}_{2}=4$ | $L{C}_{2}=3$ |

$\left(\genfrac{}{}{0pt}{}{n}{3}\right)$ | $\left\{0,0,0,1,0,0,0,1,\dots \right\}$ | ${l}_{3}=4$ | $L{C}_{3}=4$ |

$\left(\genfrac{}{}{0pt}{}{n}{4}\right)$ | $\left\{0,0,0,0,1,1,1,1,\dots \right\}$ | ${l}_{4}=8$ | $L{C}_{4}=5$ |

$\left(\genfrac{}{}{0pt}{}{n}{5}\right)$ | $\left\{0,0,0,0,0,1,0,1,\dots \right\}$ | ${l}_{5}=8$ | $L{C}_{5}=6$ |

$\left(\genfrac{}{}{0pt}{}{n}{6}\right)$ | $\left\{0,0,0,0,0,0,1,1,\dots \right\}$ | ${l}_{6}=8$ | $L{C}_{6}=7$ |

$\left(\genfrac{}{}{0pt}{}{n}{7}\right)$ | $\left\{0,0,0,0,0,0,0,1,\dots \right\}$ | ${l}_{7}=8$ | $L{C}_{7}=8$ |

Step | Op. | Seq. | Bit Position | |||
---|---|---|---|---|---|---|

0 | 4 | 8 | 12 | |||

1 | $seq$ | 0 0 0 1 | 1 1 0 1 | 1 0 0 0 | 1 0 1 1 | |

+ | $\left(\genfrac{}{}{0pt}{}{n}{3}\right)$ | 0 0 0 1 | 0 0 0 1 | 0 0 0 1 | 0 0 0 1 | |

2 | = | $seq$ | 0 0 0 0 | 1 1 0 0 | 1 0 0 1 | 1 0 1 0 |

+ | $\left(\genfrac{}{}{0pt}{}{n}{4}\right)$ | 0 0 0 0 | 1 1 1 1 | 0 0 0 0 | 1 1 1 1 | |

3 | = | $seq$ | 0 0 0 0 | 0 0 1 1 | 1 0 0 1 | 0 1 0 1 |

+ | $\left(\genfrac{}{}{0pt}{}{n}{6}\right)$ | 0 0 0 0 | 0 0 1 1 | 0 0 0 0 | 0 0 1 1 | |

4 | = | $seq$ | 0 0 0 0 | 0 0 0 0 | 1 0 0 1 | 0 1 1 0 |

+ | $\left(\genfrac{}{}{0pt}{}{n}{8}\right)$ | 0 0 0 0 | 0 0 0 0 | 1 1 1 1 | 1 1 1 1 | |

5 | = | $seq$ | 0 0 0 0 | 0 0 0 0 | 0 1 1 0 | 1 0 0 1 |

+ | $\left(\genfrac{}{}{0pt}{}{n}{9}\right)$ | 0 0 0 0 | 0 0 0 0 | 0 1 0 1 | 0 1 0 1 | |

6 | = | $seq$ | 0 0 0 0 | 0 0 0 0 | 0 0 1 1 | 1 1 0 0 |

+ | $\left(\genfrac{}{}{0pt}{}{n}{10}\right)$ | 0 0 0 0 | 0 0 0 0 | 0 0 1 1 | 0 0 1 1 | |

7 | = | $seq$ | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 1 1 1 1 |

+ | $\left(\genfrac{}{}{0pt}{}{n}{12}\right)$ | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 1 1 1 1 | |

end | = | $seq$ | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 |

$seq=\left(\genfrac{}{}{0pt}{}{n}{3}\right)+\left(\genfrac{}{}{0pt}{}{n}{4}\right)+\left(\genfrac{}{}{0pt}{}{n}{6}\right)+\left(\genfrac{}{}{0pt}{}{n}{8}\right)+\left(\genfrac{}{}{0pt}{}{n}{9}\right)+\left(\genfrac{}{}{0pt}{}{n}{10}\right)+\left(\genfrac{}{}{0pt}{}{n}{12}\right)$ | ||||||

$LC={k}_{r}+1=12+1=13$ |

seq | 0 0 0 1 | 1 1 0 1 | 1 0 0 0 | 1 0 1 1 |
---|---|---|---|---|

${\left(\genfrac{}{}{0pt}{}{n}{3}\right)}_{l}=({\left(\genfrac{}{}{0pt}{}{n}{3}\right)}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{3}\right)}_{\frac{l}{2}})$ | 0 0 0 1 | 0 0 0 1 | 0 0 0 1 | 0 0 0 1 |

${\left(\genfrac{}{}{0pt}{}{n}{4}\right)}_{l}=({\left(\genfrac{}{}{0pt}{}{n}{4}\right)}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{4}\right)}_{\frac{l}{2}})$ | 0 0 0 0 | 1 1 1 1 | 0 0 0 0 | 1 1 1 1 |

${\left(\genfrac{}{}{0pt}{}{n}{6}\right)}_{l}=({\left(\genfrac{}{}{0pt}{}{n}{6}\right)}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{6}\right)}_{\frac{l}{2}})$ | 0 0 0 0 | 0 0 1 1 | 0 0 0 0 | 0 0 1 1 |

${\left(\genfrac{}{}{0pt}{}{n}{8}\right)}_{l}=(zero{s}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{0}\right)}_{\frac{l}{2}})$ | 0 0 0 0 | 0 0 0 0 | 1 1 1 1 | 1 1 1 1 |

${\left(\genfrac{}{}{0pt}{}{n}{9}\right)}_{l}=(zero{s}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{1}\right)}_{\frac{l}{2}})$ | 0 0 0 0 | 0 0 0 0 | 0 1 0 1 | 0 1 0 1 |

${\left(\genfrac{}{}{0pt}{}{n}{10}\right)}_{l}=(zero{s}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}_{\frac{l}{2}})$ | 0 0 0 0 | 0 0 0 0 | 0 0 1 1 | 0 0 1 1 |

${\left(\genfrac{}{}{0pt}{}{n}{12}\right)}_{l}=(zero{s}_{\frac{l}{2}},{\left(\genfrac{}{}{0pt}{}{n}{4}\right)}_{\frac{l}{2}})$ | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 1 1 1 1 |

$se{q}_{16}=\left(\genfrac{}{}{0pt}{}{n}{3}\right)+\left(\genfrac{}{}{0pt}{}{n}{4}\right)+\left(\genfrac{}{}{0pt}{}{n}{6}\right)+\left(\genfrac{}{}{0pt}{}{n}{8}\right)+\left(\genfrac{}{}{0pt}{}{n}{9}\right)+\left(\genfrac{}{}{0pt}{}{n}{10}\right)+\left(\genfrac{}{}{0pt}{}{n}{12}\right)$ |

Algorithms | Length Required | Complexity | Seq. Restrictions |
---|---|---|---|

Berlekamp-Massey algorithm | $2\ast l$ | O(${l}^{2}$) | None |

BD-algorithm | $l-logl$ | O($r\xb7l$) | Length power of 2 |

Half-interval search algorithm | $l-logl$ | O(l) | Length power of 2 |

m-BD algorithm | l | O(${l}^{2}$) - $\Omega $(l) | Length power of 2 |

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**MDPI and ACS Style**

Martin-Navarro, J.L.; Fúster-Sabater, A. Review of the Lineal Complexity Calculation through Binomial Decomposition-Based Algorithms. *Mathematics* **2021**, *9*, 478.
https://doi.org/10.3390/math9050478

**AMA Style**

Martin-Navarro JL, Fúster-Sabater A. Review of the Lineal Complexity Calculation through Binomial Decomposition-Based Algorithms. *Mathematics*. 2021; 9(5):478.
https://doi.org/10.3390/math9050478

**Chicago/Turabian Style**

Martin-Navarro, Jose Luis, and Amparo Fúster-Sabater. 2021. "Review of the Lineal Complexity Calculation through Binomial Decomposition-Based Algorithms" *Mathematics* 9, no. 5: 478.
https://doi.org/10.3390/math9050478